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Towards robust dynamical decoupling and high fidelity adiabatic quantum computation

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Towards robust dynamical decoupling and high fidelity adiabatic quantum computation

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Content TOWARDS ROBUST DYNAMICAL DECOUPLING AND HIGH FIDELITY ADIABATIC QUANTUM COMPUTATION by Gregory Quiroz A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PHYSICS AND ASTRONOMY) December 2013 Copyright 2013 Gregory Quiroz Dedication To my family and friends for all their love and support throughout my pursuit of this degree. ii Acknowledgments First and foremost, I would like to thank my family for their support and guidance over the years. In particular, I would like to thank my mother for always believing in my ability to achieve my goals and showing me that anything is possible with hard work and dedication. I would also like to thank my girlfriend, Nethmi Ariyasinghe, for her love and support throughout my graduate school career. She has continually kept me motivated and positive when I needed it most. I am truly grateful to my advisor Daniel Lidar for teaching me the necessary skills required to be a successful researcher. His infinite patience and generosity will forever be appreciated. Thank you to all my friends: John Peacock, James Medina, Albert Herrera, Ben Madden, and Siddhartha Santra, for their useful advice and constructive distractions throughout my pursuit of this degree. Finally, I would like to thank all of my colleagues: Wan-jung Kuo, Kristen Pudenz, Gerardo Paz-Silva, and Zhihui Wang for the simulating technical discussions and nev- erending kindness. iii Table of Contents Dedication ii Acknowledgments iii List of Tables vii List of Figures ix Abstract xvii Chapter 1: A Brief Introduction to Quantum Computation and Decoherence 1 1.1 Quantum Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Fundamental Postulates of Quantum Mechanics . . . . . . . . . 2 1.1.2 Open System Dynamics . . . . . . . . . . . . . . . . . . . . . 7 1.2 Quantum Computation . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.1 The Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.2 Quantum Algorithms . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.3 Experimental Realizations of Quantum Computation . . . . . . 14 1.3 Decoherence Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3.1 Dephasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.3.2 Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.3.3 General Qubit Decoherence Model . . . . . . . . . . . . . . . . 23 1.4 Quantum Error Avoidance, Correction, and Suppression . . . . . . . . . 24 1.4.1 Decoherence-Free Subspaces . . . . . . . . . . . . . . . . . . . 24 1.4.2 Quantum Error Correction . . . . . . . . . . . . . . . . . . . . 25 1.4.3 Dynamical Decoupling . . . . . . . . . . . . . . . . . . . . . . 29 1.5 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Chapter 2: Dynamical Decoupling 32 2.1 Origins of Dynamical Decoupling . . . . . . . . . . . . . . . . . . . . 32 2.2 Mathematical Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.2.1 Interaction Picture . . . . . . . . . . . . . . . . . . . . . . . . 34 2.2.2 Dyson Expansion . . . . . . . . . . . . . . . . . . . . . . . . . 35 iv 2.2.3 Magnus Expansion and Average Hamiltonian Theory . . . . . . 36 2.2.4 Dynamical Decoupling Defined . . . . . . . . . . . . . . . . . 38 2.3 The Control Hamiltonian: From Ideal to Realistic Pulses . . . . . . . . 42 2.3.1 Ideal Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.3.2 Finite-width Pulse Profiles . . . . . . . . . . . . . . . . . . . . 43 2.3.3 Flip-Angle Errors . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.3.4 Generic Faulty Pulse Profiles . . . . . . . . . . . . . . . . . . . 44 2.4 Currently Available Techniques . . . . . . . . . . . . . . . . . . . . . . 44 2.4.1 Universal Dynamical Decoupling . . . . . . . . . . . . . . . . 45 2.4.2 Concatenated Dynamical Decoupling . . . . . . . . . . . . . . 50 2.4.3 Uhrig Dynamical Decoupling . . . . . . . . . . . . . . . . . . . 50 2.4.4 A Brief Summary of Additional Methods: Randomized DD to Numerical Optimization . . . . . . . . . . . . . . . . . . . . . 53 2.5 Performance Measures and Scaling . . . . . . . . . . . . . . . . . . . . 54 2.5.1 Fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.5.2 Trace-Norm Distance . . . . . . . . . . . . . . . . . . . . . . . 56 2.5.3 Kosut Distance . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.5.4 Uni-axial Error Measure . . . . . . . . . . . . . . . . . . . . . 63 Chapter 3: Analysis of Near-Optimal Quadratic DD Approach 67 3.1 QDD Evolution Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.2.1 Uni-axial error analysis . . . . . . . . . . . . . . . . . . . . . . 72 3.2.2 Intermediate uni-axial error analysis . . . . . . . . . . . . . . . 78 3.2.3 Overall QDD Performance Scaling . . . . . . . . . . . . . . . . 84 3.2.4 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . 85 Chapter 4: Dynamical Decoupling for Realizable Systems 88 4.1 Shortcomings of Current Deterministic DD Methods . . . . . . . . . . 90 4.1.1 Concatenated DD with Faulty Pulses . . . . . . . . . . . . . . . 90 4.1.2 Quadratic DD with Faulty Pulses . . . . . . . . . . . . . . . . . 91 4.2 Numerically-Optimized Dynamical Decoupling . . . . . . . . . . . . . 92 4.2.1 Ideal pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.2.2 Finite-width Pulses . . . . . . . . . . . . . . . . . . . . . . . . 101 4.2.3 Flip-angle errors . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.2.4 Finite-Width and Flip-Angle Errors . . . . . . . . . . . . . . . 115 4.2.5 Comparison with Known Deterministic Schemes . . . . . . . . 121 4.2.6 Existence of Deterministic Structure . . . . . . . . . . . . . . . 128 4.2.7 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . 131 4.3 Symmetrized QDD for Control Error Robustness . . . . . . . . . . . . 133 4.3.1 Symmetrizing Schemes for QDD . . . . . . . . . . . . . . . . . 133 4.3.2 Model Specifications . . . . . . . . . . . . . . . . . . . . . . . 136 v 4.3.3 Numerical Comparison . . . . . . . . . . . . . . . . . . . . . . 137 4.3.4 Analytical Comparison . . . . . . . . . . . . . . . . . . . . . . 146 4.3.5 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . 149 4.4 Numerically-Optimized DD vs. Symmetrized QDD . . . . . . . . . . . 151 Chapter 5: Protecting Adiabatic Quantum Computation via Dynamical Decou- pling 153 5.1 Closed System Adiabatic Quantum Computation . . . . . . . . . . . . 155 5.2 Adiabatic Quantum Computation in the Presence of Decoherence . . . . 156 5.3 Applying Dynamical Decoupling to Adiabatic QC . . . . . . . . . . . . 158 5.3.1 Stabilizer decoupling . . . . . . . . . . . . . . . . . . . . . . . 159 5.3.2 Concatenated Dynamical Decoupling . . . . . . . . . . . . . . 159 5.3.3 Quadratic Dynamical Decoupling . . . . . . . . . . . . . . . . 160 5.4 Performance of Deterministic Decoupling Schemes . . . . . . . . . . . 161 5.4.1 Grover’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 162 5.4.2 2-bit Satisfiability . . . . . . . . . . . . . . . . . . . . . . . . . 165 5.4.3 Dependence of DD Performance on Cutoff Frequency . . . . . . 167 5.4.4 Effect ofH ad (s)-induced Noise . . . . . . . . . . . . . . . . . . 174 5.5 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Chapter 6: Conclusions and Future Work 178 6.1 Analysis of Quadratic DD . . . . . . . . . . . . . . . . . . . . . . . . . 178 6.2 Robust Dynamical Decoupling . . . . . . . . . . . . . . . . . . . . . . 179 6.3 DD-Protection for Adiabatic Quantum Computation . . . . . . . . . . . 180 Chapter A: Numerical Techniques 182 A.1 Genetic Algorithm for Optimized DD . . . . . . . . . . . . . . . . . . 182 A.1.1 Chromosome structure . . . . . . . . . . . . . . . . . . . . . . 182 A.1.2 Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 A.1.3 Crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 A.1.4 Mutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 A.1.5 Necessary Convergence Accelerators . . . . . . . . . . . . . . . 187 A.2 Extracting effective error Hamiltonian scaling numerically . . . . . . . 192 References 194 vi List of Tables 3.1 Summary of the scaling for all uni-axial errors. Values ofN Ire extracted by performing a linear regression, rounded to the nearest integer, fitting the slopes of the straight line portions of the curves displayed in Figs. 3.1 and 3.2 between log 10 (J) =9 and the values of log 10 (J) indicated by the vertical lines in these figures. (a) E x , (b) E y , and (c) E z for M 1 ;M 2 2f1; 2;:::; 10g. ForN y andN z the outer sequence orderM 2 is displayed in the top row and the inner sequence orderM 1 in the first column. Each of the single-axis errors is dominated by the lowest order ofJ d , denotedN , thereforeE O[(J) N ]. Additional simulations (not shown) fully continue the trends seen in this table and summarized in Eqs. (3.16)-(3.18) all the way up toM 1 ;M 2 24. . . . . . . . . . . 76 3.2 Summary of the scaling of the overall distance measureD K with respect to inner and outer sequence orders, M 1 and M 2 , respectively. Values of N D K are extracted by performing a linear regression, rounded to the nearest integer, fitting the slopes of the straight line portions of the curves displayed in Figs. 3.1 and 3.2 between log(J) =9 and the val- ues of log(J) indicated by the vertical lines in these figures. The outer sequence order N 2 is displayed in the top row and the inner sequence orderM 1 in the first column. As expected, N D K = min(N x ;N y ;N z ). Additional simulations [not shown] fully continue the trends seen in this table and summarized in Eq. (3.31) all the way up toM 1 ;M 2 24. . . . 84 4.1 Summary of distance measure (D Kos ) scalings for each optimal GA K sequence identified in the ideal pulse limit, for a fixed pulse-interval of d . Boxed performance scalings highlight optimal sequences in each of the relevant (J;)-regimes (columns) for eachK opt . . . . . . . . . . . . 98 vii 4.2 Summary of distance measure (D Kos ) scalings for each optimalRGA K sequence located by our search algorithm, for DD evolution subjected to finite-width rectangular pulses of duration p pulses and pulse-interval d . Optimal performance scalings for each K opt are boxed for each parameter regime (column). . . . . . . . . . . . . . . . . . . . . . . . . 107 4.3 Summary of distance measure D Kos scalings for each optimal RGA K sequence located for DD pulses subjected to flip-angle errors with rota- tion error , with fixed pulse-interval d . Boxed performance scalings highlight optimal performance scaling for variousK opt . . . . . . . . . . 112 4.4 Summary of distance measureD Kos scalings for optimalRGA K sequences identified for DD evolution subjected to finite-width pulses of duration p and flip-angle errors with rotation error in the regime of system- environment interaction-dominated (J ) dynamics. The sequences with the best performance scaling in each parameter regime (column) for each K opt are boxed. Note specifically the case of strong pulses dominated by flip-angle errors (J p ) whereRGA 8a ,RGA 16a , and RGA 64a obtain the more favorable performance scaling. . . . . . . . . 118 viii List of Figures 1.1 Bloch Sphere representation of an arbitrary qubit state. The Bloch sphere is a unit sphere with the north and south poles representing the two orthonormal basis states of the qubit. . . . . . . . . . . . . . . . . . . . 11 3.1 Single-axis errors after one cycle ofU (M 1 ;M 2 ) QDD for outer sequence order M 2 = 3 and inner sequence orders M 1 = 1; 2;:::; 10 as a function ofJ, averaged over 50 random realizations of the bath operatorsB . Error bars are shown but are very small. In all our simulations I set J = 10 4 and = 10 6 . Single-axis error values are computed for log 10 (J) =9;8;:::; 2. Lines are guides to the eye. E x is desig- nated by the green squares,E y by the red circles, andE z by the black triangles. Note that log 10 (E z ) is the same in all six plots, with a slope ofM 2 + 1. The slope of log 10 (E x ), on the other hand, isM 1 + 1. The slope of log 10 (E y ) isM 1 + 2. Vertical lines denote the largest value of J utilized in the linear regression used to extract the slopeN . . . . . 73 3.2 Single-axis errors after one cycle ofU (M 1 ;M 2 ) QDD for outer sequence order M 2 = 4 and inner sequence orders M 1 = 1; 2;:::; 10 (left to right, top to bottom) as a function of J, averaged over 50 random realiza- tions of the bath operatorsB . Other details as in Fig. 3.1, except that the single-axis error suppressed by both the inner and outer sequence, E y (), exhibits a strong dependence on the parity of the inner sequence. Note that for each value ofM 1 ,E x () andE y () are essentially equal for all values ofJ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 ix 3.3 Intermediate uni-axial errors forM 2 = 3, as defined in Eq. (3.23). For givenk, the intermediate uni-axial error is computed afterk innnerZ- type UDD sequences separated byk 1 pulses. There areM 2 + 1 inner UDD sequences. The pointk = 5 is the lastX pulse at the end of last inner UDD sequence, as required for odd M 2 . Note that because the data points labeledj = 4 andj = 5 are separated by a singleX pulse, and are these pulses are instantaneous, these points have no actual time delay between them. . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.4 Intermediate uni-axial errors for M 2 = 4. As in Fig. 3.3 except that there is no finalX pulse forM 2 even, i.e., there areM 2 + 1 inner UDD sequences separated byM 2 X pulses. . . . . . . . . . . . . . . . . . . 80 3.5 Overall QDD performance after one cycle ofU (M 1 ;M 2 ) QDD;fZ;Xg for outer sequence orderM 2 = 3 and inner sequence ordersM 1 = 1; 2;:::; 6, as a function ofJ, averaged over 50 random realizations of the bath operatorsB . The performance of QDD progressively improves with increasing M 1 up toM 2 = M 1 , indicating that minfM 1 ;M 2 g dominates QDD perfor- mance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.6 Overall QDD performance after one cycle ofU (M 1 ;M 2 ) QDD;fZ;Xg for outer sequence orderM 2 = 4 and inner sequence ordersM 1 = 1; 2;:::; 6, as a function ofJ, averaged over 50 random realizations of the bath operatorsB . The dependence of the order of error suppression on minfM 1 ;M 2 g is again observed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.1 Performance ofGA K sequences forK = 4; 8; 16; 32; 64; 256, as shown in (a)-(f), respectively, as a function ofJ and. The minimum pulse- interval is fixed at d = 0:1ns and the results are averaged over 10 real- izations of B . The GA Ka sequences tend to be optimal in the range J < , whileGA Kb are optimal forJ . Sequences closely related to the deterministic structure of CDD,GA Kc , begin to appear as opti- mal sequences forJ d > 1 atK = 64. The notationnGA 4 denotes the application ofn cycles ofGA 4 and is dependent on the value ofK. For the data presented above, n = 1; 8; 16; 64 for K = 4; 32; 64; 256, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.2 Pictorial depiction for the action ofRGA 8a 0 as an Eulerian path along the Cayley graph (S;G) with verticesS =fI;X;Y;Zg and gener- ating setG = fI;X;Y;Zg. Note that unlike the EDD construction [Eq. (4.28)],RGA 8a 0 is generated by Eulerian paths, rather than cycles. 105 x 4.3 Performance of optimalRGA K sequences forK = 4; 8; 16; 32; 64; 256 shown in (a)-(f), respectively, as a function ofJ= and p = d when DD is subjected to finite pulse duration. The norm of the bath Hamiltonian is fixed at = 1kHz, whileJ=2 [10 6 ; 10 6 ]. The pulse-interval d = 0:1ns and the pulse width is varied in the range p = d 2 [10 3 ; 10 6 ]. For a givenK, the optimal sequence configuration is most sensitively depen- dent upon variations in J. Contrary to the ideal pulse case, concate- nated structures composed ofRGA 8a 0 andRGA 8c appear to be the most favorable, in particular forK 16 whereRGA 16a 0 andRGA 64c repeat- edly emerge as optimal sequences. Sequence performance saturates at K = 16, while robustness begins to diminish atK = 256.nRGA 16b 0 denotesn cycles ofRGA 16b 0;n = 1; 4; 16 forK = 16; 64; 256 respec- tively. The notation is similar formRGA 64c , wherem = 1; 4 cycles are used forK = 64; 256. . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.4 Performance ofRGA K sequences forK = 4; 8; 16; 32; 64; 256, as shown in (a)-(f), respectively, as a function ofJ= and. The minimum pulse interval is fixed at d = 0:1ns, J= 2 [10 6 ; 10 6 ], and = 1kHz, while is varied from one to twenty percent rotation error. Results are averaged over 10 realizations ofB . Sequence performance mostly increases from K = 4 to K = 64, indicating a reduction in the error terms proportional to in the effective error Hamiltonian. Successive error suppression is achieved forK = 4; 8; 16; 64, where the maximum error suppression yieldsDO( 3 J d ). forRGA 64a . Multiple cycles ofRGA 16a andRGA 64a appear as optimal sequences for variousK opt . The number of cycles for each sequence is given as follows: (d)n = 2, (e)m = 1, and (f)n = 16,m = 4. . . . . . . . . . . . . . . . . . . . . 114 4.5 Performance ofRGA K sequences forK = 4; 8; 16; 32; 64; 256, as shown in (a)-(f), respectively, as a function of and p = d . The minimum pulse- interval is fixed at d = 0:1ns, while J = 1MHz and = 1kHz. The flip-angle error is varied from a 1% to a 10% error and p is var- ied throughout a wide range of values so that p d , and p d , is explored. The majority of the parameter space is dominated by the RGA Ka sequences, even in the finite-width error-dominant regime. The robustness of RGA Ka to pulse imperfections ultimately saturates at K = 64. All simulations are averaged over 10 realizations of B . The optimal sequences that are used in multipleK opt values require the following number of cycles: (a) m = 2, (c) n = 1, (d) n = 2, (e) p =q = 1, (f)p =q = 4. . . . . . . . . . . . . . . . . . . . . . . . . . 120 xi 4.6 Comparison of performance for (a) CDD (empty symbols) and (b) QDD (empty symbols) versusGA K (filled symbols) as a function of the pulse- interval d in the ideal pulse limit. The strength of the error Hamilto- nian is chosen asJ = 1MHz and the strength of the pure-environment dynamics = 1kHz. Optimal GA sequences achieve a higher of order error suppression than CDD as the number of pulses increases; note GA 64a and GA 256a as compared to CDD 3 and CDD 4 , respectively. In contrast, QDD outperforms GA K for all sequence lengths, consistent with the expected superiority of interval-optimized schemes in the ideal pulse limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.7 Comparison of performance betweenRGA K and (a) CDD ` , or (b) QDD M , when subjected to finite pulse duration. Performance is characterized as a function of p , while d = 0:1ns. CDD ` performance is essentially the same for all `, scaling as D Kos O(J p ). RGA K achieves a sig- nificant increase in robustness over CDD ` at K = 8a 0 ; 16a 0 ; 32c; 64c, where the performance surpasses the linear scaling in p . Note the even- tual saturation in decoupling order characteristic of the rectangular pulse profile displayed by the nearly equivalent scaling of K = 32c; 64c. QDD M , M = 1; 3; 7; 15, performance becomes increasingly worse as the sequence order increases due to an accumulation in errors brought about by the finite duration of the pulses. As in the case of CDD ` , QDD M performance maintains D Kos O(J p ) for all M. All results are averaged over 10 realizations ofB . Error bars are included, but are quite small. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.8 Performance ofRGA K sequences versus (a) CDD ` and (b) QDD M as a function of flip-angle error2 [0:01; 0:2] averaged over 20 realizations ofB . The relevant parameters are chosen asJ = 1MHz, = 1KHz, and d = 0:1ns. Numerically optimal sequences are found to be highly robust against flip-angle errors, significantly outperforming CDD r for r = 1; 2; 3; 4. For QDD M , the sequence orders are chosen as M = 3; 7; 15 and directly correspond toK = 16; 64; 256. QDD is shown to be highly sensitive to flip-angle errors, decreasing in performance asM grows. Again, robust GA sequences achieve optimal performance. . . . 125 4.9 Performance of RGA K , CDD ` , and QDD M when subjected to finite pulse duration and flip-angle errors. The pulse-interval is fixed at d = 0:1ns, whileJ = 1MHz and = 1kHz.RGA K sequences significantly outperform both CDD ` and QDD M for K = 4; 16; 64; 256. The most notable region of robustness exists for < 0:04 and p < d forK = 16; 64; 256. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 xii 4.10 Comparison of GA (q) 8a , CDD ` , and GA 4 performance after one cycle as a function of the number of pulses K for ideal, zero-width pulses. The strength of the error Hamiltonian and environment dynamics are set toJ = 1MHz and = 1kHz, respectively, and the minimum pulse- interval d = 0:1ns. Results are averaged over 25 random realizations ofB , where the error bars are shown, but quite small. As expected by the results of the GA searchGA (q) 8a , q = 1;:::; 4 is indeed superior to CDD ` ,` = 1;:::; 6, andXY 4 . . . . . . . . . . . . . . . . . . . . . . . 129 4.11 Comparison ofRGA (q) 8a , CDD ` , and RGA 4 performance versus the num- ber of pulsesK for combined pulse errors (flip-angle and finite-width) after one cycle. Hamiltonian parameters,J and, are the same as those in Fig. 4.10 with the total cycle time fixed at c = 1ns, as opposed to d . Results are averaged over 25 realizations ofB with = 0:01 and p = c = 10 10 . The performance of RGA (q) 8a improves as the number of pulses is increased within a given cycle time c . For the specified parameters, CDD ` does not exhibit enhanced performance with increas- ing concatenation level. . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.12 Performance measure 1D Tr [(0);( c )] for R4QDD [shown in (a)- (e)], RQDD [shown in (f)-(j)], and QRUDD [shown in (k)-(o)] as a function of total time K c and flip-angle error averaged over 20 realizations ofH and initial conditions. The pulse duration p = 10:6s and the total number of pulses is fixed at 880 for all panels unless other- wise specified. Optimal performance is primarily observed for RQDD, most notably in the case of M = 1 and M = 3, where a regime can be defined where 1D is close to unity. R4QDD and RQDD perfor- mance is essentially equivalent for the largestM values considered (see last two columns). The remaining sequence structure, QRUDD, clearly exhibits the least robustness to pulse imperfections. . . . . . . . . . . . 139 4.13 Performance measure 1D Tr [(0);( c )] for R4QDD [shown in (a)- (e)], RQDD [shown in (f)-(j)], and QRUDD [shown in (k)-(o)] as a function of the pulse duration p and flip-angle error averaged over 20 realizations of H and initial conditions. The total time is fixed at T = 30ms and the total number of pulses at 880, unless otherwise specified. RQDD appears to be the least susceptible to variations in p , including various identifiable regimes of flip-angle error robustness for p < 10s. R4QDD exhibits similar results with relatively small regimes of robustness. Regions of high performance are nearly non- existen for QRUDD. . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 xiii 4.14 Performance measure 1D Tr [(0);( c )] for R4QDD [shown in (a)- (e)], RQDD [shown in (f)-(j)], and QRUDD [shown in (k)-(o)] as a func- tion of the average pulse delay avg and flip-angle error averaged over 20 realizations ofH and initial conditions. The pulse duration is fixed at p = 10:6s, therefore, the total cycle time c varies with d . Unless otherwise specified, K cycles of each sequence are used to generate 880 pulses. RQDD again exhibits the optimal performance for a major- ity of the parameter space considered. QRUDD performance varies the most dramatically with d and indicating a strong dependence on these parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.15 log 10 (1D Tr [(0);( c )]) for R4QDD [shown in (a)-(e)], RQDD [shown in (f)-(j)], and QRUDD [shown in (k)-(o)] as a function of the bath strength and system-environment interaction strengthJ averaged over 20 realizations of H and initial conditions. R4QDD performance is essentially-independent and out-performs RQDD, and QRUDD, when J. As sequence order increases [M > 3], RQDD overcomes the reduced performance around J and maintains a larger range of enhanced performance than R4QDD and QRUDD. See Fig. 4.12 for additional details. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 4.16 Comparison in performance between RQDD and GA-optimized sequences for fixed number of pulses with varying pulse delay and magnitude of flip-angle error. Note that GA-optimized sequences outperform RQDD for the Hamiltonian outlined in Eq. (4.74). . . . . . . . . . . . . . . . . 150 5.1 Averaged trace-norm distanceD Tr (T ) between the CDD-protected final state and the desired ground state as a function of total run timeT for Grover’s algorithm, in units of the inverse minimum gap. Cutoff fre- quency: = min =5. The ideal (solid black) and faulty (empty squares) evolutions are included for reference. Insert shows a close-up for large T , using a log scale for the vertical axis. Performance improves mono- tonically with concatenation level l, with the corresponding sequence denoted CDD l . Error bars are due to averaging over 30 random realiza- tions ofH err (t). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 xiv 5.2 Trace-norm distance between the QDD-protected final state (T ) and the desired ground state M , as a function of total run timeT for Grover’s algorithm, in units of the inverse minimum gap. Specific sequence orders are considered: M = 1; 3; 4; 6; 7; 14; 15. Odd parity sequence orders contain an equal number of pulses to those of the CDD sequences shown in Fig. 1 (main text). In contrast to CDD, QDD-protected evolu- tion does not more closely resemble closed-system evolution at largeT as the number of pulses grows. . . . . . . . . . . . . . . . . . . . . . . 164 5.3 As in Fig. 5.1, for the 2-SAT on a ring problem. The curves for ideal evolution and CDD 4 overlap to within our numerical accuracy up toT = 25= 2SAT min . Note the minimum atT 12 2SAT min for the faulty evolution, suggesting the existence of an optimal open system evolution time. This optimal time increases with concatenation level, until it disappears in the ideal case and for CDD 4 . CDD boosts the deviation by a factor of 10 fromD Tr 0:2 (atT 12:5= 2SAT min ) in the faulty case toD Tr 0:02 (atT 34:5= 2SAT min ) for CDD 4 . . . . . . . . . . . . . . . . . . . . . . 166 5.4 Same as in Fig. 5.2, for the 2-SAT on a ring problem. QDD protection extends ideal evolution as sequence order increases, ultimately saturat- ing atM = 14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 5.5 CDD 4 vs QDD 15 for the Grover problem as a function of the normalized bath correlation time 1=(T ) and normalized total run time T= G min . Increasing the bath correlation time 1= at fixedT generally results in improved performance for CDD 4 . whose performance is significantly better than QDD 15 in in the largeT regime. All results were averaged over 30 realizations ofH err (t). . . . . . . . . . . . . . . . . . . . . . . 168 5.6 Comparison between CDD 4 and QDD 14 performance for 2-SAT on a ring as a function of the normalized correlation time (T ) 1 and total run timeT . As in Fig. 3 (main text), sequence performance is nearly the same for short correlation times. Increases in correlation time result in smaller deviations between DD-protected and ideal evolution for both schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 5.7 Performance of CDD 4 for Grover’s search algorithm as a function of the pulse interval for various values of the frequency cutoff. Short correlation times ( = 200 G min ) demand a large pulse interval to obtain minimum deviation from ideal evolution. As is decreased, reduced minimum deviation is achieved at pulse intervals much smaller than the correlation time (1=). Results are averaged over 30 realizations ofH err . 170 xv 5.8 As in Fig. 5.7, for QDD 15 . The pulse interval represents the minimum delay between pulses for QDD. In contrast to CDD 4 , minimum devia- tion between DD-protected and ideal evolution does not increase with correlation time for QDD. Note also that the minimum deviations are approximately twice as large as those obtained for CDD and require a minimum pulse interval 10 times smaller. . . . . . . . . . . . . . . . . 171 5.9 Performance of CDD 4 -protected evolution for the 2SAT problem, as a function of pulse interval, for different values of the frequency cutoff ( in units of 1= to separate the curves). Total timeT = 4 4 . The minimum in each fixed curve corresponds to an optimal pulse inter- val opt (), and correspondingT opt (). Peak performanceD Tr (T opt ()) improves as is decreased, except at = 2SAT min =0:005 where self- averaging effects result from rapid fluctuations in j (t) (“motional nar- rowing”). Results are averaged over 30 realizations ofH err (t). . . . . . 172 5.10 As in Fig. 5.9, for QDD 14 . Here, the pulse interval represents the minimum delay between pulses. Minimum deviations from ideal evolu- tion are approximately twice as large as those obtained for CDD 4 . Peak performance appears insensitive to the value of. . . . . . . . . . . . 173 5.11 Performance of CDD for a constant error Hamiltonian [ j (s) 10 2 8;j in Eq. (2) (main text)]. The curves for ideal evolution, faulty evo- lution, and CDD 4 overlap up to T 4= G min , after which CDD 4 con- tinues to track the ideal evolution essentially perfectly throughout the simulated range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.12 As in Fig. 5.11, for QDD withM =f1; 3; 4; 6; 7; 14; 15g. QDD-protected evolution deviates minimally from ideal evolution for M = 15 up to T 25= G min , thereafter M = 6; 7 are the optimal sequence orders. Unlike the CDD case, increasing the number of pulses does not result in closed-system-like behavior over the entire range of T considered. M = 14; 15 evolution closely coincides with ideal evolution more so than any other M values, however also diverges faster than all order sequence orders at criticalT values. . . . . . . . . . . . . . . . . . . . 176 A.1 Complexity-reduction protocol for K = 16. The upper (blue) curves denote the linking between odd pulse sites and the lower (red) curves denote even pulse site linking. The process begins with all odd and even pulses linked, then continues by removing links between every other even pulse site. Links are removed until all even sites are uncorrelated, thereafter the odd sites undergo the same process. . . . . . . . . . . . . 189 xvi Abstract Quantum computation (QC) relies on the ability to implement high-fidelity quantum gate operations and successfully preserve quantum state coherence. One of the most challenging obstacles for reliable QC is overcoming the inevitable interaction between a quantum system and its environment. Unwanted interactions result in decoherence processes that cause quantum states to deviate from a desired evolution, consequently leading to computational errors and loss of coherence. Dynamical decoupling (DD) is one such method, which seeks to attenuate the effects of decoherence by applying strong and expeditious control pulses solely to the system. Provided the pulses are applied over a time duration sufficiently shorter than the correlation time associated with the envi- ronment dynamics, DD effectively averages out undesirable interactions and preserves quantum states with a low probability of error, or fidelity loss. In this study various aspects of this approach are studied from sequence construction to applications of DD to protecting QC. First, a comprehensive examination of the error suppression prop- erties of a near-optimal DD approach is given to understand the relationship between error suppression capabilities and the number of required DD control pulses in the case of ideal, instantaneous pulses. While such considerations are instructive for examin- ing DD efficiency, i.e., performance vs the number of control pulses, high-fidelity DD in realizable systems is difficult to achieve due to intrinsic pulse imperfections which further contribute to decoherence. As a second consideration, it is shown how one xvii can overcome this hurdle and achieve robustness and recover high-fidelity DD in the presence of faulty control pulses using Genetic Algorithm optimization and sequence symmetrization. Thirdly, to illustrate the implementation of DD in conjunction with QC, the utilization of DD and quantum error correction codes (QECCs) as a protection method for adiabatic quantum computing (AQC) is discussed. A performance compar- ison between two deterministic DD schemes is given, where preference towards one particular method is found due to sequence structure and procedure from which higher accuracy sequences are generated. xviii Chapter 1 A Brief Introduction to Quantum Computation and Decoherence Theoretical and experimental advancements in quantum computation (QC) in recent years has led to a profound interest in QC as a viable alternative to classical comput- ing techniques. Predicted computational advantages which make QC such an attractive approach are due to the intrinsic, highly-nontrivial phenomena which characterize the theory of quantum mechanics. In this chapter, the fundamental postulates of quantum dynamics are described to build the general theoretical knowledge of quantum mechan- ics necessary for subsequent chapters. Following this discussing, a brief introduction to QC is given with a summary of possible hurdles one must address in order to con- struct high-accuracy QC hardware when computational errors can occur due to intrinsic interactions between quantum systems and their environment. 1.1 Quantum Dynamics Quantum dynamics can be summarized as four fundamental postulates which relate to the mathematical representation of quantum states and compositions of systems, time- dependent dynamics of states, and how one performs measurements on quantum states. Each is discussed in the following section along with relevant notation and conceptual aspects of quantum mechanics. 1 1.1.1 Fundamental Postulates of Quantum Mechanics Postulate 1: Hilbert Space Quantum mechanical states of physical systems are characterized by vectors defined within a Hilbert spaceH, a complex vector space with an associated inner product. Each quantum state, written in Dirac notation asj i, can be expressed as a column vector j i = 0 B B B B B B B @ 1 2 . . . d 1 C C C C C C C A (1.1) defined inH such that d = dim(H) dictates the dimensionality of the states, i.e. the degrees of freedom. The normalization conditionkj ik = p h j i = 1 is imposed upon all states such that the inner product between statesj i andji is defined as h ji = 1 2 d 0 B B B B B B B @ 1 2 . . . d 1 C C C C C C C A = d X j=1 i i : (1.2) Whileh ji 0 holds for all states inH, the particular case ofh ji = 0 defines states which are orthogonal. Combined with the normalization condition, an orthogonal set of basis vectors can be used to described the quantum mechanical Hilbert space as an orthonomal basis of statesfjkig d k=1 such thatkjkik = 1 and the completeness relation P d k=1 jkihkj = I, whereI is the identity operator, are satisfied. In particular, the completeness relation is useful for expressing any statej i2H or operatorA2H 2 in terms of the basis vectors asj i = P k h jkijki and A = P k;k 0 hkjAjk 0 ijkihk 0 j, correspondingly. An alternative technique for describing states, and more generally ensembles of states, is the density operator, equivalently referred to as the density matrix. Given a system with only one possible statej i, the density operator is simply defined as =j ih j. One can show that this particular situation satisfies the condition of a pure state as the Purity() = Tr( 2 ) =kj ik = 1 by definition of normalized states. The density operator formalism is most useful when describing systems that have some classical uncertainties associated with the set of possible states, e.g. consider a system described by an ensemble of pure statesfp i ;j ig i2Z where each statej i i is associated with a classical probabilityp i , such that P i p i = 1. The density operator for this system is described by = P i2Z p i j i ih i j and the associated purity is Tr[ 2 ] = Tr X i;j2Z p i p j j i ih i j j ih j j ! = X i;j2Z p i p j jh i j j ij 2 0: (1.3) The purity reaches a maximum value of unity only if the ensemble is reduced to a single pure state, therefore 0 Tr( 2 ) 1 where the purity is strictly less than one if the state is not pure and the density matrix describes a mixed state. It is important to note that whether the state is distinguished as a pure or mixed state > 0 and Tr() = 1 are necessary and sufficient conditions for defining the density operator. Postulate 2: Compositions of Systems The second postulate of quantum mechanics pertains to the manner in which one describes Hilbert spaces, states, and operators for quantum systems which are com- posed of two or more subsystems. First consider the case of two quantum systems with respective Hilbert spacesH 1 andH 2 such thatd 1 = dim(H 1 ) andd 2 = dim(H 2 ). The 3 combined Hilbert space is defined byH =H 1 H 2 , whered = dim(H) =d 1 d 2 . The tensor product represents a generalized version of the outer product operator for vec- tors and therefore can also be used to define states of composite systems. Two quantum systems with associated statesj i2H 1 andji2H 2 combine to form j i ji = 0 B B B @ 1 . . . d 1 1 C C C A 0 B B B @ 1 . . . d 2 1 C C C A = 0 B B B B B B B B B B B B B B B B @ 1 1 . . . 1 d 2 . . . d 1 1 . . . d 1 d 2 1 C C C C C C C C C C C C C C C C A : (1.4) Operators follow similar constructions where ifA2H 1 andB2H 2 , then the combined operator which acts on both Hilbert spaces is given byO = A B where the matrix representation of this operator is essentially a generalization of the vector description. Of course, while only two Hilbert spaces are considered above, the methods can be extended to a quantum system composed ofn systems with respective Hilbert spacesH i , i = 1; 2;:::;n where the composite Hilbert space is given byH = N n j=1 H j . States and operators follow suit with similar descriptions:j i = N n j=1 j j i andO = N n j=1 A j for j j i2H j andA j 2H j . The composition of systems can also be described in terms of density matrices where 12 = 1 2 represents the combined-system density matrix for 1 2H 1 and 2 2 H 2 . Note that the trace can be utilized to extract the state of each subsystem 1 = Tr 2 [ 12 ] = P k 2 hkj 12 jki 2 and 2 = Tr 1 [ 12 ] = P j 1 hjj 12 jji 1 , where Tr j represents the partial trace of the subsystemj,H 1 = spanfjji 1 g, andH 2 = spanfjki 2 g. Although this definition appears quite trivial for the case when 12 is separable, i.e. a tensor 4 product of states, it is extremely useful for extracting states of subsystems when 12 is a nonseparable state, i.e. cannot be expressed as a tensor product of states. Such a situation will arise when considering open system quantum dynamics where the Hilbert space is partition into “system” and “environment” Hilbert spaces and only the system state is desired. Postulate 3: Dynamics The time-dependent dynamics of a quantum state are determined by the Schr¨ odinger equation i~ d dt j (t)i =H(t)j (t)i (1.5) where the HamiltonianH(t) is said to be the generator of the dynamics and~ is Plancks constant, taken as unity in all subsequent sections. Assuming the time-dependent state j (t)i can be expressed asj (t)i =U(t)j (0)i for an initial statej (0)i, Eq. (1.5 can be expressed as i~ d dt U(t) =H(t)U(t); U(0) =I (1.6) such that the time evolution propagatorU(t) is a unitary operator, i.e. U y (t)U(t) = I. Eq. (1.6) is a homogeneous first-order ordinary differential equation and therefore has the formal solution U(t) =T + exp i ~ Z t 0 H(t 0 )dt 0 (1.7) whereT + is the time-ordering operator. See Sec. 2.2 for additional information regard- ing Time-Dependent Perturbation Theory methods used to solve Eq. (1.6) and approxi- mate Eq. (1.7). 5 The definition of the time-evolved state presented above immediately leads to a def- inition for the time-evolved density operator (t) =j (t)ih (t)j =U(t)(0)U y (t) (1.8) as well, where(0) =j (0)ih (0)j is the initial density operator. The corresponding dynamical equation is d dt (t) =i~[H(t);(t)] (1.9) as verified by direct substitution of Eq. (1.8) into Eq. (1.9) or by simply differentiating Eq. (1.8) and collecting terms. Note that [A;B] =ABBA represents the commutator ofA andB. Postulate 4: Measurement One of most interesting aspects of quantum mechanics is the effect of making measure- ments on quantum states. While such actions are of fundamental importance in any scientific theory as they allow an observer to extract information about the system, a quantum mechanical measurement differs from its classical counterpart in that it ulti- mately alters the quantum state. Measurements associated with physically measurable quantities are known as observables. These operators are required to be Hermitian oper- ators such that an observable A has the spectral decomposition A = P a P where a 2 R and P are projection operators satisfying P 2 = P . Measurements of the observableA can be accomplished using the projection operators such that the measure- ment j i7!j i = P j i p p (1.10) results in measurement outcome a with a probability p =h jP j i. Naturally, an expectation value for an observableA can also be defined,hAi =h jAj i. As in case 6 of quantum evolution, the notion of measurement is also identifiable in terms of the density matrix. Given a density matrix, the state after measurement is given by 7! = P P p ; (1.11) wherep = Tr(P ). The projective measurements or V on Neumann measurements [] described above can be extended to generalized measurements operators as well. Given a set of measurement operatorsfM k g N i=1 such that P k M y k M k = I, the resulting state after a generalized measurement is described by j k i = M k j i p p k and k = M k M y k p k (1.12) withp k =h jM y k M k j i = Tr(M y k M k ) determining the probability for the associated measurement outcome. 1.1.2 Open System Dynamics An open quantum system refers to a quantum system which interacts with an external surrounding quantum system, called the environment or bath. These interactions can have a profound effect on the dynamics of the system which may result in very differ- ent time evolution dynamics than the closed-system (environment-free) case, especially when the Hilbert space dimension of the bath is much larger than the system. Perhaps the most canonical manner of displaying the effect of the environment is to consider the evolution of a system and environment with Hilbert spacesH S andH B , respectively, 7 described byU(t) =e iHt , where the HamiltonianH2H S H B . The resulting state of the system is most generally given by S (t) = Tr B U(t) SB (0)U y (t) ; (1.13) where SB (0) represents the initial state of the system and environment. Assuming that the initial joint system-environment state is separable such that SB (0) = S (0) B (0), where the initial state of the bath has the spectral decomposition B = P jihj, H B = spanfjig, the time evolved state of the system becomes S (t) = X hjU(t) [ S (0) B (0)]U y (t)ji (1.14) = X ; hjU(t)ji S (0)hjU y (t)ji (1.15) = X K (t) S (0)K y (t) (1.16) with K (t) = P p hjU(t)ji representing the Kraus operators. The resulting state of the system looks astonishingly similar to the definition of a generalized quan- tum measurements described by postulate 4 in Sec. 1.1.1. Here in lies the interesting behavior of an open quantum system as the interactions with the environment essen- tially result in measurements on the system by the bath. The dynamics no longer being unitary, nor reversible, leads to decoherence processes that produce deviations in desired system evolution and a loss of information to the environment. This mechanism is one of the most difficult hurdles to overcome as QC relies on the ability to manipulate quantum systems and preserve information regarding the state of the system with high precision. 8 1.2 Quantum Computation Despite notable progress in classical computing in recent years, inevitable limitations in classical hardware potential are a certainty. Length scales associated with classical devices will eventually saturate rendering classical computing and protection methods unreliable as one approaches the quantum regime. Futhermore, device performance is ultimately bounded by limits set by classical computational complexity theory; thereby, restraining further computational advantages. Quantum computation alleviates many of the restrictions of classical computing by exploiting quantum phenomena. Most impor- tantly, QC has the ability to offer computational advantages, i.e. reduction in computa- tional runtime, over classical computing in a number of scenarios which are outlined in Sec. 1.2.2. A large body of work has focused on circuit model QC [1], where a sequence of gate operations in the form of unitary evolution operators [See Sec. 1.1.1] are utilized to manipulate quantum systems in order to perform a desired computation. Choosing specific sets of gate operations it can be shown that universal computation, i.e. any gate operation, is achievable [1]. Physical realization of circuit model QC is dependent on the satisfaction of five criteria originally outlined by David DiVincenzo [2]: the system must by scalable with well characterized quantum bits one must have the ability to initialize the quantum bits to simple fiducial states the system must have long decoherence times relative to gate operation time, i.e. the time scale for which information is lost due to interactions between the quan- tum system and its environment must be longer than the associated QC time scale a universal set of quantum gates must be achievable one must have the ability to perform measurements on individual quantum bits. 9 Additional methods exist such as adiabatic quantum computation (AQC) where a sequence of gate operations is replaced by a continuous interpolation between two Hamiltonians [3]. Employing the adiabatic theorem of quantum mechanics, AQC per- forms computations by slowly varying physical quantities within the Hamiltonian such that the system, initially characterized by an easily prepared ground state, reaches a final state after some timeT which encodes the desired state of the computation. AQC does not stand alone as a method for QC, it has been shown to be computationally equivalent to circuit model QC [4, 5, 6, 7, 8]. Additional interesting approaches to QC also exist in the form of measurement-based QC [9, 10, 11, 12] and holonomic QC [13, 14, 15]. The former is based on the manipulation of quantum states by performing measurements on the system, while the latter exploits geometrical phases of quantum states. 1.2.1 The Qubit Classical computing implements computational algorithms and information processing by manipulating strings of binary variables where each variable takes a value of 0 or 1. QC removes this restriction by exploiting the quantum mechanical superposition prin- ciple. This key aspect of quantum mechanics states that quantum systems can exist in a physical state which consists partially of all possible system-relevant theoretical states simultaneously. In circuit model QC, quantum superposition can be utilized to generate non-classical quantum gates such as entangling gates and the possible manipulation of qubit states by indirect measurements. Furthermore, quantum superposition is essential for AQC as the ground state of the initial Hamiltonian corresponds to the superposition of all possible states; thus, allowing for the exploration of all possible solution states simultaneously throughout the interpolation process. 10 0 1    Figure 1.1: Bloch Sphere representation of an arbitrary qubit state. The Bloch sphere is a unit sphere with the north and south poles representing the two orthonormal basis states of the qubit. Mathematically, quantum superposition states can be depicted byj i = P k k jki whereH = spanfjkig. Measurements result in the collapse of the state where the asso- ciated probability of obtaining one of the possible theoretical statesjki is determined by the absolute value square of the weightsj k j 2 given to that state; this follows directly from postulate 4. The most fundamental quantum system relevant for QC is the quan- tum bit, or qubit. Defined in a similar manner to the classical bit, the qubit is a two-level system with basis statesjki2fj0i;j1ig. General superposition states of the qubit can be expressed asj i = aj0i +bj1i, wherejaj 2 +jbj 2 = 1 follows from state normal- ization. A more intuitive approach to general single-qubit states was proposed by Felix Bloch [?] and defines a general qubit state as = 1 2 (I +~ v~ ); (1.17) 11 where ~ v = (v x ;v y ;v z ) is the Bloch vector and ~ = ( x ; y ; z ) is the Pauli vector consisting of the Hermitian, traceless, and unitary Pauli matrices x = 0 @ 0 1 1 0 1 A ; y = 0 @ 0 i i 0 1 A ; z = 0 @ 1 0 0 1 1 A : (1.18) This particular representation allows for the visualization of a qubit state by the Bloch vector on the Bloch sphere illustrated in Fig. 1.1. Expressing the Bloch vector in spher- ical coordinates as ~ v = (R cos sin;R sin sin;R cos); (1.19) with2 [0;] and2 [0; 2] referring to the polar and azimuthal angles, respectively, one can discriminate pure and mixed states by the value of the real constant R. Pure states correspond to states on the surface of the sphere (R = 1) and are defined in the state representation by j i = cos 2 j0i +e i sin 2 j1i (1.20) as shown in Fig. 1.1. Mixed states are given byR< 1 and, therefore, represent all states which lie within the Bloch sphere. By the composition of states postulate, separable n-qubit states can be constructed by taking an n-fold tensor product of Eq. (1.17) or Eq. (1.20). 1.2.2 Quantum Algorithms Here a few quantum algorithms that display a computational advantage, or quantum speedup, are briefly discussed. The focus will be the Deutsch-Jozsa algorithm, Shor’s factoring algorithm, and Grover’s search algorithm as these are some of the earliest known quantum algorithms which convey the benefits of QC. 12 The Deutsch-Jozsa algorithm [16] involves the determination of whether a binary functionf(x) is constant for all valuesx or balanced, i.e. equal to 1 for exactly half of all possiblex values and 0 for the remaining half. This problem details an observer Alice selecting ann-bit numberx2f0; 2 n 1g, sending it to her companion Bob, and then receiving the value of the function back from him. In the classical scenario, 2 n =2 + 1 queries are required to determine the properties of the function, while utilizing quantum superposition and interference between then-qubit state representing the query results in a single correspondence with Bob. Consequently, an exponential speedup is achieved using the quantum Deutsch-Jozsa algorithm. A substantial speedup in computational time is also observed for Shor’s factor- ing algorithm [17], a quantum approach to prime factorizing integers. Given an integer value N that is to be factored, Shor’s algorithm performs prime factoriza- tion in a time T = O [(logN) 3 ], only requiring polynomial time. In contrast, the most efficient classical alternatives demand a sub-exponential computation time T = O e 1:9(logN) 1=3 (log logN) 2=3 . The efficiency of Shor’s algorithm, at least from the quan- tum perspective, can be attributed to performance advantages of the quantum Fourier transform which offers an exponential speedup over its classical counterpart. Search problems can also be addressed by QC as well. The first known approach to solving the problem of locating a marked item in a database consisting ofN unsorted items is Grover’s search algorithm [18]. The procedure involves making queries to an oracle which determines whether the query is indeed the desired marked item. The quantum version of Grover’s search algorithm requiresO( p N) queries to locate the marked state, as opposed toO(N) queries in the classical case. The quadratic speedup in performance is due to quantum superposition since the initial state of the system is chosen as a complete superposition state between all possiblen-qubit states representing theN = 2 n unsorted items. Simultaneously exploring all possible items in the database 13 at the commencement of the algorithm, each iteration of Grover’s algorithm increasingly localizes the search around the desired marked state. 1.2.3 Experimental Realizations of Quantum Computation Satisfying the DiVincenzo criteria presents a daunting task for experimental physicists and, while much progress has been made towards successful realization of quantum computers, issues of scalability, decoherence, and precise qubit manipulation and mea- surement still remain unsolved for many of the systems currently being examined. This includes systems such as Nuclear magnetic resonance systems [19, 20] and spin-based quantum dot systems [21, 22], which rely on the selective manipulation of the spin attributes of particles using external magnetic fields. Despite scalability issues and short decoherence times, NMR systems have demonstrated Shor’s algorithm using seven qubits to factor the number 15 [23] and later four qubits to factor the number 143 [24]. Trapped ions have also been explored as a mode for QC where electronic states of each ion compose qubit states and QC occurs via the collective quantized motion of the ions in the trap [25, 26]. On the other end of the spectrum, linear optics has been proposed as a QC approach where qubits are constructed from photonic modes and interact through various measurement-based methods [27]. Successful experiments of teleportation of quantum states in these systems has led to their continued examination as possible QC systems [28, 29]. More recently, spin qubit systems based on Nitrogen-Vacancy (NV) centers in diamond have gained some appeal as they inherently possess decoher- ence times ten or more times larger than many previously examined systems [30, 31]. However, NV centers do not immediately appear to present a scalable approach to QC. The issue of scalability has recently been tackled by systems based on superconduct- ing qubits in which the manipulation of magnetic flux through superconducting loops allows for the alteration of qubit states represented by the direction of current flow [32]. 14 Constructed by D-Wave Systems, these particular devices have been shown to scale extremely well and have led to systems containing 108 and 503 qubits. While scru- tinized in regards to the quantum nature of the device, quantum phenomena has been observed in D-Wave hardware for qubit systems composed of 16 qubits [33]. 1.3 Decoherence Mechanisms The successful implementation of QC relies on the ability to perform a computation within a period of time before information is lost from the system. Since this is a key aspect towards reliable QC, it is important to understand this concept and the decoher- ence processes from which it arises. In this section, the most common decoherence mechanism caused by interactions between the system and its environment are rigor- ously defined and discussed specifically for qubit systems. The system and environment Hilbert spaces are denoted byH S andH B , respectively. Time evolution dynamics are generated by the Hamiltonian H 0 =H S +H B +H SB (1.21) which can be partitioned into a pure system HamiltonianH S 2H S , pure bath Hamilto- nianH B 2H B , and interaction HamiltonianH SB 2H S H B . The pure system and bath Hamiltonians are defined within different Hilbert spaces, therefore [H S ;H B ] = 0. The focus of this section will beH SB which does not commute with eitherH S orH B as it is defined within the joint system-bath Hilbert space. Without loss of generality, it will be further assumed that the internal Hamiltonian can be decomposed as H 0 = X S B (1.22) 15 where S 2H S form an operator basis for the system degrees of freedom such that Tr(S S ) = ; and S 0 = I S , and B 2 H B . The notation defined here will be continuously utilized throughout the remainder of this study. 1.3.1 Dephasing Dephasing is a decoherence mechanism that results in the decay of quantum coherence. Coherence is essential for quantum computation as it allows for certain attractive char- acteristics such as quantum superposition of states. Destruction of quantum coherence leaves quantum systems in classical states where only the qubit basis statesfj0i;j1ig occur with some associated probabilities. In order to illustrate the effect of dephas- ing from a mathematical point of view, consider a single-qubit system subjected to the interaction Hamiltonian H SB = z B z (1.23) where B z is a generic bath operator. For simplicity, H S 0 and H B 0, therefore the dynamics are solely generated by the interaction Hamiltonian. In constructing the time-evolved system state S (t) = Tr B e iH SB t (0)e iH SB t (1.24) one can consider a separable initial state(0) = S (0) B (0), where the system is in a general pure state S (0) =j ih j, withj i =aj0i +bj1i, and the environment initial state has the decomposition B (0) = P jihj, without loss of generality. Upon taking into account the initial state and simplifying Eq. (1.24), the state of the system becomes S (t) = 0 @ jaj 2 ab f(t) a bf (t) jbj 2 1 A (1.25) 16 where the coherence (off-diagonal) terms contain an additional factor f(t) = P e 2itgz; when imposing the spectral decomposition B z = P g z; jihj. The environment Hilbert space is assumed to be much larger than the system Hilbert space, thus the sum over states can be extended to the continuum limit by replacing the sum by an integral to obtain f(t) = Z 1 0 e 2itgz () D()d; (1.26) whereD() is the density of states for the environment. The density of states is deter- mined by the specifications of the bath which can take various continuous functional forms to correspond to Ohmic , Gaussian , or Lorentzian approximations which well- characterize realistic bath density of states functions [34]. For simplicity, consider the case ofg z () = ~ g z andD() = D 0 ( c ) for a cut-off frequency c . The system dynamics in the long time limit are consequently given by S (t) 0 @ jaj 2 ab (~ g z t) 1 a b (~ g z t) 1 jbj 2 1 A (1.27) where the decay of the coherence terms due to the dephasing mechanism is clearly visible. The rate of decay is usually referred to as the decoherence rate, while the inverse of this quantity is known as the decoherence time. Example: Spin-boson Model The spin-boson model [34] presents a system which can be used to study the effect of decoherence due to dephasing on qubit systems. Here, the system consists of a single 17 qubit and the environment is given by a collection of bosonic particles whose dynamics are governed by H 0 = 1 2 z |{z} H S + X k ! k b y k b k | {z } H B + z X k g k (b y k +b k ) | {z } H SB : (1.28) Pure system dynamics are characterized by a energy level-splitting of (in units of ~ = 1) between the two basis states of the qubit. The environment operatorsb y k andb k correspond to the creation and annihilation operators for the kth mode of the bosonic environment, respectively, which satisfy the commutation relation [b k ;b y k 0 ] = k;k 0. The coefficient ! k characterizes the energy spectrum and resonance frequencies for the bosonic modes, while g k determines the strength of the interaction between the qubit andkth mode. A common approach to studying the effect of the system-interaction is to move into the interaction picture with respect to H S and H B , and discuss the time evolu- tion dynamics of ~ H SB (t) = U S+B (t)H SB U y S+B (t), where U S+B (t) = e i(H S +H B )t . 1 Using the commutation relation for bosonic operators, the time evolution operator for the rotated-frame dynamics written as ~ U(t) = exp ( z 2 X k h b y k k (t)b k k (t) i ) (1.29) with k (t) = 2g k ! k 1e i! k t : (1.30) 1 See Section 2.2.1 for a further discussion of the rotating frame transformation. 18 The effects of dephasing are apparent from decay of the coherence terms of the qubit density matrix S (t); hence, the focus is the time-evolution dynamics specifically for the off-diagonal terms (01) S (t) =h0jTr B h ~ U(t) SB (0) ~ U y (t) i j1i (1.31) and (10) S (t) = ( (01) S (t)) . Assuming that the qubit and the environment are initially described by the uncorrelated state SB (0) = S (0) B (0) and the environment is initially in the thermal equilibrium state B (0) = Y k (1e ! k =k B T )e ! k b y k b k =k B T (1.32) at temperatureT (k B is the Boltzmann constant), the dynamics of the off-diagonal terms can be solved exactly as (01) S (t) = (01) S (0)e (t) ; (1.33) with the decay of such terms determined by the decoherence function (t) = Z 1 0 d! coth ! 2k B T 1 cos(!t) ! 2 I(!); (1.34) displayed here in the continuum limit withI(!) = P k jg k j 2 (!! k ) representing the spectral density of the bath. Generally, the spectral density can be written in terms of the density of states as I(!) = 4jg(!)j 2 D(!) where the functional form of the cou- pling constant g(!) and D(!) depend upon the specifications of the environment, as mentioned above. Regardless the choice of such quantities, the general decay of the coherence terms is evident for the spin-boson model from Eq. (1.33); hence, represent- ing a useful physical model for studying the dephasing mechanism for systems relevant for QC. 19 1.3.2 Relaxation Decoherence can also appear in the form of relaxation processes that result in quantum states evolving to equilibrium states of the system. In many cases, these equilibrium states are classical in nature and therefore relaxation processes tend to destroy quantum superposition and coherence simultaneously. Exemplifying relaxation in single-qubit systems can be done by considering the dynamics generated by the HamiltonianH 0 = H SB , which is fully characterized by the system-environment interaction Hamiltonian H SB = x B x ; (1.35) where B x is a generic environment operator. Without loss of generality, let S (0) = j1ih1j and again consider the case of an uncorrelated initial state(0) = S (0) B (0) with B (0) = P jihj. The time-evolved state resulting from Eq. (1.24) and the specified system state can be written as S (t) = 0 @ 1 2 1 4 f(t) 1 4 f (t) 1 4 f(t) 1 4 f (t) 1 4 f(t) + 1 4 f (t) 1 2 + 1 4 f(t) + 1 4 f (t) 1 A (1.36) for f(t) = P e 2igx;t using the spectral decomposition B x = P g x; jihj. Extending the sum to the continuum limit, and utilizingg x () = ~ g x and the density of statesD() = D 0 ( c ) with cutoff frequency c , the long-time limit system state becomes S (t) 0 @ 1 2 (4~ g x t) 1 (4~ g x t) 1 1 2 1 A : (1.37) Hence, the relaxation process described by Eq. (1.35) yields a loss of coherence and an eventual relaxation of the system to a classical probabilistic state where each basis state can be individually occupied with an equal probability. 20 Two-Level System Interacting with Electromagnetic Field A canonical example of relaxation often studied in quantum optics is the two-level sys- tem, or single-qubit system, interacting with an electromagnetic field [35]. The qubit, initialized in its excited statej S (0)i =j1i, transitions to its ground state spontaneously thereby resulting in an exchange and excitation in energy within the electromagnetic field. Mathematically, this process is portrayed by the total Hamiltonian H 0 = 1 2 z |{z} H S + X k ! k b y k b k | {z } H B + x X k g k (b y k +b k ) | {z } H SB (1.38) where the pure system Hamiltonian describes the qubit system with an energy gap of between basis states and the remaining terms model the electromagnetic field and its interaction with the two-level system inH B andH SB , respectively. There are various useful approximations one can make in depicting this process. The first is accomplished by rewriting x in terms of the raising and lowering operators as x = + + and then only allowing the existence of energy conserving terms in the interaction Hamiltonian, i.e. H SB = X k g k ( b y k + + b k ): (1.39) Dropping the nonconserving terms is referred to as the rotating-wave approximation (RWA) [36] and is commonly used when terms in the Hamiltonian which rotate at high frequencies can be effectively averaged to zero. The interaction picture with respect to the pure system and environment dynamics now becomes a useful tool for focusing 21 on the dynamics generated by the interaction Hamiltonian. Moving into this frame, consider the rotated system-environment interaction Hamiltonian ~ H SB (t) = U y S+B (t)H 0 U S+B (t) = X k g k ( b y k e i( ! k )t + + b k e i( ! k )t ) (1.40) with U S+B (t) = e i(H S +H B )t . The second assumption to be made is that the time-evolved state of the combined system and environment has the formj (t)i = c 1 (t)j1; 0i + P k c 0;k (t)j0;ki, withc 1 (0) = 1 andc 0;k (0) = 0, such that the system is either in the excited state with the electromagnetic field in a vacuum state or the qubit has relaxed to the ground state with an excitation in the various k modes of the field. Solving for the coefficientsc 1 (t) andc 0;k (t) via the Schr¨ odinger equation leads to d dt c 1 (t) = i X k g k e i( ! k )t c 0;k (t) (1.41) d dt c 0;k (t) = ig k e i( ! k )t c 1 (t): (1.42) Combining the two equations into a linear differential-integral equation and making the Weisskopf-Wigner approximation, the excited state is found to decay according to c 1 (t) = c 1 (0)e t=2 where the decay constant is proportional to the qubit transition frequency and the interaction strength. 2 Note that the decoherence time is given by T = 1= for this process. 2 The Weisskopf-Wigner approximation assumes that the! k frequencies of the field are approximately equal to the transition frequency of the qubit states. 22 1.3.3 General Qubit Decoherence Model In systems where the dephasing time is much shorter than the relaxation time, the dephasing model can be used to effectively describe the system-environment interac- tions. Many situations however require both processes to effectively represent the inter- action between the system and its environment due to comparable decoherence times. Collective decoherence, involving both dephasing and relaxation, is commonly repre- sented by H SB = + B + + B + z B z (1.43) for a single-qubit system, where the environment operators satisfy B = B y + . This model can also be expressed in terms of the standard Pauli basis as H SB = x B x + y B y + z B z (1.44) where the relationship between Eq. (1.43) and (1.44) is established by + = 1=2( x + i y ),B x = 1=2(B + +B ), andB y =i=2(B + B ). Although formally equivalent, the latter formalism will encompass the remainder of this study as the general decoherence model for a single-qubit system when summed withH B =I B I . 3 Although this model appears to not be completely general, lackingH S , it easy to see thatH S +H SB has a similar representation to Eq. (1.44) when choosing the environment operators appropri- ately. For example. H S = 2 z is obtained by simply definingB z = 2 I +B 0 z . Such encompassing definitions of the pure system and system-environment Hamiltonian will become relevant in Chapters 3 and 4 where both Hamiltonians are considered to be decoherence-generating. 3 In regards to notation,I is used to represent the 2 2 identity operator whileI S denotes a general d S d S identity operator, whered S = dim(H S ). 23 1.4 Quantum Error Avoidance, Correction, and Sup- pression Combating decoherence effectively can be accomplished in a variety of ways. The first to be discussed will be avoidance of decoherence by exploiting certain symmetries in H SB . In cases where such symmetries do not exist error correction methods based on their classical counterparts can be developed to correct errors after or during a quantum computation. As a more preemptive strike against computational errors, quantum error suppression methods seek to reduce the effects of system-environment interactions in order to lower the occurrence probability of computational errors. While each method has its advantages and disadvantages, ultimately a culmination of two or more of these techniques will be required for high-accuracy QC. It is important to keep this idea in mind as each method is briefly discussed below. 1.4.1 Decoherence-Free Subspaces Error avoidance refers to the ability to perform an open system QC in such a way that decoherence does not effect the computation. This invulnerability to decoherence can be achieved by decoherence-free subspaces and its generalization to decoherence-free subsystems (DFSs) [37, 38, 39, 40] by exploiting symmetries in the system-environment interaction Hamiltonian. By identifying states or sets of states which are invariant under the action of H SB , one can essentially build general qubit states from computational basis states which evolve unitarily in the presence of decoherence. As an example of the DFS construction, consider a two-qubit system interacting with its environment via collective dephasing described by H SB = ( z I +I z ) B z (1.45) 24 where B z is again a generic environment operator. The objective is to identify states which are not effected by the interaction, i.e. eigenstates ofH SB . For this particular col- lective dephasing model, three different DFSs can be identified, however all are single- state spaces except forfj01i;j10ig. Constructing logical basis statesj0 L i =j01i and j1 L i =j10i from this DFS one can define any logical qubit state asj i =aj0 L i+bj1 L i. Attaining universal computation within this DFS requires the use of logical bit-flip and phase-flip operators x L and z L , respectively. The operators x L = x x and z L = z I performs the desired logical operations for the two-qubit collective dephas- ing model. The construction discussed here can also be extended toN-qubit collective dephasing [39] and collective decoherence models [37] as well. Ideally, the most attractive feature of the DFS construction is that after identification of such states computations can occur within the subspace (or subsystem) without the need for continuous correction or suppression of decoherence. However, in situations where such symmetries in the interaction Hamiltonian do not immediately exist DFSs can be difficult to locate and it may be more beneficial to use additional methods, such as Dynamical Decoupling [See Sec. 1.4.3], to generate symmetries inH SB [41, 42, 43]. Furthermore, one can expect imperfect logical operations which lead to leakage out of the DFS; therefore, requiring error-correction methods to return the computation to the DFS [44]. Despite such possible hurdles, experimental studies have shown that coherence times can be extended by a factor of 5 in ion trap systems when utilizing DFS encoding schemes alone [45]. 1.4.2 Quantum Error Correction Classical methods for combating computational errors have predominately taken the form of error-correction schemes. Logical bits are constructed from a collection of physical bits where computations occur in the logical space. Upon the completion of a 25 set of desired operations syndrome measurements are performed to determine if errors have occurred on any of the physical bits. Those which require correction are altered accordingly to preserve the desired state of the logical bits. The success of classical error-correction schemes immediately begs the question of whether such methods can be extended to the quantum regime. While the answer is: yes, classical schemes alone are not sufficient to protect QC from errors generated by general quantum errors. Fur- thermore, one must abide by the no-cloning theorem and use caution when performing syndrome measurements since measurements result in the collapse of quantum states by postulate four [see Sec. 1.1.1]. Quantum error-correction codes (QECCs) were first discovered by Peter Shor in 1995 [46]. The Shor code, as it’s referred to, involves the construction of a logical qubit state from nine highly entangled physical qubits with logical computational basis states: j0 L i = 1 2 p 2 (j000i +j111i) 3 (1.46) j1 L i = 1 2 p 2 (j000ij111i) 3 (1.47) and logical operators: x L = ( x ) 9 and z L = ( z ) 9 , where A n denotes an n- fold tensor product of A. Correctable errors of the Shor code correspond to single- qubit bit-flip and phase-flip errors x i j i and z i j i, respectively, for any logical qubit statej i = aj0 L i + bj1 L i. Simultaneous bit and phase-flip errors also fall within the correctable error set as well as general ith qubit errors of the form E i = P 2fI;x;y;zg c i , c 2 C. Detection of computational errors is completed by per- forming syndrome measurements on the qubit states. Bit-flip errors are detected by measuring the phases between qubits in each three-qubit block using z i z j , where 26 (i;j) =f(1; 2); (1; 3); (4; 5); (4; 6); (7; 8); (7; 9)g for theith andjth qubit pair and cor- rected using the appropriate x i operator. Phase-flip errors are detected using the opera- tors x x x x x x I I I (1.48) x x x I I I x x x (1.49) and corrected using the appropriate z i operator. Bit-phase-flip errors can be corrected by performing both procedures independently. The advantage of the Shor code over classical error-correction schemes such as the 3-bit repetition code, wherej0 L i =j000i andj1 L i = j111i, is that it also corrects for quantum errors such as phase-flip and bit-phase-flip errors. Numerous QECCs have been developed since the emergence of the Shor code, including the Steane code [47] and further generalizations known as stabilizer codes [48]. QECC codes are usually denoted as [[n;k;d]] codes in which n physical qubits are used to encode k logical qubits. The number of correctable errors is given by the distance of the coded = 2t + 1 wheret corresponds to the number of errors that can be detected and corrected by the code. Whether the QECC is the [[9; 1; 3]] Shor code or a more sophisticated stabilizer code constructions, all QECCs are described by the Knill- LaFlamme theorem [49] which states that a recover (syndrome and correction) process R for a noise channelN () = P i E i E y i can be found for a codeC = spanfj L ig if and only if PE y i E i P = ij P; 8i;j (1.50) where P = P j L ih L j is a projector onto the codespaceC and ij are constants. Provided that such a condition can be satisfied, then perfect error correction can be 27 accomplished. Note that the concept of the noise channelN () is essentially a gener- alization of the open quantum system description of Eq. (1.16) which may also include errors generated by faulty physical gate operations in addition to decoherence generated by system-environment interactions. The Shor code encodes a single logical qubit and supplies perfect error correction for arbitrary independent single-qubit errors. Codes which improve upon the Shor code have also been discovered which set the limit at a minimum of five physical qubits required to protect one logical qubit against general single-qubit errors. More sophis- ticated stabilizer codes have also been developed which increase the efficiency of the first known codes by minimizing the number of required physical qubits and increas- ing the code distance, simultaneously. The most detrimental types of errors for QECCs are correlated errors which may result in uncorrectable logical qubit errors. In order to overcome correlated errors QECCs can be concatenated to increase the distance of the code at the cost of an exponential increase in physical qubit number. An advantage of QECCs over the DFS construction is that the method is chosen to fit the system rather than the converse. Unfortunately, in many cases where large distance codes are required for correlated noise models, the syndrome measurement operators (or stabilizers) may contain a high degree of non-locality which does not easily facilitate experimental stud- ies. Nevertheless, the use of QECC codes have been demonstrated in Nuclear Magnetic Resonance (NMR) systems for up to ten qubits [50], ion trap systems [51] and super- conducting qubits systems [52]. Furthermore, numerous studies have combined DFS theory [44] and Dynamical Decoupling [53, 54, 55] with QECCs to reduce the overhead required to address correlated errors. 28 1.4.3 Dynamical Decoupling An alternative approach to addressing decoherence is to suppress the effects of system- environment interactions before or during a quantum computation in an attempt to reduce decoherence and the probability of computational errors occurring. Dynam- ical decoupling (DD) is one such method, which seeks to attenuate the effects of decoherence by applying strong and expeditious control pulses solely to the system [56, 57, 58, 59, 60, 61]. Provided the pulses are applied over a time duration suffi- ciently shorter than the correlation time associated with the environment dynamics, DD effectively averages out undesirable interactions and preserves quantum states with a low probability of error, or fidelity loss. 4 The degree of effective averaging is deter- mined by the order of time-dependent perturbation theory (TDPT) to which the system- environment interactions are suppressed. Many sophisticated schemes have been devel- oped which offer arbitrary order suppression when certain limitations are imposed on the DD pulses. However, an unfortunate feature of these DD schemes is the exponential overhead required to mitigate correlated errors in multi-qubit systems. One important advantage of DD over QECCs is that it is an open-loop technique, i.e., does not require feedback or syndrome measurement. Furthermore, DD techniques are relatively inde- pendent of the environment operators which comprise the system-environment interac- tion Hamiltonian and can be adapted to fit the most general decoherence models, unlike DFSs. DD has also been widely experimentally studied in a number of systems, includ- ing ion traps [62, 63, 64], NMR [65, 66, 67], solid state quantum dots [68], and nitrogen vacancy (NV) centers in diamond [69, 70, 71]. For an in-depth study of DD and its mathematical formulation see Chapter 2. 4 The correlation time of the environment dynamics is determined by the spectral density of the envi- ronment. Referring back to Sec. 1.3.1, if the high-frequency cutoff is not sufficiently sharp and contribu- tions from the high-frequency regime are substantial, then DD effectiveness can be drastically reduced. 29 1.5 Motivation The destruction of superposition states and emergence of undesirable system dynamics during QC due to decoherence necessitates methods which implement quantum error avoidance, correction, and suppression in order to facilitate reliable QC. Error suppres- sion will be the primary focus of this work where a variety of topics are discussed, included a hybrid quantum error correction/suppression technique. Initially, a highly efficiency DD scheme is discussed and analyzed as a function of the number of pulses in Chapter 3. This particular scheme allows for nonuniform error suppression which presents an attractive approach to combating decoherence when, e.g., only dephasing or relaxation dominates, however both are necessary to effectively model the system. Employing numerical simulations, a comprehensive examination of the error suppression characteristics for this scheme are obtained specifically for a single- qubit system subjected to general decoherence. Although it is most certainly true that some combination of error avoidance, correc- tion, and suppression methods will ultimately be required to engineer the necessary symmetry for DFSs, reduce physically (or temporally) correlated errors in the case of QECCs, or limit the overhead of multi-qubit DD, an intrinsic robustness of each approach to additional errors such as faulty gate operations or DD pulses is necessary to achieve maximum efficiency. 5 In the setting of QEC correctable errors are deter- mined by the distance of the code and hence the efficacy of this approach is limited by the number of simultaneous errors which can occur within a complete step of QEC. When the number of simultaneous errors is larger than those allowed by the code a com- mon approach is to concatenate the code to limit the propagation of multiple physical qubit errors; thus, dramatically reducing the probability of logical errors and eventually 5 Faulty gate operations include: imperfect DFS logical operators which take states out of the DFS or faulty encoding, syndrome measurement, or decoding gate operations. 30 achieving a regime of fault-tolerant QEC if the physical error rate is below a speci- fied threshold value. In contrast to QEC, intrinsic robustness to faulty DD pulses is nearly non-existent. While certain fundamental DD sequences offer some reduction of such errors, the majority of sophisticated DD protocols are dramatically hindered by the propagation of faulty pulse errors. The question becomes: how does one achieve robustness and construct fault-tolerant DD schemes which continuously limit the prop- agation of faulty pulse errors as the number of pulses in the DD sequence increases? This is the question which is addressed in Chapter 4 using numerical optimization and a symmetrization technique. There it is shown that DD sequences can be constructed which suppress specific types of pulse errors to seemingly arbitrary orders in TDPT and therefore exhibit fault-tolerant-like behavior. As detailed above, the culmination of two or more decoherence protection tech- niques is most likely essential for realistic QC systems. Recent studies confirm the advantages of using hybrid techniques in circuit model QC where DFSs and DD [41, 43], as well as QECCs and DD [54, 55], are utilized to protect quantum gates. In light of these results, a problem of interest is whether such techniques can be extended to alter- native QC methods such as AQC. In Chapter 5, this problem is addressed by imple- menting a hybrid QECC+DD technique. There, known deterministic DD constructions which exhibit arbitrary order suppression for time-independent Hamiltonians are com- pared It is shown that DD effectively suppresses decoherence such that closed-system- like behavior is achievable in the AQC setting for certain DD protocols. 31 Chapter 2 Dynamical Decoupling The primary focus of this study is quantum error suppression via dynamical decoupling. Comprehension of this technique both from a mathematical and theoretical point of view are provided in this chapter to supplement the subsequent studies discussed in Chapters 4 and 5. Initially, the origins of this technique and its mathematical formalism are described along with various specifications that characterize DD schemes and their ability to suppress errors due to system-environment interactions and intrinsic pulse imperfections. Advancements in DD have taken a number of forms built upon numerical optimization or deterministic recursion. The most in-depth discussions will focus on the deterministic methods with a general summary of additional techniques as well. Finally, this chapter ends with an overview of canonical distance measures that can be utilized to quantify DD performance and compare schemes both analytically and numerically. 2.1 Origins of Dynamical Decoupling While currently a topic of research in quantum computation and information process- ing, the general notion of dynamical decoupling originated in NMR [36]. Referred to as refocusing, this method was originally proposed as a way for reducing inhomogeneous broadening in NMR spectroscopy experiments and, therefore, increasing the ability to characterize material compositions based on fourier resonance frequency analysis. The Hahn echo, or spin echo, experiments [72] performed by Erwin Hahn are the earliest works which influenced the construction of DD refocusing schemes in NMR. There the 32 system was initialized in a state in the x-y plane of the qubit Bloch sphere, allowed to evolve for a period of time where the state of the system precessed about the plane due to dephasing, and thereafter a pulse rotation of 180-degrees was applied to the system in the x-y plane (perhaps orthogonal to the prepared initial state). After a period of time equal to the initial waiting time between initialization and the 180-degree pulse rotation, the initial state of the system would re-emerge for a period of time. Thus, it was found that decohered states could be returned, or refocused, to desired system states. Beyond the Hahn echo sequence, additional, more sophisticated methods were proposed which further improved refocusing and NMR spectroscopy characterization including the Carr- Purcell-Meiboom-Gill (CPMG) [73] and XY family [74] sequences. One interesting feature of the latter is that the XY family sequences also addressed systematic errors found in experimentation which may reduce the improvement in refocusing by applying 180-rotations along two orthogonal axis of the qubit Bloch, rather than only one as in the Hahn echo and CPMG cases. Since the emergence of these techniques DD meth- ods have become increasingly more sophisticated and have drastically improved NMR spectroscopy, while also finding their way into quantum information as an approach for protecting quantum systems relevant for quantum computation from decoherence. 2.2 Mathematical Formalism Building a rigorous mathematical definition for DD error suppression requires knowl- edge of the interaction picture transformation and few mathematical techniques uti- lized to solve the time-dependent Shr¨ odinger equation. As will be shown below, mov- ing into the interaction picture with respect to the time-dependent control Hamilto- nian which implements the DD pulses is a useful approach to analyzing the effect of 33 DD on the system-environment interaction Hamiltonian. Due to the intrinsic time- dependent nature of the DD Hamiltonian, along with the manifested time-dependence of the rotated-frame transformation, time-dependent perturbation theory (TDPT) must be used to exhibit the overall effect of DD on the system dynamics. Approximating the evolution of the system can be completed using TDPT via the Dyson or Magnus expan- sions, each of which are discussed below in conjunction with a formal definition for the interaction picture and the desired result for DD evolution. 2.2.1 Interaction Picture The interaction picture, or rotating frame, transformation presents a convenient approach to studying open quantum systems. Decoherence processes are commonly studied within the interaction picture with respect to the pure system and environment dynamics thus leaving the interaction Hamiltonian as the primary concern for the dynamics as this Hamiltonian is responsible for decoherence. The interaction picture is also useful in the context of DD where the action of the DD scheme on the interaction Hamiltonian is the focus. While the study of decoherence and DD are examples where the interaction picture is utilized frequently, the formalism of the interaction picture can be generalized by considering the Hamiltonian H(t) =H C (t) +H 0 (t): (2.1) whereH C (t)2H andH 0 (t)2H are defined within the same Hilbert space. Without loss of generality, consider the dynamics ofH 0 (t) in the interaction picture with respect toH C (t). The rotated frame Hamiltonian ~ H 0 (t) is given by ~ H 0 (t) =U C (t)H 0 (t)U y C (t) (2.2) 34 such that U C (t) =T exp i R T 0 H C (t 0 )dt 0 and the rotated evolution operator ~ U 0 (t) satisfies the Schr¨ odinger equation i d dt ~ U 0 (t) = ~ H(t) ~ U 0 (t): (2.3) The relationship between ~ U 0 (t) and the evolution operator U(t), which satisfies the Schr¨ odinger equation under the action ofH(t), is captured by ~ U 0 (t) =U y C (t)U(t). 2.2.2 Dyson Expansion Time-dependent Hamiltonians may appear naturally when describing quantum systems or as the result of interaction picture transformations such as those described by Eq. (2.3) in Sec.2.2.1. Dynamics generated by such Hamiltonians are generally solved using perturbative power series expansions when closed-form solutions are not feasible. One approach to solving the time-dependent Schr¨ odinger equation i d dt U(t) =H(t)U(t) (2.4) in the ranget2 [0;T ] is to integrate Eq. (2.4) to obtain U(T ) =I S I B i Z T 0 H(t)U(t)dt: (2.5) Continuing the process iteratively the expansion of U(T) is given by U(T ) =I S I B + 1 X k=1 U k (T ) (2.6) 35 where U k (T ) = (i) k Z T 0 dt 1 Z t 1 0 dt 2 Z t k1 0 dt k TH(t 1 )H(t 2 )H(t k ); (2.7) witht 1 > t 2 > > t k , represents thekth order expansion term of the Dyson series. One important aspect of this approach is that the convergence of the Dyson series is guaranteed for a boundedH(t) on any finite time interval. 2.2.3 Magnus Expansion and Average Hamiltonian Theory An additional method for solving quantum dynamics generated by a time-dependent Hamiltonian is the Magnus Expansion. Beginning with the Schr¨ odinger equation given in Eq. (2.4) forU(t), the formal solution U(T ) =T exp i Z T 0 H(t)dt (2.8) fort2 [0;T ] can be shown to equivalent to the exponential expansion U(T ) = exp 1 X n=1 (n) (T ) ! ; (2.9) with the anti-Hermitian operator (n) (T ) representing thenth term in the Magnus oper- ator expansion after a total timeT . The leading terms of the expansion are (1) (T ) = i Z T 0 ~ H 0 (t 1 )dt 1 ; (2.10) (2) (T ) = 1 2 Z T 0 dt 1 Z t 1 0 dt 2 h ~ H 0 (t 1 ); ~ H 0 (t 2 ) i ; (2.11) 36 while thenth order Magnus term (n) (T ) = n1 X j=1 B j j! X k 1 ++k j =n1 k 1 1;:::;k j 1 Z T 0 ad k 1 (t) ad k 2 (t) ad k j (t) H(t)dt; n 2: (2.12) is constructed recursively as a sum of (n 1)-fold commutators. The coefficientsB j are the Bernoulli numbers and ad H = [ ;H]. A sufficient condition for convergence of the Magnus expansion is Z T 0 kH(t)kdt< (2.13) for a bounded operatorH(t). In comparison to the Dyson series expansion, the Magnus expansion is subjected to a more stringent convergence condition for bounded operators. However, unitarity is preserved by the Magnus expansion at any order of truncation, a property not upheld by the Dyson expansion. Furthermore, the Magnus expansion can be utilized in con- junction with Average Hamiltonian Theory (AHT) [36, 75] to construct an effective time-independent Hamiltonian H(T ) which generates dynamics that are formally iden- tical to the original time-dependent evolution generated byH(t) on a finite time interval. Hence, time evolution operatorU(T ) is formally equivalent to U(T ) =e iT H(T ) (2.14) such that the effective time-independent Hamiltonian H(T ) = i T 1 X n=1 (n) (T ) (2.15) 37 is constructed from the Magnus operators. AHT is most notably useful in DD where the effective Hamiltonian essentially details the structure of the leading-order system- bath interactions or pulse imperfection errors that remain as the dominant decoherence mechanisms at the termination of the DD evolution. Throughout this study both the Magnus operator expansion and AHT will be utilized to calculate the effective Hamil- tonian resulting from DD evolution in order to analyze the structure of the dominant error-producing terms and sequence performance. 2.2.4 Dynamical Decoupling Defined Consider an open quantum system described by the Hamiltonian H(t) =H 0 +H C (t): (2.16) The time-independent termH 0 governs the internal dynamics of the system and envi- ronment, while H C (t) is responsible for the time-dependent DD control fields. The HamiltonianH 0 is resolved further into H 0 H S +H B +H SB ; (2.17) where H S 2H S is the pure system Hamiltonian, H B 2H B is the pure environment Hamiltonian, andH SB 2H S H B represents the system-environment interaction. For brevity, let H err H S I B +H SB (2.18) denote the error Hamiltonian, where the pure system and system-environment interac- tion Hamiltonians constitute the sources of undesired system evolution and decoherence, respectively. Removal of undesired system evolution is particularly relevant when DD 38 is utilized for high fidelity quantum memory storage [56, 76, 77] or in a “decouple- then-compute” approach to quantum gate construction [53, 54, 78]. Storing quantum memory requires initial state preservation, hence the desired system evolution is trivial action on the system. Any form of system dynamics present after the DD evolution would alter the initial state, resulting in storage errors. In a similar manner, gate errors can be acquired during the application of a nontrivial quantum gate if undesired system dynamics remain upon the completion of the gate operation. Undesired system evolu- tion must also be removed in alternative DD-protected gate construction strategies, such as “decouple-while-compute” [53, 55, 78], in particular to prevent leakage errors when a DFS or stabilizer code is used to enable computation while DD pulses are applied [41, 43]. It is assumed that the control Hamiltonian H C (t) applies the DD pulses solely to the system. H C (t) consequently acts trivially on the pure environment Hamiltonian, [H C (t);H B ] = 08t, and nontrivially on H err . The manner in which H C (t) operates onH err ultimately determines the effectiveness of DD in suppressing the contributions of the error Hamiltonian to the system evolution. Demanding that each pulse operator anticommute with at least one term comprising H err , the system is driven in such a way thatH err can be effectively averaged out for sufficiently short time durations. How short can be elucidated in the interaction (“toggling”) picture with respect to H C (t) [36, 57, 78, 79], where ~ H 0 (t) = U y C (t)H 0 U C (t) (2.19a) = ~ H err (t) +H B (2.19b) 39 with the control unitary U C (t) =T exp i Z t 0 H C (t)dt : (2.20) The unitary time evolution operator ~ U 0 (t) satisfies the Schr¨ odinger equation i @ @t ~ U 0 (t) = ~ H 0 (t) ~ U 0 (t); ~ U 0 (0) =I S I B (2.21) and ~ U 0 (t) = U y C (t)U(t), where U(t) is the time evolution operator generated by Eq. (2.16). Employing TDPT via the Magnus expansion [see Sec. 2.2.3] to solve Eq. (2.21) the rotated time evolution operator can be written as ~ U 0 ( c ) = exp 1 X n=1 (n) ( c ) ! ; (2.22) where, in accordance with AHT, yields an effective Hamiltonian H 0 = i c 1 X n=1 (n) ( c ) = H B ( c ) + H err ( c ): (2.23) AHT applies here since the interest is the joint system-environment dynamics at the end of each stroboscopic DD period of c . The effective Hamiltonian H 0 is partitioned into an effective pure environment term and a sum of effective error Hamiltonians H err ( c ) i c 1 X n=1 (n) err ( c ); (2.24) where (n) err ( c ) is thenth order Magnus operator containing non-trivial system operators, while H B ( c ) contains only terms with trivial action on the system. 40 In light of Eq. (2.24) it is found that that DD facilitates an effective suppression of H err by suppressing H err ( c ) up to some order in the Magnus expansion.When the firstN terms of the expansion of H err ( c ) vanish one speaks of “Nth order decoupling.” Assuming Nth order decoupling has been achieved, the toggling frame evolution is given by ~ U 0 ( c ) = e ic[ H B (c)+ H err(c)] (2.25a) = e ic H B (c)+O[(k H 0 err kc) N+1 ] ; (2.25b) where the evolution is predominately dictated by the effective pure environment Hamil- tonian H B ( c ) whenk H 0 err k c and N 1. Here, H 0 err = i c P 1 n=N+1 (n) err ( c ) denotes the remaining effective error Hamiltonian andkAk is the sup-operator norm of A (largest singular value): kAk = sup j i kAj ik kj ik : (2.26) Thus the effectiveness of DD is dependent upon intrinsic properties, in particular the strength of the interaction and pure environment Hamiltonians. In situations where the internal dynamics are sufficiently fast, a short DD cycle time is desirable to maintain Eq. (2.13). Furthermore, it is also desirable to achieve high order error suppression, N 1, so that the effects of H 0 err are less consequential. Both conditions are made use of in order to examine DD in the presence of various strengths of internal dynamics in the analyses presented in Chapters 4 and 5. In particular, in Secs. 4.3 and 4.2 the effective error Hamiltonian (2.24) will display significant explanatory power. 41 2.3 The Control Hamiltonian: From Ideal to Realistic Pulses In general, the control Hamiltonian which implements the DD pulses is described by H C (t) = X V (t)Q (2.27) where V (t) is the control field associated with the Q 2H S degree of freedom of the system. All of the essential information regarding the DD sequence is contained withinV (t), i.e., pulse timings and amplitude profiles. Variations in pulse timings result in an alteration in the structure of the DD sequence, while varying amplitude profiles accounts for various types of pulses which describe idealized rotations or situations of pulse imperfections. The former will be discussed in Sec. 2.4 and the latter will be the focus of the subsequent subsections. Four different pulse amplitude profiles are considered which are customary to DD studies and realistic systems: zero-width and rectangular, finite-width pulses, both with the inclusion of rotation errors to model the existence of systematic errors brought about by faulty control fields. 2.3.1 Ideal Pulses The first type of pulse to consider is an idealized, zero-width control field V (t) = X j 0 (tt j ) (2.28) where the Dirac Delta function(t) constitutes the pulse profile. The pulses are applied at times t j and the angle of rotation is given by 0 . In the case the control fields are restricted so that t j 6= t j for all j, they are said to act uni-axially. For single-qubit 42 systems,S = =2 with =I;x;y;z, this condition can be visualized in terms of the qubit Bloch sphere as allowing pulses solely along one of the three axes. The uni-axial condition is impose throughout the remainder of this study, unless otherwise specified. 2.3.2 Finite-width Pulse Profiles Since zero-width pulses are experimentally impossible, it is customary to relax the ideal pulse assumption and consider pulses of finite duration as well. The finite duration pulse can be modeled by V (t) = X j A (tt j ) (tt p ) ; (2.29) representing a piecewise continuous control field with a rectangular profile [?]. The pulse amplitude is denoted by A and the pulse duration is p , so that A p = 0 . The Heaviside Theta function, (t), dictates the pulse profile where it is assumed that the time to turn the pulse “on” and “off” is negligible, therefore, the pulse is well approxi- mated by a square wave. 2.3.3 Flip-Angle Errors An additional form of systematic error to consider is that of an over- or under-rotation in the angle 0 , commonly referred to as a flip-angle error [36]. This particular type of error is relevant, e.g., in nuclear magnetic resonance (NMR), where inhomogeneity of the control field across the sample results in qubit rotation errors [66]. However, flip-angle errors are also prevalent in other systems such as donor electron spins in Si systems [80, 81]. 43 In the case of zero-width pulses, the control field takes the form V (t) = X j 0 (1)(tt j ) (2.30) where denotes the error in the rotation angle and the +() refers to an over-(under- )rotation. By modeling the control field in this manner, it is assumed that the pulses are applied along their respective axes with zero or negligible error. The inclusion of rotation-axis errors has been previously studied for some common deterministic DD schemes [82], but is not included in this present study. 2.3.4 Generic Faulty Pulse Profiles As a worst case scenario one can consider the combined effect of flip-angle errors for finite-width pulses. Assuming that the error in the pulse duration is negligible, a flip- angle error can be thought of as an error in the pulse amplitude. The combined error control field is modeled by V (t) = X j A(1) (tt j )(tt p ) : (2.31) Note that this particular form is one of the most prevalent pulse profiles encountered in experimental settings [67, 80, 82, 83, 84]. 2.4 Currently Available Techniques Dynamical decoupling has built upon many of the techniques originally discovered in NMR to create highly sophisticated decoupling methods which are able to produce high order decoupling. One specific case known as Concatenated DD (CDD) directly builds 44 upon sequence structures found within the XY-family sequences and permits arbitrary order decoupling, i.e. Nth order decoupling using a specified number of pulses, in a deterministic manner. More efficient sequences have recently become available which improve upon the CDD construction to increase the available decoupling order using far less pulses. Beyond the deterministic approach, DD schemes based on randomization and numerical optimization have also shown to be useful under certain conditions. In this section, the currently available deterministic DD schemes are discussed in addition to a brief summary of the randomized and numerical optimization techniques. In regards to the control Hamiltonian Eq. (2.27), the focus will be the selection of timing structure ofV (t) and the choice ofS rather than the profile itself. In fact, it will be assumed throughout this section that all pulse profiles correspond to ideal, instantaneous rotations as described by Eq. (2.28). The issue of non-ideal pulses is address in Chapter 4. 2.4.1 Universal Dynamical Decoupling One of the most fundamental DD sequences, CPMG, is essentially designed to combat one-dimensional system-environment interactions by applying two 180-degree rotations to the quantum system, each separated by a fixed time interval d . In order for the sequence to be effective it is essential for the pulse operatorsQ to anti-commute with the system operator comprisingH SB = P S B , i.e. fS ;Q g = 0 (2.32) must hold for some combination of and for all timet2 [0; c ], where c is the total time of one DD sequence cycle. Provided that this condition is satisfied, CPMG is found to achieve first order decoupling (N = 1) for one-dimensional system-environment Hamiltonians. 45 As a general example of CPMG implementation consider the dephasing model described in Sec. 1.3.1 with the inclusion of possible errors generated by the pure sys- tem evolution, i.e. H err = H S +H SB = z B z . The anti-commutation condition can be achieved by choosing the pulse operatorQ = x =2, which yields the control Hamiltonian H C (t) = 2 x [(t d ) +(t 2 d )] (2.33) in the case of ideal-pulses and a total cycle time c = 2 d . Considering the case where H S 0 for simplicity, the resulting rotated frame Hamiltonian is given by ~ H 0 (t) = 8 < : H B +H err : 0t< d H B + x H err x : d t< 2 d : (2.34) First-order decoupling can now be verified using the first-order Magnus term given in Sec. 2.2.3 to yield the effective evolution operator ~ U 0 ( c = 2 d ) =e 2i d H B +O(kH errk 2 2 ) : (2.35) It is important to note that a common alternative approach to calculating the decoupling order is to consider the full (non-rotated) evolution operator U CPMG ( d ) =XU f ( d )XU f ( d ) (2.36) whereX =i x denotes the ideal-pulse operator andU f (t) =e iH 0 t represents “free evolution” dynamics. Using the propertyAe B A 1 =e ABA 1 and the Baker-Campbell- Hausdorff (BCH) formula log e A e B =A +B + 1 2 [A;B] + 1 12 ([A; [A;B]] [B; [A;B]]) + (2.37) 46 the evolution operator can be written as U CPMG ( d ) =e i( x H 0 x ) d e iH 0 d =e i d ( x H 0 x +H 0 )+O( 2 ) =e 2i d H B +O( 2 d ) (2.38) where higher order system-environment interaction terms can be confirmed by comput- ing additional terms in the BCH expansion. When applied to particular models, such the spin-boson model, the effect of multiple cycles of CPMG is essentially capture by the modified decoherence function [56] P (N; d ) = Z 1 0 d! coth ! 2k B T 1 cos(!N d ) ! 2 I(!)j1f(!;N; d )j 2 (2.39) whereN idealX-pulses are applied successively and the additional modulation function f(!;N; d ) = 2 (!;t) (!; 2N d ) N X n=1 e 2i(n1)! d (2.40) is a result of the DD evolution. 1 Considering the limit whereN!1 and d ! 0, it can be shown that P (N; d )! 0. Thus, in the limit of continuous flipping decoherence is completely eliminated at any temperature and for any spectral densityI(!). Of course, such a limit is impossible to attain and ultimately only partial decoupling is achievable in practice. However, if d can be made small compared to the fastest characteristic time scale, i.e. c d . 1 for a spectral cutoff c , then one approaches the limit of continuous flipping. While one-dimensional error models can be used to describe a number of realizable systems, situations certainly exist where system-environment interactions are more gen- eral. For example, as discussed in Sec. 1.3.3, dephasing and relaxation processes may 1 The function(!;t) is a continuum version of k (t) given in Eq. (1.30). 47 appear equally dominant such that the system-environment Hamiltonian, with additional unwanted pure system terms, is characterized by H err = x B x + y B y + z B z : (2.41) Addressing such error models requires more sophisticated DD sequences beyond the CPMG sequence since the pulse operators must be chosen such that the condition Eq. (2.32) holds. Without loss of generality, consider Q 2f x =2; y =2g, where it can be verified that each pulse operator anti-commutes with at least one system operator in Eq. (2.41). First-order decoupling can now be achieved by employing XY 4 from the XY-family sequences. Constructed in a similar manner to CPMG, the control Hamilto- nian is given by H C (t) = 2 f x [(t d ) +(t 3 d )] + y [(t 2 d ) +(t 4 d )]g (2.42) where the DD pulses are applied in an alternating fashion separated by time intervals of duration. The resulting rotated-frame Hamiltonian ~ H 0 (t) = 8 > > > > > > > < > > > > > > > : H B +H err : 0t< d H B + x H err x : d t< 2 d H B + z H err z : 2 d t< 3 d H B + y H err y : 3 d t< 4 d : (2.43) and the first-order term in the Magnus expansion can be used to confirm the first-order decoupling. Again, note that the full evolution operator U XY 4 ( d ) =YU f ( d )XU f ( d )YU f ( d )XU f ( d ); (2.44) 48 where X = i x and Y = i y , in conjunction with the BCH formula gives an equivalent result. The fact that XY 4 is able to achieve first order decoupling for general environment operators has led to its name as the universal decoupling sequence [85]. The XY 4 sequence results in first-order decoupling for a single-qubit system sub- jected to general interactions with its environment, however, the structure of the sequence does not appear to have any intuitive design at first glance. Interestingly, it turns out that the action of XY 4 can be described from the standpoint of group theory as a symmetrization ofH SB over the Pauli groupP, given by P =fI; x ; y ; z g (2.45) up to overall phases ofi [57, 58]. The symmetrization is evident from the zero-order contribution in the effective Hamiltonian 2 where H( d ) = X 2fI;x;y;zg H 0 +O( d ) (2.46) using Eqs. (2.9) and (2.15). The deep mathematical meaning behind the action of XY 4 allows one to quickly generalize the sequence beyond single-qubit systems as follows: given a unitary decoupling groupG =fg i g G i=1 such that every element in the group except the identity operator anti-commutes with at least one system operator in H SB , then symmetrization ofH SB overG yields H( d ) = G X i=1 g i H 0 g y i +O( d ) (2.47) 2 Zero-order contribution of H() corresponds to first-order contribution of TDPT. 49 and first-order decoupling using the DD sequence described by the the full evolution operator U G ( d ) = G Y i=1 g i U f ( d )g y i : (2.48) 2.4.2 Concatenated Dynamical Decoupling Extending beyond the first-order decoupling of XY 4 can be accomplished in a vari- ety of ways from a deterministic standpoint. One such approach builds directly upon XY 4 by again employing a unitary decoupling groupG =fg k g G k=1 which implements first-order decoupling by symmetrizingH SB overG. By recursively concatenating this symmetrization, Concatenated DD (CDD) achieves higher order decoupling such that each additional concatenation level averages out the remaining leading order Magnus term. Defined rigorously as U (`+1) CDD ( d ) = G Y k=1 g k U (`) CDD ( d )g y k ; (2.49) whereU (0) CDD ( d ) = U f ( d ), CDD attains`th-order decoupling usingG ` pulses. Hence, an exponential overhead in the number of pulses is required to achieve arbitrary order decoupling. Note that whenG =P, the single-qubit Pauli group, U (`+1) CDD ( d ) =YU (`) CDD ( d )XU (`) CDD ( d )YU (`) CDD ( d )XU (`) CDD ( d ) (2.50) and 4 ` pulses are required for`th-order decoupling. 2.4.3 Uhrig Dynamical Decoupling A large body of work now exists concerning DD sequences with non-uniform pulse- intervals, a technique which enables a drastic improvement over the exponential scaling 50 of CDD. Uhrig DD (UDD) is one such method, which applies DD pulses separated by unequal time intervals, at instances determined by a closed form expression originally developed in the context of the spin-boson model [77]. Using M control pulses, the UDD sequence U (M) UDD; U f () = M+1 M+1 Y k=1 U 0 ( (M) k ); (2.51) with minimum pulse delay and normalized pulse-intervals (M) k = t (M) k t (M) k1 t (M) 1 t (M) 0 (2.52) and t (M) k = c sin 2 k 2M + 2 ; j = 1; 2;:::;M + 1; (2.53) suppresses the firstM orders of the time-dependent perturbation theory expansion for all system-environment interactions terms which anti-commute with provided the bath spectral density contains a sharp high-frequency cutoff [77, 86], or for generic bounded bath operators [87]. For an analysis of the scaling properties and corresponding distance measure performance of UDD see Ref. [88]. UDD can be extended to combat general single-qubit decoherence by nesting two anti-commuting UDD sequences. The resulting quadratic DD (QDD) [89] scheme U (M 1 ;M 2 ) QDD;f 1 ; 2 g () =U (M 2 ) UDD; 2 U (M 1 ) UDD; 1 U f () (2.54) achieves min(M 1 ;M 2 )th order decoupling usingM 1 M 2 pulses in the ideal pulse limit. One interesting feature of QDD is that if 1 =X and 2 =Z, whereX andZ describe ideal pulse operators, XY 4 is recovered for M 2 = M 1 = 1. An indication that XY 4 is indeed of fundamental importance and perhaps optimal for first-order, single-qubit decoupling. Additionally, QDD is the first sequence construction which has the ability 51 to address non-uniform error models where perhaps suppression of both relaxation and dephasing is necessary, however, one is more dominant than the other. By varying the number of pulses in either nested UDD sequence, system-environment interactions can be selectively suppressed to various orders in the time-dependent perturbation theory expansion. In Chapter 3, this topic will be elaborated upon using the general single- qubit error model Eq. (2.41) to determine the decoupling order for each term in the system-environment Hamiltonian as a function of M 1 and M 2 . It will be shown that selective suppression is not trivially defined by the UDD decoupling efficiency alone and that the suppression of system-environment interactions can be altered by a nested UDD sequence comprised of pulse operators which commute with the interaction term. Further generalizations of UDD include concatenated UDD (CUDD) [90] and nested UDD (NUDD) [91]. CUDD alleviates the restriction of single-channel decoherence sup- pression, however, still suffers from the exponential scaling of CDD. NUDD essentially follows the nesting scheme of QDD, while providing a generalization to multi-qubit and multi-level systems: U (M 1 ;:::;M k ) NUDD;f 1 ;:::; k g () =U (M k ) UDD; k U (M 1 ) UDD; 1 U f () (2.55) where min(fM i g k i=1 )th order decoupling is achieved using Q k i=1 M i pulses in the ideal pulse limit for a system-environment Hamiltonian which requiresk pulse operators to satisfy the anti-commutation relation Eq. (2.32). The pulse operators themselves are required to be unitary and hermitian such that each pair of elements inf i g k i=1 either commutes or anti-commutes. As in the case of UDD and QDD, NUDD has been proven to apply to general decoherence models with bounded bath operators [92, 93]. For an analysis of the scaling properties and corresponding distance measure performance of QDD and NUDD see Ref. [94]. 52 2.4.4 A Brief Summary of Additional Methods: Randomized DD to Numerical Optimization A large number of DD studies move beyond the deterministic approach to constructing DD schemes and focus on non-deterministic methods and numerical optimization. Ran- domized DD [95] is one such non-deterministic method which randomly samples from a decoupling groupG at successive intervals of duration . Inspired by the universal decoupling sequence, randomized DD has been shown to outperform the deterministic group symmetrization approach when a large number of control operations are required or unwanted system-environment interactions vary over time scales longer than but shorter than the time required to perform exact symmetrization overG [96, 97]. Hybrid randomized-deterministic methods have also been suggested to simultaneously combat time-dependent interactions, which are more effectively addressed by randomized proto- cols, and time-independent interactions that can be dealt with by deterministic methods [98, 99]. Numerical approaches to DD build upon the results of Ref. [56] [refer to Eq. (2.39)] by considering spectral densities which are characterized by “soft” high frequency cut- offs and therefore may not contain a well-defined regime of continuous flipping. The objective is then to specify a spectral density and then numerically determine the optimal modulation function, i.e. DD pulse timings, for particular systems. Numerous methods have been suggested such as locally optimized DD (LODD) [100], bandwidth adapted DD (BADD) [101], optimized noise filtration DD (OFDD) [102], and Walsh function DD (WDD) [103]. 53 2.5 Performance Measures and Scaling Distance measures, which determine the closeness between quantum states and oper- ators, permit one to effectively quantify DD performance by the distance between the DD-evolved and target system state or DD evolution operator and desired system evolu- tion operator. From an analytical point of view, distance measures provide a method for determining bounds on DD performance as a function of internal Hamiltonian parame- ters or quantities that characterize pulse imperfections. Examining a distance measure as a function of the total cycle time c and other possible relevant quantities allows one to numerically determine the decoupling order of the sequence as well as the structure of the dominant terms in the effective system-environment Hamiltonian. In addition to presenting various DD performance measures below, general scaling for upper bounds for each measure are given in terms of =kH B k; J =kH err k; J =kB k; (2.56) wherekAk is the operator norm given in Eq. 2.26. The first quantity denotes the “strength” of the pure environment dynamics, while the remaining terms represent the “strengths” of the error Hamiltonian and individual error terms, respectively, in consid- eration of the general single-qubit decoherence model for generic bounded environment operators B . These quantities are utilized through Chapters 3 and 4 to analyze the performance of a DD scheme numerically in various parameter regimes to determine decoupling order and parameter ranges where a particular scheme is most effective. 54 2.5.1 Fidelity In classical physics one can define a measure of closeness between two classical proba- bility distributions F (fp i g n i=1 ;fq i g n i=1 ) = n X i=1 p p i q i (2.57) known as the fidelity. The density operator representation for quantum states as ensem- bles of states associated with classical probabilities naturally leads one to a quantum analog of the classical definition which determines the distance between quantum states. The quantum analog of the Fidelity measure which determines the distance between two quantum states 1 and 2 is F ( 1 ; 2 ) =k p 1 p 2 k Tr : (2.58) where kAk Tr = Tr[ p A y A] = X i j i j (2.59) represents the trace-norm ofA or, equivalently, the sum of the absolute-valued singu- lar values i for the operator A. The Fidelity is bounded between zero and one such that a maximum overlap between 1 and 2 occurs for 1 = 2 , where F ( 1 ; 2 ) = F ( 1 ; 1 ) = 1. In the case that one state is a pure state, e.g., 1 =j ih j, the Fidelity is given by F ( 1 ; 2 ) =k p j ih j p 2 k Tr = p h j 2 j i: (2.60) This situation is relevant for DD since one can consider the case where the system is initially in a pure state 1 and the resulting state after the DD evolution operator U DD (T ) is described by 2 = Tr B [U DD (T ) 1 U y DD (T )]. More fundamental cases where both 1 =j ih j and 2 =jihj are pure state leads to F ( 1 ; 2 ) =jh jij. A 55 scaling analysis for the fidelity measure is not shown here, yet indirectly determined in Sec. 2.5.2. 2.5.2 Trace-Norm Distance An additional useful distance measure which originates from classical computing theory is the trace-norm distance. Its quantum analog quantifies the distance between quantum states 1 and 2 by D Tr ( 1 ; 2 ) = 1 2 k 1 2 k Tr (2.61) where maximum overlap between states, 1 = 2 , results in D Tr = 0 and unity rep- resents an upper bound on D Tr . The trace-norm distance is intimately related to the Fidelity via 1D Tr ( 1 ; 2 )F ( 1 ; 2 ) q 1D 2 Tr ( 1 ; 2 ) (2.62) and therefore can be used to describe maximum or minimum Fidelity using the upper or lower bound expressions, respectively. In Chapter 5, the trace-norm distance is utilized to compute the distance between a desired system state and the resulting system state after DD evolution. In this way, the performance of a DD scheme may be effectively determined. Trace-Norm Distance Scaling Consider the DD-evolved system state S ( c ) = Tr B [ ~ U 0 ( c )(0) ~ U y 0 ( c )], where ~ U 0 ( c ) =T exp i Z c 0 ~ H 0 (t)dt (2.63) is the evolution generated by internal Hamiltonian ~ H 0 (t) in the rotating frame with respect to the DD control Hamiltonian H C (t) [see Sec. 2.2.4] describing ideal pulse 56 rotations and the initial state is (0) = S (0) B (0). Considering the trace-norm distance between S ( c ) and the initial system state S (0) =j ih j, one determines 2D Tr [ S ( c ); S (0)] = k S ( c ) S (0)k Tr = kTr B f ~ U 0 ( c )[ S (0) B (0)] ~ U y 0 ( c )g S (0)k Tr (1) = kTr B f ~ U 0 ( c ) [ S (0) Tr R (jihj)] ~ U y 0 ( c )g S (0) Tr B+R (jihj)k Tr (2) = kTr B+R n ~ U 0 ( c ) [ S (0) jihj] ~ U y 0 ( c ) S (0) jihj o k Tr (3) ~ U 0 ( c ) [ S (0) jihj] ~ U y 0 ( c ) S (0) jihj Tr (4) h ~ U 0 ( c )I S I B i j i ji (5) ~ U 0 ( c )I S I B (2.64) as a upper bound for between the DD-evolved and target system states. In order to obtain this bound, the following assumption and properties are made: (1) the environment initial state is assumed to be a general, possibly highly mixed, quantum state that is obtained from a larger Hilbert spaceH B H R by partially tracing the pure stateji2 H B H R overR, i.e. B (0) = Tr R (jihj), (2) the trace operator is linear, (3) the trace- norm of the partial trace satisfieskTr B+R (A)k Tr kAk Tr forA2H S H B H R , (4) the trace-norm of matrices is less than the Euclidean norm of vectors and (5) sub- multiplicativity. Expressing the unitary evolution operator ~ U 0 ( c ) as a time-dependent perturbation expansion ~ U 0 ( c ) =I S I B + 1 X n=1 ~ U (n) 0 ( c ); (2.65) 57 with Dyson operators ~ U (n) 0 ( c ) = (i) n Z c 0 dt 1 Z t n1 0 dt n n Y j=1 ~ H 0 (t j ); (2.66) one obtains, using the triangle inequality, D Tr 1 2 k 1 X n=1 ~ U (n) 0 ( c )k 1 2 1 X n=1 k ~ U (n) 0 ( c )k: (2.67) The final step is to show determine the scaling ofkU n ( c )k and thus the scaling of the trace-norm distance. This is accomplished by using (1) the triangle inequality, (2) sub- multiplicativity, and (3) unitary invariance to obtain an upper-bound on Eq. (2.66) as follows: k ~ U (n) 0 ( c )k = k(i) n Z c 0 dt 1 Z t n1 0 dt n n Y j=1 ~ H 0 (t j )k (1) Z c 0 dt 1 Z t n1 0 dt n k n Y j=1 ~ H 0 (t j )k (2) Z c 0 dt 1 Z t n1 0 dt n n Y j=1 k ~ H 0 (t j )k (3) = Z c 0 dt 1 Z t n1 0 dt n n Y j=1 kH 0 k = n c n! kH 0 k n (1) n c n! (J +) n : (2.68) As a result, Eq. (2.67) achieves the upper-bound D Tr 1 2 1 X n=1 n c n! (J +) n = 1 2 e c(J+) 1 : (2.69) 58 Now consider the case of Nth order error suppression, where all ~ U (n) 0 ( c ) for n N vanish. The bound then truncates to D Tr 1 2 1 X n=N+1 n c n! (J +) n ; (2.70) which scalings accordingly as D Tr .O N+1 c (J +) N+1 ; (2.71) when J c 1 and c 1 are satisfied. The conditions on the strengths of the interaction and pure bath terms relative to the DD cycle time are essentially equivalent to k H SB k c 1 from Sec. 2.2.4, yet perhaps more intuitive as they express the condition for DD effectiveness in terms of two independent quantities that can be used to describe a variety of open quantum systems. Note that Eq. (2.71) indirectly determines a scaling for the fidelity measure when combined with Eq. (2.62). An additional point to stress is that Eq. (2.71) is specific to ideal pulses and would contain quantities which characterize pulse imperfections if such considerations were included. 2.5.3 Kosut Distance The distance measures discussed thus far determine the distance between two quantum states. Since the dynamics of quantum states are characterized by the unitary time evo- lution operatorU(t), one can focus instead on determining the distance between unitary operators where one may represent a target evolution and the other corresponding to the 59 “actual” evolution operator produced by some Hamiltonian which describes the system and environment. The Kosut distance [104] D Kos (U;G) = 1 p d S d B min kUG k (2.72) is one way of computing such a distance specifically for unitary operators of differing dimensions. In studying open quantum systems, this distance measure can be quite useful as the operatorU may represent the evolution produced by the complete internal system-environment Hamiltonians,G describes the desired evolution of the system, and d S and d B are the dimensions of the system and the environment Hilbert spacesH S andH B , respectively. In Sec. 4.2, this distance measure is used in the context of DD to quantify the distance between the DD time evolution operator and a desired trivial system evolutionG =I S . There, the state-independence of the Kosut distance becomes very convenient as it reduces the overhead of averaging over various initial system and environment states for a numerical optimization study of DD. The norm associated with Eq. (2.72) can be chosen in may different ways. In the case that the Frobenius norm kAk F = p Tr[A y A] = s X i 2 i (2.73) is chosen, where i are the singular values of the operatorA, the minimization can be performed to obtain the closed form expression D Kos (U;G) = r 1 1 d S d B kk Tr (2.74) with = Tr S [U(G y I B )]. It can be easily shown from this expression that if U = G I B , = d S I B and the Kosut distance yields a minimum value ofD Kos = 0. 60 Alternatively, a maximum value ofD Kos = 1 is reached by noting thatkk Tr 0 and thus 0 kk Tr d S d B . Sincekk Tr = d S d B corresponds to minimum distance between the unitaries U and G,kk Tr = 0 can be expected to result in maximum dis- tance. Takingkk Tr = 0 quickly conveys a maximum Kosut distance of unity. Kosut Distance Scaling Similar to the trace-norm distance, an upper-bound scaling can also be obtained for the Kosut distance in the ideal pulse limit. In fact, the scaling of this distance measure is equivalent to Eq. (2.71) and given by D Kos .O N+1 c (J +) N+1 : (2.75) Proving this expression for the Kosut distance first requires showing that the general distance measureD Kos (U;G) [see Eq. (2.72)] satisfies D Kos (U;G) 1 p 2 kUG I B k: (2.76) The upper-bound is obtained by utilizing the steps originally taken in Ref. [104] to obtain the closed form expression ofD Kos (U;G) given in Eq. (2.74), where initially it is shown that, for satisfying y =I B , D Kos (U;G) = 1 p 2d S d B min kUG k F = min r 1 1 d S d B RefTr [U(G y y )]g = min r 1 1 d S d B RefTr B [Tr S [U(G y I B )] y ]g = min r 1 1 d S d B Re[Tr( y )]: (2.77) 61 It is then noticed that computing the minimization problem of Eq. (2.72) is equivalent to finding the maximum value of Re[Tr( y )] over all unitary . In order to complete the proof, the singular value decomposition (SVD) =W V y is invoked, whereW;V are unitary and = diag(s 1 ;:::;s d B ) is a real diagonal matrix containing the singular valuess 1 s 2 s d B 0. The relevant expression becomes Tr( y ) = Tr(W V y y ) = Tr[(V y y W )] (2.78) and the final details of the proof involve showing that RefTr[(V y y W )]g is essentially maximized if and only ifV y y W =I B , or equivalently when =WV y . Note that RefTr[(V y y W )]g Tr() (2.79) holds for all that differ fromWV y . It is now straightforward to obtain the bound expressed in Eq. (2.76) since by choos- ing = I B one is satisfying the lower bound expression of Eq. (2.79) and generating the upper-bound D Kos (U;G) 1 p 2d S d B kUG I B k F (2.80) 1 p 2 kUG I B k: (2.81) Note that the inequalitykTk F p d S d B kTk has been used in order to transform from the Frobenius norm [Eq. (2.80)] to the sup-operator norm [Eq. (2.81)]. 62 Choosing G = I S , as specified by the desired action of the DD evolution on the system, and the toggling frame evolution operator U = ~ U 0 ( c ), the distance measure upper-bound becomes D Kos 1 p 2 k ~ U 0 ( c )I S I B k: (2.82) Noting that this expression is nearly the same as Eq. (2.64), up to an overall constant, a similar process is followed to obtain the upper-bound of Eq. (2.75) in terms of the strength of the pure environment dynamics and interaction Hamiltonian. 2.5.4 Uni-axial Error Measure The scaling of the performance measures discussed so far is essentially dictated by the leading order error term (lowest order error suppression) resulting from the DD evolu- tion. Although one may argue that the lowest decoupling order provides ample infor- mation for characterizing the error suppression properties of a sequence, the fact is that it is difficult to draw any conclusions concerning the achievable decoupling order for individual error terms in relevant situations; most notably in DD constructions designed to combat nonuniformly-distributed decoherence, e.g., QDD and NUDD. The method discussed here allows one to overcome this constraint and extract the decoupling order for any error term remaining at the end of the DD evolution. In order to describe this technique consider the general decoherence model of Eq. (1.22) and the DD rotated-frame evolution operator at the termination of the DD sequence expressed in terms of the system operator basis: ~ U 0 ( c ) =I S I B + X S B 0 ( c ): (2.83) 63 The effective environment operatorsB 0 ( c ) are calculated analytically using TDPT via the Dyson or Magnus expansion and, in general, not equivalent to B . An important feature of these operators is that they encompass all of the remaining error terms associ- ated withS and therefore quantifying the decoupling order for eachS error requires a projection onto that particular system degree of freedom. The uni-axial error measure E ( c ) =kB 0 ( c )k F (2.84) where B 0 ( c ) = Tr S [ ~ U 0 ( c )S ]; (2.85) performs this task and expresses the magnitude of the error in terms of the Frobenius norm. This approach to DD error suppression analysis is most convenient as a numerical approach to extracting the decoupling order for a particular error term as it only requires examination of the uni-axial error measure as a function of c rather than direct calcula- tion of the effective Hamiltonian. The uni-axial error is specifically applied in Chapter 3 in the analysis of the nonuniform decoherence suppression properties of QDD. Uni-axial Error Scaling The uni-axial error measure E ( c ) is dominated by the lowest nonvanishing order of c . Hence,E O( N +1 c ) provided that the firstN terms of the effective Hamiltonian associated withS vanish. In terms of the effective environment operators described in Eq (2.83), this statement corresponds to B 0 ( c ) = 1 X j=N B 0(j) j c ; (2.86) 64 such that B 0(j) = X ~ j r 1 2 ::: j B 1 B 2 B j (2.87) with~ =f 1 ; 2 ;:::; j g such that j indexes all possible values of for the operator basisfS g and = Q j j follows the product rules associated with that particular operator basis. If considering a single-qubit system S 2 fI; x ; y ; z g and k j 2 fI;x;y;zg with following the Pauli product rulesxy = z,zx = y,yz = x. In this manner a convenient notation is created to identify the environment operators present in each uni-axial error. For example, for a single-qubit system withE z O( 2 c ) then the possible summands comprising B 0(2) z will be proportional to B I B z , B z B I , B x B y , and B y B x . The constituent operators,B j , are the environment operators initially defined in Eq. (1.22). The only terms of interest are those proportional to N c , since these are the dominant terms ofB 0 ( c ). Using Eqs. (2.84), (2.86), (2.87), and the submultiplicativity property of unitarily invariant norms [105], the uni-axial error is E () kB 0(N ) N c k F N c X ~ N jr 1 ::: N jkB 1 k F kB N k F = N c X ~ N ~ r 1 ::: N J 1 J N ; (2.88) where only the leading order contribution in c is retained. The coupling strength param- eters defined byfJ;g have been incorporated into the sum along withJ =kB k, such thatJ I = andJ J86=I. The parameters ~ r 1 ::: N account for a conversion fac- tor between the Frobenius and sup-operator norms. Factoring outJ N out from the sum, the desired functional form of the uni-axial error is log(E )N log(J c ) + log ( ); (2.89) 65 with ` = X ~ N ~ r 1 ::: N 1 ` 2 ` N ` (2.90) and with j ` =J j ;` =J 1: (2.91) Eq. (2.89) is expressed as a function of the dimensionless parameterJ c , as opposed to c , in anticipation of a uni-axial error analysis in the J regime whereJ sets the relevant environment time scale. Hence, J c . 1 should be a necessary condition for DD to be beneficial over uncontrolled free evolution; a restatement of the condition for DD effectiveness for this measure. The quantity ` does not depend on c and hence will play the role of a constant offset. 66 Chapter 3 Analysis of Near-Optimal Quadratic DD Approach The Quadratic DD approach, which is based on the pulse-interval-optimized Uhrig DD scheme, possesses certain features which perhaps make it more attractive than other deterministic methods when combating nonuniform decoherence. Already offering improved decoupling efficiency over CDD, QDD also permits selective error suppres- sion which can be used to suppress general single-qubit errors in a nonuniform and highly efficient manner when processes like dephasing and longitudinal relaxation are both present yet not uniformly contributing. While much of the work on QDD has focused on the error suppression capabilities of QDD and it is known that this method can be utilized for nonuniform error suppression, the relationship between the number of pulses in each nested UDD sequence and decoupling order for each , = x;y;z, error type remains unclear. In this chapter, the error suppression properties for QDD are characterized as a function of the inner and outer sequence ordersN 1 andN 2 , respec- tively. The system is described by the general single-qubit decoherence model, H 0 =I B I + X =x;y;z B (3.1) where the single-qubit system is interacting with its environment via generic environ- ment operatorsB . Extraction and subsequent analysis of the individual error types is accomplished via the uni-axial error measure described in Sec. 2.5.4. The decoupling 67 order is extracted directly from the scaling of the uni-axial error measure as a func- tion of either the total cycle time or the minimum pulse interval, thereafter allowing for possible constraints on QDD effectiveness to be accurately identified. The uni-axial error analysis is complemented by an examination of the intermediate uni-axial errors, which are taken at intermediate points throughout the sequence rather than solely at its commencement. QDD intermediate uni-axial errors provide additional insight into the interim dynamics of the sequence and prove to be very useful for explaining the results presented in Chapter 5 where QDD is employed to protect adiabatic evolution. Finally, the overall performance of QDD will be compared to the uni-axial error results using the Kosut distance given by Eq. (2.72). The primary objective of this study is to obtain the ability to determine which combination of sequence orders provides the largest order of error suppression for realistic QIP systems that are intrinsically subjected to nonuniform decoherence, such as solid state NMR systems [65, 66, 84]. 3.1 QDD Evolution Analysis In order to provide some motivation for employing the uni-axial error in the case of nonuniform QDD, consider the QDD evolution operator U (M 2 ;M 1 ) QDD;fZ;Xg () =U (M 2 ) UDD;X U (M 1 ) UDD;Z U 0 (); (3.2) which contains all the information regarding decoherence suppression for each uni-axial error, and utilizes the ideal instantaneous DD pulse operators X(Z) =i x(z) (3.3) 68 to satisfying the anti-commutation condition of DD. The sequence comprised of Z pulses will be referred to as the inner sequence, while the UDD sequence ofX pulses denotes the outer sequence. Construction of the final evolution operator is accomplished by first considering the inner sequence evolution, U (M) UDD;Z U 0 (), which for conve- nience will be referred to as U (M) UDD;Z () below. Let Eq. (3.1) be partitioned such that H 0 =H + +H , where H + = x B x + y B y (3.4) and H =I B I + z B z : (3.5) Clearly, the pulse operators comprisingU (M 1 ) UDD;Z () anti-commute withH + ,fH + ;Zg = 0, and commute withH , [H ;Z] = 0. Hence, theZ-type UDD sequenceU (M 1 ) UDD;Z () is effective against the anti-commuting Hamiltonian, H + and completely ineffective against unwanted interactions within H . Any additional errors associated with H must be addressed by the additional nested UDD sequence. The inner sequence evolu- tion can be expanded in terms ofH [87], U (M 1 ) UDD;Z () =e i[(1) M 1 H + +H ] (M 1 ) M 1 +1 e i[H + +H ] (M 1 ) 2 e i[H + +H ] (M 1 ) 1 ; (3.6) where the anti-commuting and commuting properties of H , respectively, have been used. Transforming into the interaction picture with respect to H , U (M 1 ) UDD;Z () = U (M 1 ) ()U (M 1 ) z (), such thatU (M 1 ) () =e iH S (M 1 ) , withS (M) = P M+1 j=1 (M) j , and U (M 1 ) z () = ^ T exp i Z S (M 1 ) 0 f z (t)H + (t)dt ! : (3.7) 69 The modulation function f z (t) = (1) j1 is defined for t 2 [ P j1 `=1 (M 1 ) ` ; P j `=1 (M 1 ) ` ]. H + in the rotating frame with respect to H takes on the form of a power series expansion int, H + (t) =U (M 1 )y (t)H + U (M 1 ) (t) = 1 X k=0 H (k) + t k : (3.8) The power series form of H + (t) is useful (though not essential [91]) for the proof of UDD and therefore the suppression of error associated withH + [87, 88, 106, 107]. All constants of the expansion are condensed withinH (k) + , along with thek-fold commutator [ k H ;H + ] = [H ; [H ; [H ;H + ] ]]: (3.9) Using TDPT,U (M 1 ) z () is expanded in the Dyson series U (M 1 ) z () = 1 X p=0 X kp H (kp) + H (k 1 ) + F (M 1 ;kp) z (); (3.10) wherek p =fk 1 ;:::;k p g withk i = 0; 1;::: for alli, and all of the time-dependence of the expansion has been placed in F (M 1 ;kp) z () = (i) n Z S (M 1 ) 0 Z t p1 0 Z t 2 0 p Y j=1 dt j f z (t j )t k j j (3.11) The proof of UDD is completed by parametrizing t j as t j = sin 2 ( j =2), Fourier expanding f z (t j ), and showing that F (M 1 ;kp) z () = 0 for all odd values of p when p + P p j=1 k j M 1 [87]. All even orders of the expansion are proportional to unity or z , and are therefore not associated withH + . The expansion ultimately yields U (M 1 ) UDD;Z () =e iH 0 ()S (M 1 ) +O[(kH 0 + ()k) M 1 +1 ] ; (3.12) 70 whereH 0 + ( d ) is a generic single-qubit system-bath Hamiltonian and H 0 () =B 0 I () + z B 0 z () (3.13) is composed of environment operators dependent on the minimum pulse interval. Note that these environment operators are not the same operators defined in Eq. (3.1), but combinations of the original Hamiltonian operators resulting from the perturbation expansion. Left with only dephasing errors, terms proportional to z , the process is continued again by defining ~ H + = z B 0 z ( d ) and ~ H =I B 0 I ( d ) for the outerX-type UDD sequence of Eq. (5.11). ~ H is defined in this way such that [ ~ H ;X] = 0, a similar condition to that required for the inner Z-type sequence. The resulting evolution of Eq. (5.11) can be summed up as U (M 1 ;M 2 ) QDD () =I B 00 I () + X 2fx;y;zg B 00 (): (3.14) Once again, the time-dependent environment operators, B 00 (), differ from the previ- ously defined environment operators. EachB 00 (),2fx;y;zg, contains the uncom- pensated decoherence along each of the qubit Bloch sphere axes. Bounds on the order of error suppression are derived analytically in Ref. [108]. Unlike the UDD evolution operatorU (M 1 ) UDD;Z ( d ), the decoupling order for each error type is not known for generalfM 1 ;M 2 g once the outer UDD sequence has been applied to form the QDD sequence. The uni-axial error provides a method for obtaining such information numerically by analyzing the scaling of the error measure as a function of a relevant time parameter, here the minimum pulse interval. As previously discussed in Sec. 2.5.4, it is not possible to use a distance measure to extract such a scaling since any viable distance measure only scales with the minimum decoupling order. Thus, the 71 uni-axial error measureE ( d ) yields additional information that can otherwise only be obtained by computing the effective error Hamiltonian. 3.2 Numerical Results In this section a numerical analysis of QDD is presented based on the uni-axial errors and Kosut distance measure D K (U;G). The environment is specified as a four-qubit environment where the environment operators are given by B = X i6=j X ; c i j ; (3.15) wherei;j index the bath qubits,;;2fI;x;y;zg, andc ; 2 [0; 1] are random coef- ficients chosen from a uniform probability distribution. The construction ofB permits at most two-body interactions between environment qubits and three-body interactions between the system and the environment. While the number of environment qubits are limited to only four, increasing this value further does not indicate an alteration in any of the following results. Note that Eq. (3.15) includes terms proportional to I i I j , which account for the pure system Hamiltonian that also constitutes error-generating dynamics. 3.2.1 Uni-axial error analysis Figures 3.1 and 3.2 display the uni-axial errors as a function of J for M 2 = 3 and M 2 = 4, respectively, with the inner sequence order varying fromM 1 = 1 toM 1 = 10. The pure bath and error Hamiltonian strengths are adjusted such that the Hamiltonian is dominated by the error term, J. Each data point corresponds to a single cycle of the QDD sequence averaged over 50 random instances of the parametersc appearing 72 Figure 3.1: Single-axis errors after one cycle ofU (M 1 ;M 2 ) QDD for outer sequence orderM 2 = 3 and inner sequence ordersM 1 = 1; 2;:::; 10 as a function ofJ, averaged over 50 random realizations of the bath operatorsB . Error bars are shown but are very small. In all our simulations I setJ = 10 4 and = 10 6 . Single-axis error values are computed for log 10 (J) =9;8;:::; 2. Lines are guides to the eye. E x is designated by the green squares,E y by the red circles, andE z by the black triangles. Note that log 10 (E z ) is the same in all six plots, with a slope ofM 2 + 1. The slope of log 10 (E x ), on the other hand, is M 1 + 1. The slope of log 10 (E y ) is M 1 + 2. Vertical lines denote the largest value ofJ utilized in the linear regression used to extract the slopeN . in Eq. (3.15). Since the minimum pulse interval is kept fixed, the total sequence duration increases with increasingM 1 andM 2 . The first thing to notice about Figures (3.1) and (3.2) is that they match the prediction of Eq. (2.89) very well in the regime of small J d . Namely, in all cases observed a constant slope, untilJO(1). This is also in agreement with the result of Ref. [89]. A summary of the scalings forE (J) are given in Table 3.1 for all combinations of M 1 ;M 2 2f1; 2;:::; 10g. The values ofN are extracted by performing linear regres- sions for log 10 [E (J)] between log 10 (J) =9 and the values of log 10 (J) indicated by the vertical lines in Figs. 3.1 and 3.2, and rounding to the nearest integer (in all cases the deviation from an integer value was at most in the third significant digit). Consider now the effect of varying the inner and outer sequence orders on the uni- axial errors. WhenM 2 > M 1 , higher order suppression is expected for the errors that 73 Figure 3.2: Single-axis errors after one cycle ofU (M 1 ;M 2 ) QDD for outer sequence orderM 2 = 4 and inner sequence ordersM 1 = 1; 2;:::; 10 (left to right, top to bottom) as a function of J, averaged over 50 random realizations of the bath operators B . Other details as in Fig. 3.1, except that the single-axis error suppressed by both the inner and outer sequence,E y (), exhibits a strong dependence on the parity of the inner sequence. Note that for each value ofM 1 ,E x () andE y () are essentially equal for all values ofJ. correspond to the system basis operators which anti-commute with pulse operator com- prising the outerX-type sequence. Thus the uni-axial errorsE y () andE z () are most heavily suppressed. Since only the outer sequence can suppressz-axis, orZ-type errors [recall Eq. (3.2)], E z (J) only gains additional error suppression if the outer sequence order is increased. In Fig. (3.1),M 2 = 3 andE z (J)O[(J) 4 ] for allM 1 , exhibiting error suppression of the firstM 2 terms of the errors proportional to z . Thus QDD operates with UDD efficiency for error suppression by the outer nested sequence alone. In a similar manner toE z (J), the behavior ofE x (J) can be attributed to one of the two nested sequences. Namely, the error measured by E x is associated with x , which only anti-commutes with the pulse operator present in the innerZ-type sequence; determined solely by the inner sequence order,E x (J)O[(J) N 1 +1 ]. Essentially, the outer sequence has no effect on the order of error suppression for E x (J), as can be seen from Fig. 3.1 forM 2 = 3. The interpretation for E y (J) is not as simple, since this uni-axial error is com- pensated by both the inner and outer sequences. One might expect both the inner and 74 outer sequence to contribute to Y -type error suppression, i.e., E y (J) to scale with (J) max(M 1 ;M 2 )+1 . However, if this were the case then, e.g., the case M 1 = 1 would display an equal order of error suppression for both E y () and E z ( d ). Instead it is found that E y (J)O[(J) M 1 +2 ] for M 2 = 3. Thus the suppression of E y (J) is constrained byM 1 , even whenM 2 >M 1 , though it is larger by one order of magnitude than UDD error suppression efficiency for the inner sequence. Similar observations apply for all odd-order outer sequences analyzed [not shown here]. Odd-order sequences are anti-symmetric with respect to time-reversal, and the conclusions concerning the caseM 2 = 3 can be generalized as follows: when the outer sequence is anti-symmetric, terms in the QDD evolution operator which anti-commute with only one pulse operator are suppressed with UDD efficiency, determined by the order of the nested sequence composed of the corresponding anti-commuting pulse oper- ator (this applies to x and z ). In contrast, terms that anti-commute with both pulse operators are suppressed to one order beyond UDD efficiency, dictated exclusively by the inner sequence order (this applies to y ). Below it is shown how this observation is modified when considering larger values ofM 2 . Comparing the case of the anti-symmetric outer sequence of Fig. 3.1 to that of the symmetric sequence of M 2 = 4 in Fig. 3.2, one notices immediately that there is a qualitative difference. The uni-axial errorE y (J), the component anti-commuting with both the inner and outer sequences, y , fluctuates strongly as a function of M 1 . A similar effect is observed for other even values ofM 2 [not shown here]. Only the outer sequence order has been changed, therefore this characteristic is entirely dependent on the fact that the outer sequence is now symmetric. Analogous to the anti-symmetric outer sequence, Fig. 3.2 shows that the single- axis error E z (J) is independent of the inner sequence order. The scaling E z (J) 75 (a)N x =M 1 + 1 forM 1 ;M 2 2f1;:::; 10g (b) N y M 1 M 2 = 1 M 2 = 2 M 2 = 3 M 2 = 4 M 2 = 5 M 2 = 6 M 2 = 7 M 2 = 8 M 2 = 9 M 2 = 10 1 3 2 3 2 3 2 3 2 3 2 2 4 3 4 5 6 7 8 9 10 11 3 5 4 5 4 5 4 5 4 5 4 4 6 5 6 5 6 7 8 9 10 11 5 7 6 7 6 7 6 7 6 7 6 6 8 7 8 7 8 7 8 9 10 11 7 9 8 9 8 9 8 9 8 9 8 8 10 9 10 9 10 9 10 9 10 11 9 11 10 11 10 11 10 11 10 11 10 10 12 11 12 11 12 11 12 11 12 11 (c) N z M 1 M 2 = 1 M 2 = 2 M 2 = 3 M 2 = 4 M 2 = 5 M 2 = 6 M 2 = 7 M 2 = 8 M 2 = 9 M 2 = 10 1 2 3 4 4 4 4 4 4 4 4 2 2 3 4 5 6 7 8 9 10 11 3 2 3 4 5 6 7 8 8 8 8 4 2 3 4 5 6 7 8 9 10 11 5 2 3 4 5 6 7 8 9 10 11 6 2 3 4 5 6 7 8 9 10 11 7 2 3 4 5 6 7 8 9 10 11 8 2 3 4 5 6 7 8 9 10 11 9 2 3 4 5 6 7 8 9 10 11 10 2 3 4 5 6 7 8 9 10 11 Table 3.1: Summary of the scaling for all uni-axial errors. Values ofN Ire extracted by performing a linear regression, rounded to the nearest integer, fitting the slopes of the straight line portions of the curves displayed in Figs. 3.1 and 3.2 between log 10 (J) = 9 and the values of log 10 (J) indicated by the vertical lines in these figures. (a)E x , (b)E y , and (c)E z forM 1 ;M 2 2f1; 2;:::; 10g. ForN y andN z the outer sequence order M 2 is displayed in the top row and the inner sequence orderM 1 in the first column. Each of the single-axis errors is dominated by the lowest order ofJ d , denotedN , therefore E O[(J) N ]. Additional simulations (not shown) fully continue the trends seen in this table and summarized in Eqs. (3.16)-(3.18) all the way up toM 1 ;M 2 24. O[(J) 5 ] holds for allM 1 . Thus the single-axis errorE z (J) again exhibits UDD effi- ciency, independent of the parity of the outer sequence. Similarly, again E x (J) O[(J) M 1 +1 ] in Fig. 3.2. However, when considering theN z results for all values ofM 1 andM 2 it is found that there are exceptions to this simple behavior. As can be seen from Table 3.1, when M 1 = 1 andM 2 4, the value ofN z is fixed at 4. The same phenomenon is observed forM 1 = 3 andM 2 8. 76 On the basis of the numerical data one can summarize the scaling of the X and Z-type uni-axial errors as follows: N x = M 1 + 1; (3.16) and N z = 8 > > > < > > > : M 2 + 1 :M 1 even M 2 + 1 :M 1 odd; M 2 < 2M 1 + 2 2M 1 + 2 :M 1 odd; M 2 2M 1 + 2 (3.17) Qualitatively, it is expected that when the inner sequence works imperfectly, as is the case forM 1 odd, the lowest order sequence will determine the scaling of the single axis error, and this is what is stated in Eq. (3.17). As is clear from Fig. 3.2, the scaling ofE y (J) is dependent on the parity ofM 1 . If the inner sequence is of odd parity thenE y (J)O[(J) M 1 +2 ] whenM 2 is odd as well, orE y (J)O[(J) M 1 +1 ] whenM 2 is even. Thus the scaling ofE y is dominated by the inner sequence order whenM 1 is odd. The situation changes whenM 1 is even. Now, ifM 2 is odd the sequence is still anti-symmetric, however there is an immediate improvement in error suppression,E y (J)O[(J) max(M 1 +1;M 2 )+1 ]. If the complete sequence is fully symmetric (bothN 1 andN 2 even) a scaling is found that is dependent on both the inner and outer sequence orders, E y (J)O[(J) max(M 1 ;M 2 )+1 ]. Thus the suppression of error terms which anti-commute with both pulse operators depends 77 sensitively on the parity of the inner sequence order, and is summarized forE y (J) O[(J) Ny ] as N y = 8 > > > > > > > < > > > > > > > : max(M 1 ;M 2 ) + 1 :M 1 even; M 2 even max(M 1 + 1;M 2 ) + 1 :M 1 even; M 2 odd M 1 + 1 :M 1 odd; M 2 even M 1 + 2 :M 1 odd; M 2 odd : (3.18) The dependence upon the symmetry of the inner sequence, the parity ofM 1 , was first noted by Wang & Liu in the context of overall QDD performance [91]. The dependence on the outer sequence symmetry, however, was not noted previously. These results show that the symmetry of the outer sequence impacts the efficiency of y error suppression as well. The efficiency of QDD error suppression is directly related to the efficiency of UDD. Each interaction that anti-commutes with at least one pulse operator is expected to achieve UDD efficiency. Fully symmetric QDD, i.e., even order M 1 and M 2 , recov- ers the efficiency of UDD for all uni-axial errors. Consequently, QDD performs with optimal efficiency when it is fully symmetric. 3.2.2 Intermediate uni-axial error analysis Rather than considering the single-axis errors just at the end of the QDD sequence, one can consider the single-axis errors prior to the application of each X-type outer sequence pulse. Let such errors be denoted as “intermediate uni-axial errors” since they are extracted during the QDD evolution, unlike those presented in Figures 3.1 and 3.2 which are extracted at the end of the complete evolution. By studying this intermediate time-dependence of the errors another interesting perspective is given on the manner in which the QDD sequence suppresses decoherence. 78 Figure 3.3: Intermediate uni-axial errors forM 2 = 3, as defined in Eq. (3.23). For given k, the intermediate uni-axial error is computed after k innner Z-type UDD sequences separated byk 1 pulses. There areM 2 + 1 inner UDD sequences. The pointk = 5 is the lastX pulse at the end of last inner UDD sequence, as required for oddM 2 . Note that because the data points labeledj = 4 andj = 5 are separated by a singleX pulse, and are these pulses are instantaneous, these points have no actual time delay between them. Let the set of “intermediate QDD” sequences be defined as U (M1;j) QDD;fZ;Xg X j Y k=1 X h U (M 1 ) UDD;Z U 0 ( (M 2 ) k ) i (3.19) where j2f1;:::;M 2 + 1g. Thus, except for j = 1, U (M 1 ;j) QDD contains j 1 X-type pulses sandwiched betweenjZ-type UDD sequences. Whenj = 1 U (M 1 ;1) QDD;fZ;Xg U (M 1 ) UDD;Z U 0 ( (M 2 ) 1 ) (3.20) is just the UDD sequence. Consider separately the definition U (M 1 ;M 2 +2) QDD;fZ;Xg U (M 1 ;M 2 ) QDD;fZ;Xg =X M 2 U (M 1 ;M 2 +1) QDD;fZ;Xg ; (3.21) 79 Figure 3.4: Intermediate uni-axial errors forM 2 = 4. As in Fig. 3.3 except that there is no finalX pulse forM 2 even, i.e., there areM 2 + 1 inner UDD sequences separated by M 2 X pulses. i.e., the complete QDD sequence, Eq. (3.2). Note that U (M 1 ;M 2 +2) QDD;fZ;Xg contains a final X pulse ifM 2 is odd, but not ifM 2 is even. Similar to the error expansion Eq. (3.14), the intermediate error expansion can be expressed as U (M 1 ;j) QDD;fZ;Xg =I B (j) I () + X 2fx;y;zg B (j) (): (3.22) Furthermore, in analogy to Eq. (2.84) an intermediate uni-axial errors can be defined as E (j) ( d ) =kB (j) ( d )k F ; (3.23) where k2f1;N 2 + 2g. Note that for odd M 2 the errors E (M 2 +1) and E (M 2 +2) differ by a single instantaneousX pulse (which is significant, as the simulations results will demonstrate), while for M 2 even E (M 2 +1) = E (M 2 +2) , so that below E (M 2 +2) is not displayed in the even case. 80 Figures 3.3 and 3.4 display the intermediate single-axis errors forM 2 = 3 andM 2 = 4, respectively, withM 1 = 1; 2;:::; 6. The coupling parameters are fixed atJ = 10 4 and = 10 6 as in the previous figures. Several features are noteworthy in these figures. (i) E (1) x and E (1) y are equal and substantially smaller than E (1) z , and the difference grows as N 1 is increased. This is because the inner Z-type sequence only suppresses the X and Y -type errors, and the point j = 1 does not include the first X-type outer sequence pulse. Formally, this is expressed by E (1) () = kB (1) k F ( (M 1 ) ) M 1 +1 ; 2fx;yg E (1) () = kB (1) k F 1; 2fI;zg: (3.24) (ii) The intermediate uni-axial errors all fluctuate throughout the QDD evolution. This is due to a reshuffling of the errors after each outer sequence X-type pulse is applied, a simple consequence of the rules of Pauli matrix multiplication. To see why in some detail, consider the effect of the firstXi x I B pulse: X U (M 1 ;1) QDD;fZ;Xg = x B (1) I + X 2fx;y;zg x B (1) (3.25) = I B (1) x + x B (1) I + y B (1) z + z B (1) y ; where the factors ofi have been dropped. The reshuffling effect is clear: for example, the uni-axial z-type error now comes from B (1) y . To explain the j = 2 behavior it is necessary to consider the effect of multiplying X U (M 1 ;1) QDD;fZ;Xg by the next inner UDD sequenceU (M 1 ) UDD;Z ( (M 1 ) 2 ). Theith inner UDD sequence has the expansion U (M 1 ) UDD;Z ( (M 1 ) i ) =I B I(i) + x B x(i) + y B y(i) + z B z(i) ; (3.26) 81 where similarly to Eq. (3.24) kB (i) k F ( (M 1 ) i ) M 1 +1 ; 2fx;yg kB (i) k F 1; 2fI;zg: (3.27) Using this to carry out the multiplication to the next order U (M 1 ;2) QDD;fZ;Xg = U (M 1 ) UDD;Z ( (M 1 ) 2 )X U (M 1 ;1) QDD;fZ;Xg (3.28) = [I B I(2) + x B x(2) + y B y(2) + z B z(2) ] [I B (1) x + x B (1) I + y B (1) z + z B (1) y ] Consequently E (2) x () = kB I(2) B (1) I +B x(2) B (1) x +B y(2) B (1) y +B z(2) B (1) z k F 1 E (2) y () = kB I(2) B (1) z +B x(2) B (1) y +B y(2) B (1) x +B z(2) B (1) I k F 1 E (2) z () = kB I(2) B (1) y +B x(2) B (1) z +B y(2) B (1) I +B z(2) B (1) x k F 2( (M 1 ) 1 ) M 1 +1 + 2( (M 1 ) 2 ) M 1 +1 ; (3.29) whereE (2) x () is dominated byB I(2) B (1) I andE (2) y () is dominated byB I(2) B (1) z , neither of which is suppressed, whence the 1 result. On the other hand every one of the terms inE (2) z () is suppressed. Hence, as can be seen in Figures 3.3 and 3.4, atj = 2 both the X and Y -type errors have increased relative to j = 1, while the Z-type error has 82 decreased. One can similarly understand the remaining oscillations of the intermediate single-axis errors in terms of this reshuffling of error types. (iii)E x andE z oscillate out of phase, whileE y oscillates in phase withE x for even M 2 , but not necessarily for oddM 1 . This is again a consequence of error reshuffling. TheY -type error behaves differently from the other two since it experiences suppression from both the inner and outer sequences. For the same reason it is always found that E (j) y <E (j) x . (iv) E x attains its minimum for j = 1 and then slowly increases, though while maintaining its suppression order. This is because theX-type error is suppressed only by the inner sequences, and these are simply applied to it with fixed order (M 1 ), a total ofM 2 orM 2 +1 times. Repeated application of the inner UDD sequence is similar to the periodic DD (PDD) protocol, whose performance is well known to deteriorate as time grows [53, 76]. The reason is that the error accumulates over time, without a mechanism for reducing it. (v) There does not appear to be much of a difference between even and odd values ofM 2 in terms of the intermediate uni-axial errors. One difference is thatE (j) y tends to be more erratic for oddM 2 at highj values. A simple explanation for this behavior can not be found. Another difference is that for evenM 2 all uni-axial errors have the same final value whenM 1 = M 2 , but for oddM 2 theX-type error is always slightly worse at the end of the sequence, thus setting the bottleneck. Perhaps additional pulse interval optimization can remove this asymmetry. (vi) Only at the very end are all three uni-axial errors simultaneously small. Thus, while suppression of one error type can be achieved in the middle of the QDD sequence, one must wait until its completion to suppress all errors. 83 n D Kos M 1 M 2 = 1 M 2 = 2 M 2 = 3 M 2 = 4 M 2 = 5 M 2 = 6 M 2 = 7 M 2 = 8 M 2 = 9 M 2 = 10 1 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 2 3 4 4 4 4 4 4 4 4 4 2 3 4 5 5 5 5 5 5 5 5 2 3 4 5 6 6 6 6 6 6 6 2 3 4 5 6 7 7 7 7 7 7 2 3 4 5 6 7 8 8 8 8 8 2 3 4 5 6 7 8 9 9 9 9 2 3 4 5 6 7 8 9 10 10 10 2 3 4 5 6 7 8 9 10 11 Table 3.2: Summary of the scaling of the overall distance measureD K with respect to inner and outer sequence orders,M 1 andM 2 , respectively. Values ofN D K are extracted by performing a linear regression, rounded to the nearest integer, fitting the slopes of the straight line portions of the curves displayed in Figs. 3.1 and 3.2 between log(J) =9 and the values of log(J) indicated by the vertical lines in these figures. The outer sequence orderN 2 is displayed in the top row and the inner sequence orderM 1 in the first column. As expected, N D K = min(N x ;N y ;N z ). Additional simulations [not shown] fully continue the trends seen in this table and summarized in Eq. (3.31) all the way up toM 1 ;M 2 24. 3.2.3 Overall QDD Performance Scaling While the uni-axial error analysis presented in the previous two subsections helps in unraveling the mechanism of QDD performance, it does of course not tell the whole story. A complete quantitative description of QDD performance can be given by ana- lyzing a distance measure, here chosen as the Kosut distance D Kos (U;I) [Eq. (2.72)]. By the definition of DD and distance measure scaling described by the upper-bound on D Kos , the overall performance of QDD can be expected to be dictated by the lowest order of present in the final evolution operator, Eq. (3.14), i.e., D Kos O[(J) N D K ] (3.30) where N D Kos = min(N x ;N y ;N z ): (3.31) 84 Overall QDD performance for M 2 = 3 and M 2 = 4 is shown in Figures 3.5 and 3.6, respectively. The outer sequence orderM 2 is fixed and the inner sequence orderM 1 is varied from 1 to 6. These results are for the same model considered in the previous subsection. A summary of the distance scaling results is presented in Table 3.2. Considering M 2 = 3 first (Fig. 3.5), when M 2 > M 1 the overall order of error suppression is hindered by the inner sequence order. This is evident by the increasing order of error suppression as M 1 increases. In this regime the lower sequence order is determined by the inner sequence, therefore the scaling ofD Kos is equivalent to that of E x , i.e., D Kos O[(J) M 1 +1 ]. As M 1 passes M 2 , there is a saturation of error suppression corresponding to a performance bounded by the lower outer sequence order. The amplitude of performance increases slightly beyondM 2 = M 1 , however begins to decrease whenM 1 >M 2 , as evidenced not by the slope but by the offset of the distance curves. Namely, the ordering, from worst to best, is M 1 = 6; 5; 4. The latter is an interesting feature not easily deduced from the uni-axial errors. Increasing the inner sequence order results in an accumulation of error for the uni-axial error dominating the performance; whenM 1 >M 2 + 1 this corresponds toE z . The results are similar forM 2 = 4, as shown by Fig. 3.6. The order of error suppres- sion, given by the slope, increases until M 1 = M 2 in correspondence with an overall performance dominated by the lowest order of present inE . In addition to the satura- tion of the order of error suppression, an offset-related deterioration is again observed. Namely,M 1 = 6 is slightly worse thanM 1 = 5. 3.2.4 Summary of results In this chapter, a comprehensive numerical analysis of the error suppression characteris- tics of QDD is presented. This was achieved by isolating the uni-axial errors associated with each system basis operator comprising the error Hamiltonian portion of Eq. (3.1). 85 Figure 3.5: Overall QDD performance after one cycle ofU (M 1 ;M 2 ) QDD;fZ;Xg for outer sequence orderM 2 = 3 and inner sequence ordersM 1 = 1; 2;:::; 6, as a function ofJ, averaged over 50 random realizations of the bath operatorsB . The performance of QDD pro- gressively improves with increasingM 1 up toM 2 =M 1 , indicating that minfM 1 ;M 2 g dominates QDD performance. The order of error suppression was determined by computing the uni-axial error as a function of the minimum pulse interval. The analysis was performed for a model in which the dynamics generated by the error Hamiltonian dominated the internal bath dynamics, in order to study study the properties of the uni-axial errors in the regime where DD is most beneficial. Considering a QDD sequences with M 1 Z-type pulses comprising the inner sequence, andM 2 X-type pulses comprising the outer sequence, the system-bath interaction term proportional to x was found to be suppressed with UDD efficiency for all values ofM 1 andM 2 [Eq. (3.16)]. The decoherence-generating terms proportional to z and y both exhibit parity effects [Eqs. (3.17), (3.18)] whose origins are the symmetry or anti-symmetry of the inner and outer UDD sequences. Of course, permuting the pulse types of the inner and outer sequences will correspondingly modify these conclusions. 86 Figure 3.6: Overall QDD performance after one cycle ofU (M 1 ;M 2 ) QDD;fZ;Xg for outer sequence orderM 2 = 4 and inner sequence ordersM 1 = 1; 2;:::; 6, as a function ofJ, averaged over 50 random realizations of the bath operatorsB . The dependence of the order of error suppression on minfM 1 ;M 2 g is again observed. An analysis of the intermediate time-dependent performance of QDD was also per- formed. There it was found that the uni-axial errors are strongly time-dependent, oscil- lating between outer-sequence pulses, until they all converge to nearly the same value after the final outer-sequence pulse. The closest convergence occurs for QDD sequences with equal inner and outer orders. Finally, the overall performance of QDD was computed using the state-independent Kosut distance measure, and then reconciled with the scaling of the uni-axial errors. The overall QDD error suppression was shown to scale with the lowest order of uni-axial error suppression, i.e., the first non-vanishing contribution appears at order min(M 1 ;M 2 )+1. QDD accomplishes this by applying (M 1 +1)(M 2 +1) pulses. While this study focuses primarily on the single-qubit case, the definition of the uni-axial error measure naturally permits for a similar analysis for NUDD performance characteriza- tion as well. 87 Chapter 4 Dynamical Decoupling for Realizable Systems As the techniques of DD sequence construction have become increasingly sophisticated, the issue of robustness to control pulse imperfections has remained one of the prominent restrictions of sequence performance in experimental settings. Systematic errors such as rotation-angle or rotation-axis errors, and finite pulse duration errors brought about by bandwidth constraints, can generate additional decoherence that quickly destroys the decoupling efficiency of all known DD schemes. Robustness to such errors has been addressed by the XY-family of sequences and CDD, and in a systematic manner— for pulse-width errors—by Eulerian DD (EDD) [109] and its generalization to logic gates, known as dynamically corrected gates (DCG) [110, 111]. A concatenated ver- sion of DCG (CDCG) has been shown to be capable in principle of achieving arbitrarily accurate gates using finite width pulses [112].Protocols based on pulse-interval opti- mization, such as UDD and its variates, have been shown experimentally to be highly sensitive to pulse imperfections, thus forfeiting their ideal-pulse decoupling efficiencies [83]. Certain numerical optimization techniques such as LODD, BADD, OFDD, and WDD (see Sec. 2.4.4), exhibit a degree of robustness to finite pulse duration, however, the relationship between sequence performance and rotation errors is unclear. A more recent approach for combating pulse imperfections, known as Knill DD (KDD), utilizes a sequence of variable phase-pulses separated by fixed pulse-intervals to generate an effective sequence of four-pulses centered around a specified axis with an additional 88 overall accumulated phase [84]. In contrast to the XY-family and XY 4 -based CDD, KDD exhibits robustness to finite-width and flip-angle errors; however, this robustness is somewhat limited in the original construction as applying standard concatenation protocols to generate a hierarchy of KDD sequences does not appear to offer further improvement in sequence robustness. In this chapter, the problem of sequence robustness is addressed by two approaches specifically for a single-qubit system. The first seeks to generate robustness against pulse imperfections for known deterministic decoupling schemes, in particular QDD. Utilizing a technique known as time symmetrization, robust variations of QDD are con- structed and analyzed in order to determine the most viable approach to maximizing the suppression of control errors in the pulse-interval optimized case. Time symmetrization is expected to offer an advantage since all odd orders in TDPT are automatically sup- pressed, even in the case of control errors. Determining the correct manner in which to apply time symmetrization to the QDD scenario is essentially the objective for this por- tion of the study. The second approach seeks to address the issue of control error robust- ness from a numerical optimization point of view. Using genetic algorithms (GAs), brand new DD sequences are constructed which are essentially optimized for particular types of control errors. In one specific setting, it will be shown that optimal sequences can be constructed which display an arbitrary decoupling-like behavior. Thus far no such sequences have been known to exist simply be manipulating sequence structure alone. As a final step, the results of both robustness studies are compared under condi- tions which effectively characterize a realistic system relevant for QC, namely, a solid state NMR quantum system. 89 4.1 Shortcomings of Current Deterministic DD Meth- ods The attractive decoupling efficiencies of the deterministic methods discussed in Sec. 2.4.2 and 2.4.3 greatly diminish when control errors are introduced into the pulse operators. Arbitrary order decoupling no longer becomes achievable in this setting and eventual saturation in the decoupling order is observed as the number of pulses increase. The lack of robustness to such errors is illustrated here via an analysis of the effective error Hamiltonian H err [see Sec. 2.2.4] for both CDD and UDD-based methods under the condition of faulty pulses described by the pulse profile of Eq. (2.31) and pulse operators X(Y;Z) =e ip[A(1+) x(y;z) +H 0 ] (4.1) whereA p = and H 0 =I B I + X =x;y;z B ; (4.2) the general decoherence model for a single-qubit system. 4.1.1 Concatenated DD with Faulty Pulses Consider the CDD sequence U (`) CDD ( d ) described by Eq. (2.50) which implements DD protection on a single-qubit system and therefore symmetrizes the error Hamil- tonian over the single-qubit Pauli groupP given in Eq. (2.45). In the ideal pulse case, CDD offers `th order decoupling using ` levels of concatenation which trans- lates into the effective evolution operator in the toggling frame scaling accordingly as ~ U 0 ( d ) = e i H B +O( `+1 d ) and therefore the effective error Hamiltonian scaling as H err O( ` d ). Upon introducing control errors in the form of finite amplitude and pulse 90 rotation errors, the efficiency of this scheme is dramatically reduced and the effective error Hamiltonians are summarized by: H (` odd) err (`) 2 z + p z ( (`) x B x + (`) y B y ) + X ; ( (`) ; d + (`) ; p )( B ); (4.3) H (` even) err ~ (`) z + X ; ( ~ (`) ; d + ~ (`) ; p )( B ) (4.4) Parity dependence in CDD performance is due variations in sequence symmetry man- ifested by the concatenation process. Even levels of concatenation result in symmetric sequences, while odd levels of concatenation yield anti-symmetric sequences. The effect of symmetry on the error suppression capabilities is clearly evident from the higher order suppression of flip-angle (rotation) errors or finite-amplitude errors for anti-symmetric and symmetric CDD, respectively. Note that in the calculation of Eqs. 4.3 and 4.4 con- secutive pulse operators resulting from concatenation are simplified according to the Pauli multiplication rules: XY =Z,YZ =X, andZX =Y with additional property that pulse operators square to identity. These rules essentially define the multiplication properties of ideal pulses, however, can also utilized in the non-ideal case to represent the effective rotation which is to be captured by consecutive pulse operators. Of course, one can also consider the case where pulses are not simplified and therefore are applied successively where application. The effective error Hamiltonian for the unsimplified case is essentially described by Eq. (4.3) for all`. Hence, it is more beneficial to make pulse substitutions for this particular DD scheme. 4.1.2 Quadratic DD with Faulty Pulses Similar saturations in error suppression are also observed for QDD. Consider, for example, the QDD sequenceU (M;M) QDD;fX;Yg ( d ) following the structure of Eq. (5.11) with M 1 = M 2 = M. For evenM values the resulting sequence is described byX andY 91 pulses, while oddM leads to an outer nested UDD layer ofYX pulses. Initially, con- sider the case where the substitutionYX = Z is made in accordance with the Pauli mul- tiplication rules. The operator Z denotes the 180-degree phase-rotatedZ-pulse mathe- matically described by Eq. (4.1) withA!A. Incorporating such definitions into the QDD evolution, the following effective Hamiltonians are obtained for odd and evenM values: H (M odd) err (M) 2 y + p (M) y B x + X ; ( (M) ; d + (M) ; p )( B ); (4.5) H (M even) err ~ (M) x x + ~ (M) y y + p X ; ~ (M) ; ( B ) + p X ; ~ (M) ; ( B ): (4.6) Contrary to CDD, the error suppression properties for QDD do not appear to depend upon sequence parity for finite-width amplitude errors, as seen by the linear dependence on p in both effective Hamiltonians. Parity dependence does exist for flip-angle errors where oddM values result in first-order suppression in and evenM values are unable to manifest such suppression. As in the CDD case it is also possible to consider unsim- plified pulses as well, which effects only odd M sequences. Sequence performance diminishes when considering successiveY andX pulses. Hence, pulse substitution is more favorable for the QDD scheme when considering pulse imperfections. 4.2 Numerically-Optimized Dynamical Decoupling While CDD achieves arbitrary order decoupling in the ideal pulse limit, the analysis above conveys that a saturation in CDD performance develops which is merely depen- dent on the parity of the concatenation level. The inherent non-robustness of CDD is perhaps not an indication of an inability to achieve control error robustness in the 92 case of fixed pulse-intervals, but perhaps a matter of sequence construction. The suc- cess of certain XY-family sequences which effectively address control errors and are far superior to CDD suggests that additional sequence constructions may exist. Employing genetic algorithms (GAs), optimal sequences based on fixed pulse-intervals are located in this section which are robust to certain control errors, are far superior to CDD, and essentially build upon the XY-family sequences previously located. As an additional result, optimal sequence are obtain that improve upon the CDD decoupling efficiency and exhibit a deterministic structure. The focus will be numerically optimal -pulse sequences developed for a single- qubit system described by the general decoherence model of Eq. (3.1) withB operators describing the qubit environment of Eq. (3.15). Initially, the case of ideal zero-width pulses is considered. Then, finite-width rectangular profiles, flip-angle errors, and finally the culmination of both types of errors are account for. For all pulse profiles the number of pulses is varied fromK = 1; 2;:::; 256 with a pulse-interval d = 0:1ns. 1 Due to the piecewise continuous form of H C (t), the general structure of the sequences is described by U( c ) =P K U f ( d )P K1 U f ( d )P 2 U f ( d )P 1 U f ( d ); (4.7) whereP j is the unitary evolution operator achieved by thejth pulse and c =K d . The pulse operators are defined such thatP j 2D, whereD denotes a discrete set of allowable control pulses that depends upon the choice ofV (t). The total sequence time c is also dictated by the choice of V (t) since the finite duration of the pulse contributes when applicable, e.g., for a sequence ofm d pulse delays andm p nontrivial pulses c =m d d + 1 The choice of units is arbitrary but is meant to be commensurate with electron spin qubits in, e.g., quantum dots. 93 m p p . While Eq. (4.7) is not the most general DD evolution operator for fixed pulse- intervals, since it does not permit consecutive pulses without free evolution periods, it still captures a majority of the known sequences and additional highly robust sequence constructions. Further details regarding the algorithm can be found in Sec. A.1. The value ofK was varied over a significant range in the simulations, however, it is found that only specific values of K are relevant for successive error suppression. In particular,K opt = 4; 8; 16; 32; 64; 256 correspond to the minimum number of pulses required to observe an increase in error suppression or significant improvement in per- formance. All of the remaining values of K result in a sequence performance upper bounded by the performance of the previousK opt . For example, the optimal sequences for K = 17; 18;:::; 31 exhibit a performance proportional to that of K = 16, if not worse. In order to reduce the overhead of initial state averaging, the Kosut distance [Sec. 2.5.3] is adopted as the appropriate performance measure. The values of K opt were obtained by analyzing the scaling of the performance of the optimal sequences identified at each value ofK in the ideal pulse limit. ForK 12, optimal sequences were located by an exhaustive search, whileK > 12 demands the use of the GA algo- rithm discussed in Sec. A.1. Upon locating the optimal sequences, the performance measureD Kos is analyzed as a function of c . The order of error suppression,N, is then determined from Eq. (2.75), which also holds forD Kos , by N = log 10 (D Kos ) log 10 [(J +) c ] 1: (4.8) Values ofK whereN is found to increase ultimately correspond to those identified as values ofK opt . 94 The scaling method described above is a convenient numerical method for deter- mining the structure of the dominant term in the effective error Hamiltonian for each of the optimal sequences obtained for a given K opt . In order to fully characterize the scaling of H err it is necessary to analyze the distance measureD K as a function of each relevant parameter. In the case of ideal pulses, this would correspond to analyzing the performance as a function offJ;; d g in the regimes of interaction-dominated dynam- ics (J ) and environment-dominated dynamics (J ) since each regime may exhibit different scalings. When finite pulse duration and flip-angle errors are included the number of parameters increases to either a subset offJ;; d ; p ;g or the entire set if both forms of pulse errors are present. Further details regarding this procedure can be found in Sec. A.2. It is then necessary to analyze the scaling ofD K in each of the various parameter regimes in addition to the interaction or environment-dominated regimes, e.g., pulse-width dominated ( p d ), free evolution dominated ( p d ), flip-angle error dominated ( J d ), etc. In the subsequent analysis, this technique is made use of in conjunction with direct calculation of the effective error Hamiltonian to fully characterize sequence performance for each pulse profile. The effective Hamil- tonian calculation is utilized to provide additional insight into the structure of H err that can not be observed from the scaling method, most notably in situations where multiple sequences exhibit identical performance scalings. 4.2.1 Ideal pulses Here the optimal sequence structure of ideal, zero-width pulses with respect to the strengths of the internal dynamics,J and, in the regime whereJ d ; d 2 [10 10 ; 10 2 ] 95 are examined. Under the condition of uni-axial pulses, the set of possible control pulses D =fI;X;Y;Zg, whereI is the identity operator and X(Y;Z) =i x(y;z) (4.9) describe -pulse unitary operators generated by Eq. (2.27) when H C (t) is non-zero. Neither the error Hamiltonian, nor pure environment Hamiltonian, is present during the pulse evolution in the limit of zero-width (infinite amplitude) pulses. Characteristics such as the dimension of the reduced search spaceN R (K) [see Sec. A.1 for definition] are determined by the number of elements in the pulse setD. Under the conditions of ideal-function pulses,N R (K) = 4 K1 atl = l max . Conse- quently, the initial search space only contains 16 possible sequence configurations for allK using the complexity-reduction procedure described in A.1.5. All 16 are chosen to represent the initial population at the commencement of the algorithm and the size of the population is kept constant throughout. Summary of Numerical Search Initially the algorithm is benchmarked atK = 4, where it is confirmed that the universal decoupling sequence discussed in Sec. 2.4.1 XY 4 =YU f ( d )XU f ( d )YU f ( d )XU f ( d ); (4.10) along with its obvious generalization GA 4 :=P 2 U f ( d )P 1 U f ( d )P 2 U f ( d )P 1 U f ( d ); (4.11) 96 whereP 1 6= P 2 2fX;Y;Zg, is indeed optimal over the range ofJ; specified above. The optimality of this particular sequence is attributed to its achievement of first order error suppression for general single-qubit decoherence [57], which can be confirmed numerically by analyzing the scaling of Eq. (2.75) with respect tofJ;; d g, where D 4 8 < : O(J 2 d ) : J O(J 2 2 d ) : J : (4.12) Alternatively, first order error suppression can be validated by calculating the effective error Hamiltonian for a specific choice ofP j , e.g., H XY 4 err i d x [B 0 ;B x ] + i 2 d z ([B 0 ;B z ]ifB x ;B y g): (4.13) As mentioned above, the next interesting result occurs atK = 8 where second order decoupling is first observed. The optimal sequences located at this particular value of K can be partitioned into two general structures denoted asa-type andb-type sequences such that GA 8a := IP 1 P 2 P 1 IP 1 P 2 P 1 ; (4.14) GA 8b := (P 3 P 2 )P 1 P 2 P 1 (P 3 P 2 )P 1 P 2 P 1 : (4.15) Note that the free evolution periods have been dropped for convenience of notation and the pulses which are not separated by free evolution periods have been highlighted with parentheses. In addition to the obvious structural differences between the two sequences, essentially described by whetherP 3 = P 2 is satisfied, a contrast is also observed from the standpoint of the effective error Hamiltonian. Since it is possible to effectively 97 Sequence J J Name Description GA 4 P 1 P 2 P 1 P 2 O(J 2 ) O(J 2 2 ) GA 8a IP 1 P 2 P 1 IP 1 P 2 P 1 O(J 2 3 ) O(J 3 3 ) GA 8b P 3 (GA 4 )P 3 (GA 4 ) O(J 2 3 ) O(J 2 3 ) GA 16a P 3 (GA 8a )P 3 (GA 8a ) O(J 2 3 ) O(J 3 3 ) GA 16b GA 4 [GA 4 ] O(J 2 3 ) O(J 2 3 ) GA 32a GA 4 [GA 8a ] O(J 3 4 ) O(J 2 2 4 ) GA 32b GA 8a [GA 4 ] O(J 3 4 ) O(J 2 2 4 ) GA 64a GA 8a [GA 8a ] O(J 4 5 ) O(J 3 2 5 ) GA 64b GA 8b [GA 8b ] O(J 4 5 ) O(J 3 2 5 ) GA 64c GA 4 [GA 4 [GA 4 ]] O(J 3 4 ) O(J 2 2 4 ) GA 256a GA 4 [GA 64a ] O(J 5 6 ) O(J 3 3 6 ) GA 256b GA 8b [GA 32a ] O(J 5 6 ) O(J 3 3 6 ) GA 256c GA 4 [GA 64c ] O(J 4 5 ) O(J 2 3 5 ) Table 4.1: Summary of distance measure (D Kos ) scalings for each optimal GA K sequence identified in the ideal pulse limit, for a fixed pulse-interval of d . Boxed per- formance scalings highlight optimal sequences in each of the relevant (J;)-regimes (columns) for eachK opt . extract the dominant terms of the effective error Hamiltonian by examining the scaling of the performance measure, numerical method is utilized to determine D 8a 8 < : O(J 2 3 d ) : J O(J 3 3 d ) : J (4.16) D 8b O(J 2 3 d ) 8J;: (4.17) Clearly, the difference occurs in the regime of interaction-dominated dynamics, where terms in the effective error Hamiltonian that solely comprise products of the interaction Hamiltonian begin to govern the scaling ofGA 8a ’s performance. 98 In addition to providing insight into the general structure of optimal sequences for K = 8, the results also show an immediate correspondence with known 8-pulse sequences, namely, XY 8 =IXYXIXYX: (4.18) Known for providing second order error suppression for general single-qubit decoher- ence [74], XY 8 gains its decoupling attributes from its structure: the XY 4 sequence followed by a time-reversed copy. An alternative perspective ofXY 8 is that of a con- catenated sequence composed ofXY 4 (GA 4 = XY 4 ) and CPMG = PU f ( d )PU f ( d ) (P =X). In general, depending on the choice of the CPMG pulses two different varia- tions arise:GA 8a orGA 8b . Interestingly, the latter viewpoint is perhaps the most useful for sequence characterization since all remaining optimal sequences from K = 16 to K = 256 can be interpreted as concatenations of various combinations of CPMG,XY 4 , and both K = 8 optimal sequences. This result not only conveys the importance of these sequences as fundamental building blocks for arbitraryK optimal sequences, but also the significance of concatenation in achieving high, perhaps even arbitrary, order error suppression in the regime of fixed-pulse intervals. Now consider the remaining sequences and focus on sequence lengths, and gener- alized sequence constructions, that yield additional orders of error suppression. It is found find that in order to achieve third, fourth, and fifth order decoupling a minimum of 32, 64, and 256 pulses are required, respectively. The sequences responsible for these effects are: GA 32a :=GA 4 [GA 8a ]; GA 32b :=GA 8a [GA 4 ]; GA 64a :=GA 8a [GA 8a ]; GA 64b :=GA 8b [GA 8b ]; GA 256a :=GA 4 [GA 64a ]; GA 256b :=GA 8b [GA 32a ]; 99 (a)K = 4 (b)K = 8 (c)K = 16 (d)K = 32 (e)K = 64 (f)K = 256 Figure 4.1: Performance ofGA K sequences forK = 4; 8; 16; 32; 64; 256, as shown in (a)-(f), respectively, as a function ofJ and. The minimum pulse-interval is fixed at d = 0:1ns and the results are averaged over 10 realizations ofB . TheGA Ka sequences tend to be optimal in the rangeJ <, whileGA Kb are optimal forJ . Sequences closely related to the deterministic structure of CDD,GA Kc , begin to appear as optimal sequences forJ d > 1 atK = 64. The notationnGA 4 denotes the application of n cycles of GA 4 and is dependent on the value of K. For the data presented above, n = 1; 8; 16; 64 forK = 4; 32; 64; 256, respectively. where the brackets are used to denote a concatenated structure, e.g., GA 32a =GA 4 [GA 8a ] =P 2 (GA 8a )P 1 (GA 8a )P 2 (GA 8a )P 1 (GA 8a ): (4.19) The performance of both 32-pulse sequences scales asD 32a;b O( 4 d ). Similarly, the 64 and 256-pulse sequences obtain performance scalings ofD 64a;b O( 5 d ) andD 256a;b O( 6 d ). A more detailed characterization of the performance scaling for each sequence is given in Table 4.1. Note that we also includeK = 16 sequences even though they do not achieve additional orders of error suppression. Their significance will become apparent in subsequent discussions on pulse imperfections presented in the main text and 100 appendix. In regards to the remaining optimal sequences, one may notice the absence of XY 4 -based CDD which contains sequences ofK = 16; 64; 256 pulses. The generalized sequences defined above in fact out perform CDD sequences and obtain an additional order of decoupling at each of theK = 16; 64; 256 sequence lengths. Characterization ofGA K Sequences in (J;)-space In Figure 4.1, the space of optimal sequences is characterized as a function of J d and d for each of the optimal sequence lengths discussed above. A general trend is observed for each sequence type, where a-type sequences tend to be optimal when J < and b-type sequences dominate the J > regime. Note that when the bath dynamics are dominant such that d 1, repeated cycles of GA 4 become the pre- ferred sequence. Here, bath self-averaging effects are more prominent and sequence effectiveness is reduced dramatically for more sophisticated sequence structures which possess cycle times that are longer than 1 . It is important to note that additional sequences beyond those discussed above also appear: GA 64c = GA 4 [GA 4 [GA 4 ]] and GA 256c = GA 4 [GA 64c ]. These structures correspond to generalized XY 4 -based CDD sequences which only appear to be optimal when J d 1. This result is consistent with the performance scaling equations presented in Table 4.1, where the quadratic scal- ing inJ for c-type (as opposed to cubic scaling for a,b-type) sequences is clearly more favorable whenJ d 1. 4.2.2 Finite-width Pulses The analysis is now extended to include errors due to finite-width rectangular pulses of duration p . It is noted that in this case EDD is a known way to achieve first order pulse- width error suppression, with the added assumption that pulse shaping is possible [109]. 101 Each optimal sequence construction is examined with respect toJ=2 [10 15 ; 10 3 ] for = 1kHz and p = d 2 [10 6 ; 10 3 ] for d = 0:1ns. The set of allowable control pulses D =fI;X;Y;Z; X; Y; Zg (4.20) contains three additional pulsesf X; Y; Zg corresponding to 180-degree phase flips of the non-trivial pulsesfX;Y;Zg, which are generated by takingA!A in Eq. (2.29). Note that in contrast to the ideal pulse case, the identity operatorIe ipH 0 so that for a givenK the cycle time is equivalent for all sequences. The remaining unitary pulse operators are generated from Eq. (2.29) and given by X(Y;Z) =e ip(A x(y;z) +H 0 ) ; (4.21) where the internal Hamiltonian H 0 is now included due to the fact that the pulses are now finite in amplitude. This form can be reconciled with the ideal pulse operators given in Eq. (4.9) by considering infinite pulse amplitude (A!1) and a pulse duration that is effectively zero ( p ! 0) so thatkH C (t)k p ==2 is maintained andkH 0 k p ! 0. The finite amplitude assumption essentially defines a pulse duration that differs from zero, so thatH 0 becomes a relevant contributor to the dynamics generated by the pulse. Of course we must still make the strong pulse assumptionkH C (t)kkH 0 k in order to reduce the additional errors generated byH 0 during the pulse. We enforce this assump- tion in the following section and utilize it to calculate effective pulse dynamics where the contributions ofH 0 are essentially perturbations toH C (t). The effective pulse oper- ators are then used to calculate effective error Hamiltonians for each finite-width-robust GA sequence. The dimension of the reduced search space is increased by the additional pulses toN R (K) = 7 K1 at l max . Consequently, the initial search space contains 7 2 = 49 102 configurations at all values of K. We find that not all 49 sequences are required to obtain optimal sequences. Surprisingly, an initial population of 16 sequences chosen at random from the 49 is sufficient to converge on global extrema for allK. Summary of Numerical Search The first case to consider isK = 4, where it is found that RGA 4 := P 2 P 1 P 2 P 1 ; (4.22) RGA 4 0 := P 2 P 1 P 2 P 1 ; (4.23) are the only optimal sequence configurations. The two sequences result in similar effec- tive error Hamiltonians: H RGA 4 err 4 p c z B xy + H GA 4 err (4.24) H RGA 4 0 err 4 p c ( y B z + z B x ) + H GA 4 err ; (4.25) which scale linearly in p and, therefore, do not provide first order error suppression in the pulse duration. Note the difference in error distribution between the effective error Hamiltonians generated simply by reversing the phase of a single pulse. Pulse imperfections generate errors along the z -channel forRGA 4 and along the y and z channels forRGA 4 0. Depending on the form of the system-environment interaction, the difference in sequence performance could be quite drastic. For example, consider the case of uniform decoherence in the xy-plane (B x =B y ) which results in complete first order decoupling in p forRGA 4 only. 103 The above results indicate that robustness to pulse imperfections can be extremely sensitive to variations in pulse phases, even in the simplest case of uni-axial pulses. This statement continues to hold true for RGA 8a 0 := IP 1 P 2 P 1 IP 1 P 2 P 1 (4.26) RGA 8b := RGA 2 [RGA 4 ] (4.27) RGA 8c := P 1 P 2 P 1 P 2 P 2 P 1 P 2 P 1 ; (4.28) the finite-width pulse error-optimized sequences for K = 8. First, note the effective error Hamiltonian forRGA 8a 0, H RGA 8a 0 err 16 d p c y B 2 x 8 d p c y fB x ;B y g +O(J d p ); (4.29) conveys complete first order suppression in the pulse duration. RGA 8a 0 is the primary optimal sequence located atK = 8, but is also accompanied by less-robust sequences such asRGA 8b andRGA 8c .RGA 8b is a robust form ofGA 8b [see Eq. 4.14] whose lack of first order decoupling in p is particularly favorable when J . The remaining sequence, RGA 8c , denotes a generic version of the Eulerian DD (EDD) sequence and attains first order error suppression in p by traversing the Cayley graph = (S;G), whereS denotes the single-qubit Pauli group with elements denoting the graph vertices andG =fI;X;Y;Zg is the generating set comprising the edges [109]. The original construction (captured by RGA 8c ) utilized two closed Eulerian cycles on such that the second is completed by returning along the first path. However, additional paths exist which do not require closed cycles to obtain first order suppression in p . One such case is RGA 8a 0, where the initial path and its inversion are both open Eulerian paths, as illustrated in Fig. 4.2. In comparingRGA 8a 0 andRGA 8c , the most important aspect appears to be the manner in which the paths are traversed rather than their closure. Note 104 I X Y Z X Y X I I IXYXIXYX Figure 4.2: Pictorial depiction for the action of RGA 8a 0 as an Eulerian path along the Cayley graph (S;G) with verticesS = fI;X;Y;Zg and generating setG = fI;X;Y;Zg. Note that unlike the EDD construction [Eq. (4.28)],RGA 8a 0 is generated by Eulerian paths, rather than cycles. that variations in pulse phases may aid in the error suppression process, but are not necessarily required to obtain first order decoupling in p asGA 8a also perform the task. In terms of performance,RGA 8c does not match the second order error suppression in d found forRGA 8a 0, as indicated by H RGA 8c err 4 d p c x (B 2 y +fB x ;B y g) + y (B 2 x +fB x ;B y g) +O(J 2 2 d ): (4.30) This attribute of RGA 8c is ultimately the cause for the overall advantageous perfor- mance ofRGA 8a 0. BeyondK = 8, optimal sequence configurations are generally characterized by two specific sequences: RGA 16a 0 := P 3 (RGA 8a 0)P 3 (RGA 8a 0); (4.31) RGA 64c := RGA 8c [RGA 8c ]: (4.32) 105 The former emerges at K = 16; 32; 64; 256, where cycles of RGA 16a 0 are utilized to generate the correct number of pulses for each corresponding K value. Additional sequences such asRGA 16b 0 := RGA 4 [RGA 4 0] andRGA 32c := RGA 8c [RGA 4 ] appear atK = 16 andK = 32, respectively, yet neither generate the effective symmetrization of error along all three decoherence channels achieved byRGA 16a 0. Effective dynamics forRGA 16b 0 andRGA 32c are essentially described by H RGA 16a 0 err 8i d p c x [B 0 ;B x 2 B z ] + y [B 0 ;B y 2 B z ] + z [B 0 ;B z + 2 B y ] (4.33) with an additional error term ofO(J d p ) along one of the three decoherence chan- nels. The final sequence described above, RGA 64c , produces a similar effective error Hamiltonian toRGA 16a 0, H RGA 64c err 32i d p c x [B 0 ;B x + 2 B y ] + y [B 0 ;B y 2 B z ] + z [B 0 ;B z + 2 B y ] ; (4.34) and further suppresses errors ofO(J 2 2 p ) to overtakeRGA 16a 0 as an optimal sequence forK = 64; 256 in the system-environment interaction-dominant (J >) regime. A summary of the performance scaling equations for all optimal sequences discussed above is presented in Table 4.2. Note that first order error suppression, in p , is achieved for a majority of K opt . However, we are only able to demonstrate the reduction of second order decoherence operators, such as the suppression ofO(J 2 d p ) terms for certain cases, and not complete suppression ofO( p d ) orO( 2 p ) terms. This result is consistent with DD no-go theorems which prove that it is not possible to suppress decoherence operators that are manifested by the second order perturbation expansion for the pulse error evolution operator, i.e.O( p d ) andO( 2 p ) terms, when rectangular pulse profiles are utilized [113, 114]. This analysis is consistent with these theorems 106 Sequence p d p d Name Description RGA 4 P 2 P 1 P 2 P 1 O(J 2 d ;J 2 2 d ) O(J p ) RGA 4 0 P 2 P 1 P 2 P 1 O(J 2 d ;J 2 2 d ) O(J p ) RGA 8a 0 IP 1 P 2 P 1 IP 1 P 2 P 1 O(J d p ;J 2 d p ) O(J 2 p ;J 2 2 p ) RGA 8b RGA 2 [RGA 4 ] O(J p ) O(J p ) RGA 8c P 1 P 2 P 1 P 2 P 2 P 1 P 2 P 1 O(J 2 d ;J 2 2 d ) O(J 2 p ;J 2 2 p ) RGA 16a 0 P 3 (RGA 8a 0)P 3 (RGA 8a 0) O(J d p ) O(J d p ;J 2 2 p ) RGA 16b 0 RGA 4 [RGA 4 0] O(J d p ) O(J d p ;J 2 2 p ) RGA 32c RGA 8c [RGA 4 ] O(J d p ) O(J 2 p ) RGA 64c RGA 8c [RGA 8c ] O(J d p ) O(J 2 p ) RGA 256c RGA 4 [RGA 64c ] O(J p ) O(J p ) Table 4.2: Summary of distance measure (D Kos ) scalings for each optimal RGA K sequence located by our search algorithm, for DD evolution subjected to finite-width rectangular pulses of duration p pulses and pulse-interval d . Optimal performance scalings for eachK opt are boxed for each parameter regime (column). and further conveys the need to utilize pulse shaping techniques in conjunction with optimal sequence construction to achieve high order error suppression in the presence of finite-width pulses. Indeed, when liberated from the constraint of rectangular pulse profiles, pulse sequences using DCG and CDCG [110, 111, 112] may be employed when pulse-width errors are the dominant concern. Characterization ofRGA K Sequences in ( d ; p )-space In the previous section, finite-width pulse error-optimizedRGA K sequences were iden- tified for various values ofK. In Figure 4.3, these results are summarized using numer- ical simulations to characterize the regions of optimal performance for each sequence as a function ofJ= and p = d . All results are averaged over 10 random realizations of 107 (a)K = 4 (b)K = 8 (c)K = 16 (d)K = 32 (e)K = 64 (f)K = 256 Figure 4.3: Performance of optimal RGA K sequences for K = 4; 8; 16; 32; 64; 256 shown in (a)-(f), respectively, as a function ofJ= and p = d when DD is subjected to finite pulse duration. The norm of the bath Hamiltonian is fixed at = 1kHz, while J= 2 [10 6 ; 10 6 ]. The pulse-interval d = 0:1ns and the pulse width is varied in the range p = d 2 [10 3 ; 10 6 ]. For a given K, the optimal sequence configuration is most sensitively dependent upon variations inJ. Contrary to the ideal pulse case, con- catenated structures composed ofRGA 8a 0 andRGA 8c appear to be the most favorable, in particular for K 16 where RGA 16a 0 and RGA 64c repeatedly emerge as optimal sequences. Sequence performance saturates at K = 16, while robustness begins to diminish at K = 256. nRGA 16b 0 denotes n cycles of RGA 16b 0; n = 1; 4; 16 for K = 16; 64; 256 respectively. The notation is similar formRGA 64c , wherem = 1; 4 cycles are used forK = 64; 256. H 0 with fixed d = 0:1ns and = 1kHz. The system-environment interaction strength is varied within the rangeJ=2 [10 6 ; 10 6 ] and p = d 2 [10 3 ; 10 6 ]. Variations in the regions of optimal performance are primarily dependent upon the value ofJ for a givenK. Sequences which obtain a favorable performance scaling for a givenJ tend to maintain their dominance throughout a wide range of p values extend- ing from the strong pulse to pulse-width error dominant regimes. As a function ofK, 108 optimal performance eventually saturates atK = 16 whereRGA 16a 0 maintains regions of optimal performance withinK = 32; 64; 256 forJ <. BeyondK = 64,RGA 16a 0 is accompanied byRGA 64c , which maintains its region of optimal performance within J > forK = 64; 256. Saturation in optimal sequence configuration and performance clearly agrees with the results of DD no-go theorems related to the achievable order of error suppression for finite-width pulse errors generated by rectangular pulse profiles. 4.2.3 Flip-angle errors An additional form of pulse error considered is that of a flip-angle error. The control pulse setD =fX;Y;Z; X; Y; Zg , with the pulse profile defined in Eq. (2.30). The resulting unitary pulse operators are given by X(Y;Z) =e i=2 (1+) x(y;z) (4.35) andf X; Y; Zg = fX y ;Y y ;Z y g. The analysis is symmetric with respect to over or under-rotations, therefore, our focus on over-rotations does not result in a loss of gener- ality. Summary of Numerical Search In contrast to the finite-width analysis, where eventual saturation in performance was observed, flip-angle error-optimized sequences exhibit an increase in overall decoupling order for a majority of theK opt values. Therefore, manipulation of sequence configu- ration is sufficient for acquiring robustness to this particular type of pulse imperfection. Advanced pulse shaping techniques may aid in the suppression of additional errors; however, as will be display below, sequence manipulation alone produces a surprisingly high decoupling efficiency. 109 Robustness against flip-angle errors is completely characterized by the a-type RGA K sequences. For K = 4, 2RGA 2 , RGA 2 = PU f ( d )PU f ( d ), is predom- inately the optimal choice, although RGA 4 := P 2 P 1 P 2 P 1 (4.36) does appear optimal when J . The dominance of RGA 2 follows from itsO( 2 ) decoupling, shown by H RGA 2 err x B x 2 ( y B z z B y ) + 2 2 4 ( y B y + z B z ): (4.37) RGA 4 does not achieve such a decoupling, H RGA 4 err 2 2 8 d z 2 z B xy + 2 2 4 [ x B xy + y (B y 2B x )]; (4.38) and, therefore, is not the preferred optimal sequence forK = 4 until the effects of flip- angle errors are negligible compared to the errors generated by free evolution (J d ). The sole optimal sequence forK = 8 is identified as RGA 8a :=I P 1 P 2 P 1 IP 1 P 2 P 1 : (4.39) The structure is similar toGA 8a , differing only by pulse phases, and identical to a time- symmetrized version ofRGA 4 sequence, namelyRGA 4 RGA 4 . Time-symmetrization has long been known to be beneficial for DD sequence construction since all odd-order terms in the effective error Hamiltonian are averaged out [54, 115], even in the case of pulse errors [67]. The effect of symmetrization is apparent within H RGA 8a 2 z B xy + 2 2 4 y B xy 2 2 4 y (2B x B y ); (4.40) 110 where the dominant error term scales asO(J d ). Comparing RGA 8a to RGA 2 , the primary difference is the second order suppression ofO(J d ) terms acquired byRGA 8a . WhileRGA 2 achieves a similar robustness against flip-angle errors, it fails to address errors created by free evolution. In the case ofK = 16; 32, all optimal sequences are characterized in terms of RGA 16a := P 3 (RGA 8a )P 3 (RGA 8a ): (4.41) As one may notice from the definition of the sequence given in Table 4.4, there is some freedom in the choice of theP 3 pulse. Considering the usual case offP 1 ;P 2 g =fX;Yg discussed so far, the following effective error Hamiltonians emerge for eachP 3 6=I: H RGA 16a err;P 3 =X 2 2 4 [ x B xy y B xy ]; (4.42) H RGA 16a err;P 3 =Y 2 2 4 [ x B xy y (2B x B y )]; (4.43) H RGA 16a err;P 3 =Z 2 z B xy : (4.44) Interestingly, the decoupling order for terms proportional to isP 3 -dependent. Choos- ingP 3 to be orthogonal toP 1 ;P 2 is clearly the least favorable choice, withP 3 =P 1 being the optimal choice, most notably in the case of uniform decoherence in the xy-plane. From the effective error Hamiltonians above, we find that optimalRGA 16a performance is determined byO( 2 J d ) terms. Additional sequence structures such asRGA 32a do not achieve similar performance and, in fact, suffer from the presence ofO( 2 ) terms. The decoupling order again increases atK = 64, where RGA 64a :=RGA 8a [RGA 8a ] (4.45) 111 Sequence J d J d Name Description RGA 2 PP O(J d ) O(J d ) RGA 4 P 2 P 1 P 2 P 1 O(J 2 d ;J 2 2 d ) O( 2 ) RGA 8a I P 1 P 2 P 1 IP 1 P 2 P 1 O(J d ) O(J d ) RGA 16a P 3 (RGA 8a )P 3 (RGA 8a ) O( 2 J d ) O( 2 J d ) RGA 32a RGA 4 [RGA 8a ] O( 2 ) O( 2 ) RGA 64a RGA 8a [RGA 8a ] O( 3 J d ) O( 3 J d ) RGA 256a RGA 4 [RGA 64a ] O( 2 ) O( 2 ) Table 4.3: Summary of distance measure D Kos scalings for each optimal RGA K sequence located for DD pulses subjected to flip-angle errors with rotation error, with fixed pulse-interval d . Boxed performance scalings highlight optimal performance scal- ing for variousK opt . attains suppression of all errors up toO( 3 J d ) terms, confirmed by H RGA 64a 3 3 4 z B y 3 3 3 8 z B x : (4.46) ComparingRGA 8a and its first level concatenation, RGA 64a , it is found that an addi- tional two orders of error suppression are achieved simply by a single level concate- nation. An obvious question that arises from this result is whether additional orders of decoupling, and possibly arbitrary order decoupling, is attainable by continuing the concatenation procedure. While the question of arbitrary order decoupling will not be addressed here, the results presented here suggest that such a scheme exists due to the O( 5 J d ) scaling acquired byRGA 512a =RGA 8a [RGA 8a [RGA 8a ]]. The final sequence length considered, K = 256, is the first instance of com- plete breakdown in performance. Optimal sequences consist of cycles ofRGA 16a and 112 RGA 64a with various regions where free evolution reigns supreme. More sophisticated sequence structures, such as RGA 256a :=RGA 4 [RGA 64a ] (4.47) are not found to be optimal due to remainingO( 2 ) terms. In Table 4.3, we display the performance scaling forK = 256 along with the scalings for the remaining values of K opt discussed above. In summary, the results presented for K=K opt in the presence of flip-angle errors demonstrate that successive error suppression is achievable by concatenation only if the outer sequence maintains the same decoupling order as the inner sequence(s). Supplying a lower order decoupling outer sequence ultimately leads to an effective error Hamilto- nian that possesses dominant error terms that are intrinsic to the low-order sequence. Provided this condition is satisfied flip-angle error-optimized sequences exhibit a con- tinual increase in decoupling order with an increasing number of pulses. Uninhibited, due to the absence of no-go theorems for flip-angle errors, we believe that extending the search beyondK = 256 will result in additional sequence configurations that uti- lize concatenations ofRGA 8a , or even a more robust construction such asRGA 16a , to achieve higher order decoupling. This conclusion is supported by the increase in decou- pling order found for RGA 512a , which suggests that arbitrary order error suppression using` concatenations ofRGA 8a can be used to achieveD Kos O( 2`1 J d ). Characterization ofRGA K Sequences in (;J d )-space In this section, the results obtained for the flip-angle error-optimized sequence search are illustrated using numerical simulations to correctly identify the regions of optimal performance as a function ofJ= and [see Fig. 4.4]. The pulse delay and the strength 113 (a)K = 4 (b)K = 8 (c)K = 16 (d)K = 32 (e)K = 64 (f)K = 256 Figure 4.4: Performance ofRGA K sequences forK = 4; 8; 16; 32; 64; 256, as shown in (a)-(f), respectively, as a function of J= and . The minimum pulse interval is fixed at d = 0:1ns, J=2 [10 6 ; 10 6 ], and = 1kHz, while is varied from one to twenty percent rotation error. Results are averaged over 10 realizations ofB . Sequence performance mostly increases fromK = 4 toK = 64, indicating a reduction in the error terms proportional to in the effective error Hamiltonian. Successive error suppression is achieved for K = 4; 8; 16; 64, where the maximum error suppression yields D O( 3 J d ). for RGA 64a . Multiple cycles of RGA 16a and RGA 64a appear as optimal sequences for variousK opt . The number of cycles for each sequence is given as follows: (d)n = 2, (e)m = 1, and (f)n = 16,m = 4. of the bath dynamics are fixed at d = 0:1ns and = 1kHz, respectively. The strength of the system-environment interaction is varied within the rangeJ=2 [10 6 ; 10 6 ] and the flip-angle error2 [0; 0:2], corresponding to a 0% to 20% error in pulse rotation. All results are averaged over 10 random realizations of the bath operatorsB . In contrast to the results obtained for finite-width pulse errors, optimal sequence configuration and performance generally increases as a function ofK, up toK = 64, indicating an increase in the suppression of error terms proportional to . Saturation 114 in performance is first observed atK = 32, where two cycles ofRGA 16a is the opti- mal configuration8J; considered. This effect is actually quite brief, as an increase in error suppression returns atK = 64 viaRGA 64a . The most significant attenuation in performance is found forK = 256. Optimal sequence configurations either consist of complete free evolution or cycles of previously located sequences. As discussed in the previous section, it is expected that additional increases in performance using more sophisticated sequencs beyondK = 256, perhaps most obviously for concatenations of RGA 8a . This analysis is left for future studies. 4.2.4 Finite-Width and Flip-Angle Errors Flip-angle and finite-width pulse errors are prevalent in a variety of experimental set- tings. Therefore, it is necessary for DD sequences to be robust against both types of errors simultaneously if reliable computation is to be implemented under the protection of DD in realistic setups. Designs based on (C)DCG [110, 111, 112] do not apply in this case as they do not address flip-angle errors. The KDD sequence is applicable, but unlike the present study, it employs pulses which are not uni-axial [84]. As a final consideration, we investigate the inclusion of the two forms of pulse errors and search for optimal sequences at each K opt . By performing this search, we are essen- tially addressing the possibility of constructing fault-tolerant DD in the most convenient possible arrangement: fixed pulse-interval and rectangular pulse shape. The control pulse setD again contains 7 possible pulses, where each pulse operator is defined by X(Y;Z) =e ip[A(1+) x(y;z) +H] (4.48) 115 andf X; Y; Zg correspond to 180-degree phase rotations, i.e.A!A. The amplitudes of the error and bath Hamiltonians are chosen as J = 1MHz and = 1kHz, respec- tively. Optimal sequence performance is analyzed with respect to and p in the regime whereJ andJ d 1 to characterize sequence robustness as a function of errors generated by the DD pulses. Results of Numerical Search The first case to examine is that ofK = 4, whereRGA 4 [Eq. (4.36)] and 2RGA 2 are found to be optimal. It is perhaps not surprising that a robust version ofGA 4 appears as an optimal sequence given the results of the ideal pulse analysis. The more interesting result is the emergence of 2RGA 2 as an optimal four-pulse sequence since it does not provide complete first order error suppression. Its presence is clearly attributed to its robustness against pulse imperfections, rather than errors generated by free evolution. Determining which form of pulse error is addressed most effectively byRGA 2 , and RGA 4 for that matter, is best accomplished via direct calculation of the effective error Hamiltonian. In the case ofRGA 4 , the effective dynamics are governed by H RGA 4 err 2 2 2 c z 4 p c z B xy +O(J p ;J d ); (4.49) withB xy B x B y , while forRGA 2 , H RGA 2 err x B x 4 p c (1) z B y +O(J p ;J d ): (4.50) Here, the pulses are taken asfP 1 ;P 2 g =fX;Yg and P = X for RGA 4 and RGA 2 , respectively. Note thatRGA 4 produces first order decoupling in p and, yet does not effectively address errors generated by the finite pulse duration. The effective error Hamiltonian for RGA 2 confirms the presence ofO(J d ) terms and also displays a 116 similar lack of first order decoupling in p . The primary distinction between the two sequences is the suppression ofO( 2 ) errors provided solely by RGA 2 . In summary, the regimes of optimal performance for each sequence can be characterized as follows: RGA 4 is most advantageous when the pulse imperfections are relatively small and the free evolution periods ascribe to the primary source of decoherence (J d > J p and J p > ), whileRGA 2 is most effective when flip-angle errors are dominant ( > J p and>J d ). Continuing the search toK = 8, it is found that all 8-pulse optimal sequences which exhibit robustness to both finite pulse-width and flip-angle errors can be described by RGA 8a andRGA 8c . Again, a correspondence betweenRGA K andGA K sequences is observed withRGA 8a , as it is essentiallyGA 8a with-phase adjustments required for additional robustness against pulse imperfections. The latter sequence did not appear as an optimal sequence in the ideal pulse analysis; however, it is identified as a generalized version of the well-known Eulerian DD (EDD) sequence which is known for producing first order decoupling in the pulse duration [109]. The effective error Hamiltonian forRGA 8a , H RGA 8a err 4 p c [ x B z + z (B x 2B z )] 4 d c z B xy ; (4.51) displays an improvement in performance over both K = 4 optimal sequences in its ability to produce second order decoupling in both d and . Further improvement in robustness is observed forRGA 8c , where H RGA 8c err ( x + y ) 3 3 2 c 4 p c (B x +B y ) +O(J d ;J 2 2 d ) (4.52) confirms the first order decoupling in p expected from EDD sequences and, interest- ingly, displays a second order decoupling in as well. WhileRGA 8c appears to address 117 Sequence J p J p Name Description RGA 2 PP O(J d ) O(J p ) RGA 4 P 2 P 1 P 2 P 1 O(J 2 2 d ; 2 ) O(J 2 2 d ;J p ) RGA 4 0 P 2 P 1 P 2 P 1 N/A N/A RGA 8a I P 1 P 2 P 1 IP 1 P 2 P 1 O(J d ) O(J p ) RGA 8c P 1 P 2 P 1 P 2 P 2 P 1 P 2 P 1 O(J d ; 2 ) O(J 2 2 d ;J 2 d p ) RGA 16a P 3 (RGA 8a )P 3 (RGA 8a ) O(J p ) O(J p ) RGA 16b 00 RGA 4 0[RGA 4 0] O( 2 ) O(J p ) RGA 32a RGA 4 [RGA 8a ] O( 2 ) O(J p ) RGA 32c RGA 8c [RGA 4 ] O( 2 ) O(J p ) RGA 64a RGA 8a [RGA 8a ] O(J p ) O(J p ) RGA 64c RGA 8c [RGA 8c ] O( 2 ) O(J p ) RGA 256a RGA 4 [RGA 64a ] O( 2 ) O(J p ) Table 4.4: Summary of distance measureD Kos scalings for optimalRGA K sequences identified for DD evolution subjected to finite-width pulses of duration p and flip-angle errors with rotation error in the regime of system-environment interaction-dominated (J) dynamics. The sequences with the best performance scaling in each parameter regime (column) for each K opt are boxed. Note specifically the case of strong pulses dominated by flip-angle errors (J p ) whereRGA 8a ,RGA 16a , andRGA 64a obtain the more favorable performance scaling. pulse imperfections more effectively thatRGA 8a , its optimality is limited to the regime where p > d andJ p > . Below, we will illustrate this result by analyzing perfor- mance numerically as a function of and p . The remaining optimal sequences obtained forK = 16; 32; 64; 256 do not appear to offer any additional robustness. Each sequence is either robust against finite pulse-width or flip-angle errors, but not both forms of pulse errors simultaneously. The effective error Hamiltonians are generally defined as H a err p c X ; B +O(J p ;J d ) (4.53) 118 fora-type sequences and H c err 2 c X +O(J p ;J d ) (4.54) for c-type sequences, with the worse possible case of H b err = H a err + H c err appearing primarily forb-type sequences. Table 4.4 presents all remaining sequences located, and outlines the scaling of the effective error Hamiltonians for each sequence. In summary, the results presented here essentially indicate that attaining robustness for a wide range of p , d , and solely by manipulating the sequence configuration is insufficient. However, this does not invalidate the sequences obtained here since if a particular parameter regime is achievable, namely, the strong-pulse regime where flip- angle errors are dominant (J p J d ), sequences such asRGA 8a ,RGA 16a and RGA 64a still exhibit a high level of robustness. This is not only evident from their ability to suppress errors solely proportional to and d [see 4.2.3 for a discussion of RGA K excusively for flip-angle errors], but also from the fact that error accumulation forO(J p ) terms is not observed as the number of pulses is increased. Therefore, if the pulse duration can be made small relative to the pulse-interval then the finite-width errors are less consequential and it is still possible to maintain some form of robust- ness without the need for additional techniques. When such a regime is not attainable it may be necessary to utilize pulse shaping techniques to aid flip-angle error robust sequences in the suppression of finite-width pulses, or to exploit composite pulses to suppress flip-angle errors for sequences highly robust to finite-width pulses. Ultimately, a combination of sequence configuration, pulse shaping, and composite pulses is most likely the path forward to constructing fault-tolerant DD sequences for a wide range of parameter regimes. 119 (a)K = 4 (b)K = 8 (c)K = 16 (d)K = 32 (e)K = 64 (f)K = 256 Figure 4.5: Performance ofRGA K sequences forK = 4; 8; 16; 32; 64; 256, as shown in (a)-(f), respectively, as a function of and p = d . The minimum pulse-interval is fixed at d = 0:1ns, whileJ = 1MHz and = 1kHz. The flip-angle error is varied from a 1% to a 10% error and p is varied throughout a wide range of values so that p d , and p d , is explored. The majority of the parameter space is dominated by theRGA Ka sequences, even in the finite-width error-dominant regime. The robustness ofRGA Ka to pulse imperfections ultimately saturates at K = 64. All simulations are averaged over 10 realizations ofB . The optimal sequences that are used in multipleK opt values require the following number of cycles: (a)m = 2, (c)n = 1, (d)n = 2, (e)p =q = 1, (f)p =q = 4. Characterization ofRGA K Sequences in (;J p )-space In Figure 4.5, the regions of optimal performance for theRGA K sequences are charac- terized with respect to magnitude of the flip-angle error and the ratio of the pulse dura- tion p to pulse delay d . For eachK, an evident partitioning in the space is observed depending on the form of the prevailing pulse error indicating that it is not possible to combat both forms of pulse imperfections simultaneously by solely manipulating sequence configuration. The a-type sequences, along with RGA 2 , are more effective 120 against flip-angle errors and offer increasing performance for K = 16; 32; 64; 256 as p = d ! 0. Upon reaching p = d , the optimal sequence structure is highly dependent upon the relationship betweenJ p and. Sequences of a-type continue to be the pre- ferred structure for > J p , while b-type and additionally defined (c-type) sequences are optimal when < J p . These result are essentially illustrations of the analysis given in the previous section where sequence effectiveness is described in terms of the effective error Hamiltonian. 4.2.5 Comparison with Known Deterministic Schemes In this section,GA K andRGA K optimal sequences are compared to CDD and QDD, for each pulse profile. The CDD and QDD sequences are constructed as described in Sec. 4.1.1 and Sec. 4.1.2 withf 1 ; 2 g =fX;Zg, respectively. The notated CDD ` will be utilized to denote a CDD sequence of` concatenation levels, while QDD M denotes a QDD sequence withM =M 1 =M 2 pulses on each nested level. CDD represents a fair comparison to the numerically optimal sequences since in both cases the pulse-intervals are fixed and the error suppression properties are dictated only by the sequence structure. Schemes which rely on optimized pulse delays to gain decoupling efficiency, such as QDD, have much better scaling and error suppression properties than fixed delay schemes in the ideal pulse limit. However, theGA K optimal sequences can be expected to prevail in the case of non-ideal pulse profiles, where no robust version of QDD currently exists. Ideal Pulses In Figure 4.6(a), the performance of CDD ` to GA K are compared with respect to J d 2 [10 12 ; 10 2 ] forJ = 1MHz and = 1kHz in the ideal pulse limit. Numerically optimal sequences first coincide with CDD ` at K = 16, where both achieve second 121 (a) (b) Figure 4.6: Comparison of performance for (a) CDD (empty symbols) and (b) QDD (empty symbols) versusGA K (filled symbols) as a function of the pulse-interval d in the ideal pulse limit. The strength of the error Hamiltonian is chosen asJ = 1MHz and the strength of the pure-environment dynamics = 1kHz. Optimal GA sequences achieve a higher of order error suppression than CDD as the number of pulses increases; note GA 64a and GA 256a as compared to CDD 3 and CDD 4 , respectively. In contrast, QDD outperformsGA K for all sequence lengths, consistent with the expected superiority of interval-optimized schemes in the ideal pulse limit. order error suppression. The main advantage ofGA 16a over CDD 2 is a reduction in the error amplitude by a factor of approximately 10 3 . Significant improvement in error sup- pression is observed forK = 64 andK = 256, where GA optimal sequences offer an additional order of error suppression over corresponding CDD sequences,r = 3; 4. The results indicate that CDD does not constitute an optimal deterministic sequence structure for fixed pulse delays. In Sec. 4.2.6, this is fact is elaborated upon and the possibility of designing a deterministic scheme to correctly describe the optimal GA sequences is explored. Pulse interval optimized sequences have been shown to far surpass the decoupling efficiency of any known fixed pulse-interval scheme in the ideal pulse limit, requiring only a quadratic increase in the number of pulses to suppress an additional order of the Magnus expansion. The above statement is validated by comparing QDD M to the opti- mal GA sequences in Fig. 4.6(b) for M = 1; 3; 7; 15; an equivalent number of pulses 122 (a) (b) Figure 4.7: Comparison of performance betweenRGA K and (a) CDD ` , or (b) QDD M , when subjected to finite pulse duration. Performance is characterized as a function of p , while d = 0:1ns. CDD ` performance is essentially the same for all `, scaling as D Kos O(J p ). RGA K achieves a significant increase in robustness over CDD ` at K = 8a 0 ; 16a 0 ; 32c; 64c, where the performance surpasses the linear scaling in p . Note the eventual saturation in decoupling order characteristic of the rectangular pulse profile displayed by the nearly equivalent scaling of K = 32c; 64c. QDD M , M = 1; 3; 7; 15, performance becomes increasingly worse as the sequence order increases due to an accumulation in errors brought about by the finite duration of the pulses. As in the case of CDD ` , QDD M performance maintainsD Kos O(J p ) for allM. All results are averaged over 10 realizations ofB . Error bars are included, but are quite small. to K = 4; 16; 64; 256, respectively. For each set of sequence ordersfM;Mg QDD M attains a decoupling order ofM, which is beyond the ability of the GA K for an equiva- lent number of pulses. QDD M superiority is an indication that optimizing with respect to pulse-interval, in addition to pulse configuration, is necessary to obtain higher decou- pling efficiency. However, these results depend heavily on the ideal pulse limit. As will be shown in the next subsection, the story is very different when finite width and flip-angle error effects are accounted for. 123 Finite-width Pulses In Fig. 4.7(a) and (b), CDD ` and QDD M , respectively, are compared to the RGA K sequences optimized for finite pulse width:K = 4; 8a 0 ; 16a 0 ; 32c; 64c; 256c. The pulse- interval is chosen as d = 0:1ns, while p = d 2 [10 5 ; 10 7 ],J = 1MHz, and = 1kHz. All results are averaged over 10 random instances of the Hamiltonian. Optimal performance forRGA K is observed predominately forK = 32c; 64c with K = 16a 0 exhibiting a more favorable performance only in the strong-pulse regime, namely when p = d < 10 2 in Fig. 4.7. As the finite-width pulse errors contribute more substantially,O(J 2 2 p ) terms remaining in the effective error Hamiltonian forK = 16c results in a rapid decrease in performance leading toRGA 32c =RGA 64c -dominance. K = 4 maintains the lowest performance of allRGA K sequences due to its inability to suppress the first order contribution in p ,D Kos O(J p ). CDD r performance is nearly equivalent toK = 4 for p = d 10 4 , scaling asD Kos O(J p ) for all`. The most noticeable difference occurs at p = d < 10 4 , where CDD ` maintains the linear scaling in p forr = 2; 3; 4 and surpassesK = 4; 8c. In Ref. [76], an analysis of CDD r in the presence of finite pulse width is discussed as well. There it was shown that CDD ` can reduce pulse-width errors as the concatenation level increases if p d . Although the total cycle time was fixed, as opposed to the pulse-interval, the results obtained here are quite similar in the p = d < 10 4 regime and confirm the inherent robustness of CDD ` to finite-width pulse errors. As discussed in Refs. [116], UDD-based schemes are quite susceptible to finite pulse-width errors and must be implemented with specially tailored pulses to regain a portion of the UDD decoupling efficiency. This result is confirmed here for the most simplistic pulse shape: the rectangular pulse. Increasing the sequence order does not result in an increase, or sustainability, of performance; rather an accumulation of error results. As in the case of CDD ` , the performance maintains a linear scaling in p 124 (a) (b) Figure 4.8: Performance of RGA K sequences versus (a) CDD ` and (b) QDD M as a function of flip-angle error 2 [0:01; 0:2] averaged over 20 realizations of B . The relevant parameters are chosen asJ = 1MHz, = 1KHz, and d = 0:1ns. Numerically optimal sequences are found to be highly robust against flip-angle errors, significantly outperforming CDD r forr = 1; 2; 3; 4. For QDD M , the sequence orders are chosen as M = 3; 7; 15 and directly correspond toK = 16; 64; 256. QDD is shown to be highly sensitive to flip-angle errors, decreasing in performance asM grows. Again, robust GA sequences achieve optimal performance. throughout the specified range for the higher of the three sequence orders:M = 3; 7; 15. The only variation occurs atM = 1 where the performance becomes dependent upon d for p d . Although there exists a regime where both deterministic schemes out- perform K = 4; 8c, higher order RGA K sequences provide a level of robustness that cannot be matched by either CDD ` or QDD M . Flip-Angle Errors In 4.2.3, robust sequences were identified for control pulses subjected to flip-angle errors. Here, the numerically optimalRGA K sequences to CDD ` and QDD M are com- pared as a function of . The case of interaction-dominated dynamics are considered and the strengths of the environment dynamics and system-bath interaction are set to = 1kHz and J = 1MHz, respectively. The pulse delay is chosen as d = 0:1ns and all data is averaged over 20 realizations of the bath operatorsfB g. In addition to 125 K = 2; 8a; 16a; 64a, which exhibit an increase in decoupling order for terms propor- tional to,K = 4; 32a; 256a are included in the comparison as well to fully characterize RGA K performance with respect to both deterministic schemes. First focus on RGA K and CDD ` , where RGA K superiority is clearly evident for all values of K shown in Fig. 4.8(a). Optimal performance is observed for K = 2; 8a; 16a; 64a, as expected, with K = 64c providing the highest level of robustness to flip-angle errors using the smallest number of pulses. Although the lowest level of performance forRGA K occurs atK = 4; 32a; 256a, a considerable improvement over the corresponding CDD ` ,` = 1; 2; 3; 4, is seen. Optimal CDD ` performance is achieved at ` = 2; 4, where the performance can be shown to scale as D Kos O( 2 ). The remaining levels of concatenation,` = 1; 3 do not achieve first order suppression, there- fore, D Kos O(). The comparison clearly indicates that the numerically optimized sequences are highly robust to flip-angle errors and capable of dramatically outperform- ing CDD ` . Analyzing QDD M , forM = 1; 3; 7; 15, as a function of, it is found that the perfor- mance maintains aD Kos O( 2 ) scaling for allM; see Fig. 4.8(b). The performance for QDD M diminishes with increasingM, indicating an accumulation of error rather than a reduction, or sustainability as in the case of CDD ` . Although variations in performance exist between the deterministic schemes, their robustness to flip-angle errors is clearly not comparable to the optimizedRGA K sequences. Finite-width and Flip-angle Errors As a final comparison, the performance and robustness ofRGA K , CDD ` , and QDD M in the presence of both finite-width and flip-angle pulse errors is examined. In particular, the performance as a function of and p relative to the size of the pulse-interval d = 0:1ns is the focus. The flip-angle error is varied from zero to a 15% rotation error, the 126 Figure 4.9: Performance ofRGA K , CDD ` , and QDD M when subjected to finite pulse duration and flip-angle errors. The pulse-interval is fixed at d = 0:1ns, while J = 1MHz and = 1kHz. RGA K sequences significantly outperform both CDD ` and QDD M forK = 4; 16; 64; 256. The most notable region of robustness exists for< 0:04 and p < d forK = 16; 64; 256. ratio of pulse duration to interpulse delay p = d 2 [10 5 ; 10 4 ], and J = 1MHz with = 1kHz. All simulations are average over 10 realizations ofB . The performance of RGA K is shown in Fig. 4.9 in panels (a)-(d) for K = 4; 16a; 64a; 256a, respectively. Although additional optimal configurations were identi- fied at other sequence lengths, we focus on these particular values ofK to compare them directly to CDD ` , ` = 1; 2; 3; 4, and QDD M , M = 1; 3; 7; 15. RGA K performance is found to be primarily dependent upon p , only showing significant-dependence when 0:04. Below = 0:04, specifically in the region where p < d , robustness increases as the number of pulses is increased fromK = 16a toK = 256a. Error accumulation 127 within this particular range of values appears not to be an issue. CDD ` performance is displayed in panels (e)-(h) of Fig. 4.9 for ` = 1; 2; 3; 4, respectively. In contrast toRGA K , CDD r performance exhibits a strong dependence on flip-angle errors rather than finite pulse duration. Optimal performance is heavily concentrated around small values of due to the low level of robustness exhibited by CDD ` for flip-angle errors. Robustness to finite pulse duration appears to be most noticeable forr = 2; 4, although performance is still rather poor compared toRGA K . Lastly, QDD M is examined in panels (i)-(l) in Fig. 4.9, where M = 1; 3; 7; 15, respectively. The lowest sequence order, M = 1, generates the exact same sequence asRGA 4 , therefore, performance is identical. The remaining sequence orders result in continual error accumulation, which is evident from the steady decline in performance from M = 3 to M = 15. As in the case of CDD r , QDD M performance is primarily -dependent and degrades rapidly with increasing . Similarly to the additional com- parisons of numerically optimized sequences and deterministic schemes for faulty DD pulses given above,RGA K sequences significantly outperform CDD ` and QDD M . 4.2.6 Existence of Deterministic Structure Concatenation appears to play an important role in the construction of optimal DD in the case of fixed pulse-intervals. One of the goals of this study is to determine whether the optimal sequences provided above can be generalized into a deterministic concatenation scheme for arbitrary order decoupling. As suggested by the ideal pulse analysis summa- rized in Table 4.1, such a scheme is possible by utilizingGA 8a as the fundamental unit of concatenation in GA (q) 8a =GA 8a [GA (q1) 8a ]; (4.55) 128 à à à à æ æ æ æ æ æ ò ò òòòòò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò 1 2 3 4 5 6 -40 -30 -20 -10 0 log 4 K log 10 D Kos ò GA 4 æ CDD l à GA 8 a HqL Figure 4.10: Comparison ofGA (q) 8a , CDD ` , andGA 4 performance after one cycle as a function of the number of pulses K for ideal, zero-width pulses. The strength of the error Hamiltonian and environment dynamics are set to J = 1MHz and = 1kHz, respectively, and the minimum pulse-interval d = 0:1ns. Results are averaged over 25 random realizations of B , where the error bars are shown, but quite small. As expected by the results of the GA search GA (q) 8a , q = 1;:::; 4 is indeed superior to CDD ` ,` = 1;:::; 6, andXY 4 whereGA (0) 8a f d . Requiring 8 q pulses,GA (q) 8a achieves 2qth order error suppression; a quadratic improvement over the decoupling efficiency of CDD, which requires 4 2q pulses to achieve an equivalent decoupling order. The increased decoupling efficiency is facilitated by second order error suppression provided by GA 8a , which essentially boosts the efficiency by a factor of 2 at each level of concatenation. In Fig. 4.10, the performance of GA (q) 8a is compared to that of CDD ` and GA 4 in the case of zero-width pulses as a function of the number of pulsesK. The results are averaged over 25 random realizations ofB , where the pulse operators are designated byfP 1 ;P 2 g =fX;Yg for each generalized sequence. The strengths of the error Hamil- tonian and bath dynamics are given byJ = 1MHz and = 1kHz, respectively, and the minimum pulse delay d = 0:1ns. In comparingq = 1; 2;:::; 4 andl = 1; 2;:::; 6, the 129 performance ofGA (q) 8a improves dramatically as the level of concatenation increases, far exceeding that of CDD ` andGA 4 . However, the truly meaningful test of sequence performance is in the presence of pulse errors. Defining RGA (q) 8a =RGA 8a [RGA (q1) 8a ]; (4.56) to effectively combat the inclusion of flip-angle and finite-width pulse errors, the perfor- mance ofRGA (q) 8a is compared to CDD ` andRGA 4 in Fig. 4.11 forfP 1 ;P 2 g =fY;Xg. The relevant Hamiltonian parameters are equivalent to those chosen for the ideal case, while the flip-angle error = 0:01. As opposed to fixing d , a fixed cycle time c is selected to analyze the relationship between p and d as the number of pulses grows. In particular, c = 1ns and p = c = 10 10 . Robustness is expected to be most noticeable in the strong pulse regime, p d , where the primary form of pulse error is due to the flip-angle errors. Performing the analysis with a fixed cycle time allows us to examine robustness as a function of concatenation level and as p ! d , simultaneously. As expected by direct calculation of the effective Hamiltonian forRGA 4 [see Eq. (4.49)], increasing the number of pulses via multiple DD cycles does not offer an enhance- ment in sequence performance due to an immediate accumulation of error proportional to p and 2 . In contrast, CDD ` performance remains fairly consistent and oscillates between two values, in accordance with the results of Sec. 4.1.1. Combined errors are most effectively addressed byRGA (q) 8a , as can be seen by the improvements in sequence performance as concatenation level increases. Note that the performance eventually begins to show signs of saturation as the concatenation level increases. Essentially, the pulse-interval is approaching a value comparable to the pulse duration which leads to finite-width errors becoming a more significant decoherence mechanism than flip-angle errors or the error Hamiltonian. RGA 8a and its concatenated versions do not provide 130 à à à à æ æ æ æ æ æ ò ò ò òòòò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò 1 2 3 4 5 6 -12 -10 -8 -6 -4 -2 0 log 4 K log 10 D Kos ò RGA 4 æ CDD l à RGA 8 a HqL Figure 4.11: Comparison ofRGA (q) 8a , CDD ` , and RGA 4 performance versus the num- ber of pulsesK for combined pulse errors (flip-angle and finite-width) after one cycle. Hamiltonian parameters,J and, are the same as those in Fig. 4.10 with the total cycle time fixed at c = 1ns, as opposed to d . Results are averaged over 25 realizations ofB with = 0:01 and p = c = 10 10 . The performance ofRGA (q) 8a improves as the number of pulses is increased within a given cycle time c . For the specified parameters, CDD ` does not exhibit enhanced performance with increasing concatenation level. protection against finite-width errors, and therefore their performance becomes hindered by the presence of terms that are first order in p . 4.2.7 Summary of Results In this work it has been showed that numerically optimal DD sequences can be con- structed using a genetic algorithm in conjunction with a simulated annealing conver- gence accelerator and a novel complexity-reduction technique. The search focused on sequences containingK = 1; 2;:::; 256 pulses, however, optimal performance is iden- tified atK = 4; 8; 16; 32; 64; 256 and compared each sequence to known deterministic schemes, such as CDD and QDD. The ideal-pulse analysis showed that, under the con- straint of a fixed pulse interval, optimal sequences can be constructed which outperform CDD, yet fall short of the decoupling efficiency realized by QDD. Optimization proved 131 to be quite beneficial in the case of finite-width and flip-angle errors, where numerically optimal sequences obtained a level of robustness that could not be reached by either CDD or QDD. The culmination of this study is centered around the identification of a determinis- tic sequence structure that obtains a high decoupling efficiency in the ideal limit and robustness to errors generated by pulse imperfections. It is determined that RGA 8a [Eq. (4.56)] is the favored generating sequence for a majority of the pulse profiles con- sidered. Concatenating this particular sequence, it is possible to suppress the first 2q terms in the Magnus expansion using 8 q pulses. Compared to the 4 q pulses required for the widely used original version of CDD [117], the concatenated version ofRGA 8a utilizes quadratically fewer pulses to obtain the same decoupling order. Although it is not possible to obtain such high levels of decoupling in the presence of pulse imperfec- tions,RGA 8a contains an inherent robustness that continues to aid the error suppression process as the level of concatenation increases. This result is most apparent in our final study of faulty DD pulses, which includes both finite duration and flip-angle errors; we found the concatenatedRGA 8a construction to be the most robust scheme available for fixed pulse-interval DD sequences. The importance of optimizing over pulse-intervals was clearly displayed in the ideal pulse analysis, where sequence configuration optimization alone could not supply the decoupling efficiency achieved by QDD. However, in agreement with previous work, it has been shown that pulse-interval optimized sequences fail to be robust against addi- tional errors generated by faulty DD pulses. 132 4.3 Symmetrized QDD for Control Error Robustness In light of the substantial diminishing error suppression of QDD in the presence of control errors, an immediate question arises: is it possible to robustize QDD in such a way to recover, or at least partially recover, the attractive decoupling efficiency of this scheme beyond the ideal pulse limit? Addressing such a question effectively can be accomplished by sequece symmetrization. Here a variety of methods for QDD symmetrization are investigated including the known approach of time-symmetrization where given a sequenceU DD () defined in terms of the minimum inter-pulse delay, a time-symmetrized version of this sequence is given by U sym DD () = U DD () T U DD () (4.57) where the bar denotes a 180-degree phase rotation of all pulses and the superscript T denotes the transpose of the sequence. 4.3.1 Symmetrizing Schemes for QDD Three different variations of QDD symmetrization are considered for this study. The first involves a symmetrization of the pulse phases for UDD, while the latter two employ time-symmetrization at either the UDD or QDD level. Details regarding the robust QDD constructions are discussed below. 133 Robust 4k-pulse QDD (R4QDD) One approach to possibly generating robustness to control errors in the setting of QDD is to symmetrize each nested UDD sequence with respect to the phases of the pulses according to U (M) R4UDD; U f () := M Y j=3M=4+1 U f ( (M) j ) 3M=4 Y j=M=4+1 U f ( (M) j ) M=4 Y j=1 U f ( (M) j ): (4.58) whereM is taken to be multiples of four in order to generate complete symmetrization. Essentially this scheme is designed in such a way so that the set of pulses betweenM=4 and 3M=4 are applied with 180-degree phase rotations of the original pulse operator. Nesting Eq. (4.58) accordingly, one obtains the candidate robust QDD sequence U (M 1 ;M 2 ) R4QDD;f 1 ; 2 g () :=U (M 2 ) R4UDD; 2 U (M 1 ) R4UDD; 1 U f (): (4.59) An attractive feature of this particular approach compared to the latter two discussed below is that it utilizes an equivalent number of pulses as the standard QDD scheme. Time-symmetrization requires twice as many pulses per sequence and therefore many result in as much as a factor of four increase in required pulses. Robust QDD (RQDD) Due to the nesting protocol of QDD it is possible to consider two variations of time symmetrization. The first is U (M 1 ;M 2 ) RQDD;f 1 ; 2 g := U (M 1 ;M 2 ) QDD;f 1 ; 2 g T U (M 1 ;M 2 ) QDD;f 1 ; 2 g (4.60) 134 where the QDD sequence itself is time symmetrized. The total number of pulses required to perform this particular evolution increases by an overall factor of two over the standard QDD definition. Note that in this construction errors are accumulated by the complete QDD sequence thereafter addressed by the second half of time-symmetrized sequence. Quadratic Robust UDD (QRUDD) The second form of time symmetrization that can be implemented on QDD is a time symmetrization of UDD U (M) RUDD; U f () := U (M) UDD; T U (M) UDD; (4.61) and then a subsequent nesting of the time-symmetrized sequences to obtain U (M 1 ;M 2 ) QRUDD;f 1 ; 2 g :=U (M 2 ) RUDD; 2 U (M 1 ) RUDD; 1 U f (); (4.62) a quadratic robust UDD (QRUDD) scheme. Since each time-symmetrized UDD sequence utilizes 2M pulses, the nesting process results in approximately 4M 1 M 2 pulses for this particular method. In terms of functionality, QRUDD essentially attempts to address errors which are generated by each individual UDD sequence using the time symmetrization approach and then subsequently nesting the error-reduced sequences. Of course an important question to address in this study is: which method for time symmetrization is more beneficial as a method to combat errors due to pulse imper- fections? Employing effective Hamiltonian theory and a numerical study of sequence performance this question is accurately answered from an analytical and system-specific numerical standpoint. 135 4.3.2 Model Specifications The system of interest is a single-qubit which interacts with its environment via the dephasing mechanism. Specifically the total internal Hamiltonian is defined as H 0 =I X i<j;j=1 b ij 3 z i z j ~ i ~ j | {z } H B + z X j d j z j | {z } H err (4.63) where the environment is represented by a qubit system interacting through the dipole- dipole interaction and the second term in Eq. (4.63) expresses the dephasing interaction between the system and environment. The operator ~ j = ( x j ; y j ; z j ) represents the Pauli spin-1/2 vector for the jth environment qubit. The coefficients b ij ;d j 2 [0; 1] are randomly chosen from a uniform probability distribution. The strength of the bath dynamics and error Hamiltonian are chosen as = 3:5kHz andJ = 7:5kHz, respec- tively, in order to effectively represent an NMR system for Adamantane where the 13 C atoms serve as the system qubits and 1 H spins take the role of the environment. Since the interactions between 13 C atoms are negligible compared to the interactions between the system and environment spins, the 13 C atoms essentially describe single-qubit sys- tems; thus, the single-qubit Hamiltonian model. The designated system and environ- ment qubits for Adamantane interact via the dipole-dipole interaction, however, the lon- gitudinal relaxation time for this system is much longer than the dephasing time. For this reason, the error Hamiltonian is effectively characterized by dephasing only. The initial state of the combined system and environment is assumed to be a product state(0) = S (0) B (0), where S (0) =j ih j such that j i = cos 2 j0i +e i sin 2 j1i (4.64) 136 andfj0i;j1ig represent the standard computational basis states of the single qubit sys- tem (eigenstates ofH S ). The initial state of the environment is given by B (0) = X ; ; jihj; (4.65) wherefjig form a basis for the environment states. For the numerical analysis, the basis statesji = N 4 j=1 j j i withj j i2fj0i;j1ig. The coefficients ; are chosen such that average polarization of the environment spins is equal to zero hI i = X j Tr I j B (0) = 0; =x;y;z (4.66) Subsequent results are averaged over initial conditions with2 [0;],2 [0; 2], and 2 [1; 1], all of which are chosen randomly from a uniform distribution. Sequence performance is evaluated with respect to the trace-norm distance [Eq. (2.61)] between the initial system state S (0) = Tr B [(0)] and final DD-evolved system state S (T ) = Tr B [U X (T )(0)U y X (T )], X 2 fRQDD; QRUDD; R4QDDg, resulting from a DD evolution of total time T which may not be equivalent to the DD cycle time c . All subsequent numerical data displays the performance quantity log 10 (1 D Tr [ S (0); S (T )]), which yields an effective lower-bound estimate to the Uhlmann fidelity [see Sec. 2.5.2]. 4.3.3 Numerical Comparison While varying parameters such as pulse timings and magnitudes of pulse imperfections in order to discriminate scheme performance is constructive, such a comparison is essen- tially only beneficial when utilizing an equal number of pulses to generate an ”apples- to-apples” comparison. For the QDD variations discussed here, the number of pulses 137 utilized by each scheme does not differ by multiplicative factor, therefore it is impos- sible to obtain an equal pulse-number comparison. Instead, the number of pulses are made approximately equivalent where two cycles ofU (20;20) R4QDD (880 pulses) is utilized as a reference point, i.e. numerous cycles of each sequence are applied such that 880 pulses are obtained. The specified K cycles for all schemes and sequences order are chosen accordingly with this convention holding for all results displayed in Figs. 4.12-4.15 whereM =M 1 =M 2 for all cases. The first performance comparison is shown in Fig. 4.12, where the total time T =K c and magnitude of flip-angle error are varied over a range of values. Select- ingK such that each sequence utilizes approximately 880 pulses, varying total time is equivalent to varying sequence cycle time c . The pulse duration is fixed at p = 10:6s in accordance with the NMR system described above, therefore implying that increas- ing sequence order results in a reduction in the minimum pulse delay. The first five relevant sequence orders are considered for each scheme and are meant to capture, and therefore summarize, the performance characteristics of each scheme as a function of sequence order. In Fig. 4.12(a)-(e), the results for R4QDD indicate a moderate dependence on primarily for M = 4. Remaining sequence orders exhibit an increased robustness to flip-angle errors with an approximately uniform performance for all T and for M = 8; 16; 20. As a function of sequence order, only minor variations in performance are observed; hence, performance saturates with increasing sequence order. In con- trast, more extensive deviations in sequence performance can be found for RQDD, as shown in Fig. 4.12(f)-(j). Dependence on total sequence time is essentially negligible for N = 1; 2 where the dominant feature is the dependence on flip-angle error. High fidelity regimes (D Tr < 0:01, designated in Fig. 4.12 as red and red-orange color) exist for less than a 10% ( = 0:10) flip-angle error forM = 1 and for less than 1% ( = 0:01) 138 Figure 4.12: Performance measure 1D Tr [(0);( c )] for R4QDD [shown in (a)-(e)], RQDD [shown in (f)-(j)], and QRUDD [shown in (k)-(o)] as a function of total timeK c and flip-angle error averaged over 20 realizations ofH and initial conditions. The pulse duration p = 10:6s and the total number of pulses is fixed at 880 for all panels unless otherwise specified. Optimal performance is primarily observed for RQDD, most notably in the case ofM = 1 andM = 3, where a regime can be defined where 1D is close to unity. R4QDD and RQDD performance is essentially equivalent for the largestM values considered (see last two columns). The remaining sequence structure, QRUDD, clearly exhibits the least robustness to pulse imperfections. forM = 2. Such regimes also exist forM = 3; 4, however they appear to be dependent upon T as well. An interesting feature of RQDD is the apparent overall performance deviations with sequence order parity. Odd sequence orders tend to perform more favor- ably within the parameter regimes considered; although ultimately increasing sequence order generally does not appear to be beneficial. High fidelity regimes also observed for QRUDD, however they are substantially smaller than those found for RQDD; see Fig. 4.12(k)-(o). In the case ofM = 1,D Tr < 0:01 exists only within a small region aroundT . 12ms and. 0:01. Note the drastic reduction in performance with increasingT and forM = 1; far worse than all other sequence orders and schemes displayed. Upon increasing sequence order, performance considerably increases and high fidelity regimes moderately increase, relative toM = 1. 139 Figure 4.13: Performance measure 1D Tr [(0);( c )] for R4QDD [shown in (a)-(e)], RQDD [shown in (f)-(j)], and QRUDD [shown in (k)-(o)] as a function of the pulse duration p and flip-angle error averaged over 20 realizations of H and initial con- ditions. The total time is fixed at T = 30ms and the total number of pulses at 880, unless otherwise specified. RQDD appears to be the least susceptible to variations in p , including various identifiable regimes of flip-angle error robustness for p < 10s. R4QDD exhibits similar results with relatively small regimes of robustness. Regions of high performance are nearly non-existen for QRUDD. As in the case of RQDD, sequence order parity-dependence is again observed. For QRUDD, even parity sequence orders appear to be more favorable. As an alternative to using total time to evaluate sequence performance, consider the case of varying average pulse delay avg = c n p = P i;j (M 2 ) i (M 1 ) j n p ; (4.67) where =f1; 2; 4g for R4QDD, RQDD, and QRUDD, respectively,n p is the number of pulses in one cycle of DD, and is the minimum pulse interval. In Fig. 4.14, the performance for each robust QDD scheme is shown for avg 2 [20s; 120s] and again ranging from a zero to 20% flip-angle error. The pulse duration is chosen as p = 10:6s, therefore c , andT , increase directly with variations in avg . 140 Figure 4.14: Performance measure 1D Tr [(0);( c )] for R4QDD [shown in (a)-(e)], RQDD [shown in (f)-(j)], and QRUDD [shown in (k)-(o)] as a function of the average pulse delay avg and flip-angle error averaged over 20 realizations of H and initial conditions. The pulse duration is fixed at p = 10:6s, therefore, the total cycle time c varies with d . Unless otherwise specified, K cycles of each sequence are used to generate 880 pulses. RQDD again exhibits the optimal performance for a majority of the parameter space considered. QRUDD performance varies the most dramatically with d and indicating a strong dependence on these parameters. Performance comparisons for R4QDD, shown in Fig. 4.14(a)-(e), generally con- vey dependence only on avg until avg 1 ( avg 60s) where performance decays quickly thereafter with average pulse delay and for allM. DD tends to perform more favorably when the bath correlation time (proportional to 1 ) is much larger than the (average) delay time, essentially captured by avg 1. Hence, the sudden incline inD can be attributed to the average delay time approaching the value of the bath correlation time. Note that this effect occurs in a rather uniform manner across all values of M. Furthermore, observe that R4QDD performance remains fairly constant with increasing sequence order, similar to the previous comparison shown in Fig. 4.12. Significant improvements in sequence performance are found for RQDD, most notably for odd parity sequence orders; see Fig. 4.14(f)-(j). In the case of M = 1, high fidelity is achieved for avg . 80s and . 0:15. The remaining parameter 141 Figure 4.15: log 10 (1D Tr [(0);( c )]) for R4QDD [shown in (a)-(e)], RQDD [shown in (f)-(j)], and QRUDD [shown in (k)-(o)] as a function of the bath strength and system-environment interaction strengthJ averaged over 20 realizations ofH and ini- tial conditions. R4QDD performance is essentially -independent and out-performs RQDD, and QRUDD, when J. As sequence order increases [M > 3], RQDD overcomes the reduced performance around J and maintains a larger range of enhanced performance than R4QDD and QRUDD. See Fig. 4.12 for additional details. regimes exhibit more significant dependence on avg and such that RQDD remains more favorable than R4QDD at the lowest sequence order; compare Fig. 4.14(a) and Fig. 4.14(f). Similar results are found for the remaining values of M with the most noticeable enhancements in sequence performance appearing forM = 3; 5. QRUDD again exhibits the poorest performance of the three robust QDD schemes in Fig. 4.14(k)-(o). The lowest overall performance is obtained for the smallest order sequence, M = 1, while the highest level of robustness to variations in avg and is found for even parity sequence orders,M = 2; 4. The highest performance forM = 1 is achieved within avg . 40s and . 0:01. For M > 1, high fidelity regimes are essentially characterized by avg < 1 ( avg . 60s) and . 0:1 for the sequence orders considered in this study. Fig. 4.13 presents a comparison of sequence performance as a function of p and . The total time is fixed atT = 30ms so that the minimum, and average, pulse delay 142 decrease with increasing pulse duration. Thus, the relationship between pulse duration and pulse delay can be effectively analyzed for each scheme in addition to the depen- dence of performance on pulse duration and flip-angle error. In Fig. 4.13(a)-(e), the performance of R4QDD withM = 4; 8; 12; 16; 20 conveys a predominant dependence on pulse width rather than flip-angle error. While this result indicates that R4QDD obtains a higher degree of robustness to flip-angle errors, the rather minor reductions in performance with increasing pulse width also convey finite- width errors are also mitigated quite successfully. Robustness to finite-width errors is captured essentially by the improving performance with increasing sequence order. In agreement with the previous comparisons, RQDD obtains the most robustness to variations in parameters governing pulse imperfections and sequence timing; see Fig. 4.13(f)-(j). The smallest sequence order, M = 1, maintains high fidelity for all p considered, up to approximately a 15% flip-angle error. Variations in performance become more dependent on both p and with increasingM, yet high fidelity regimes are still observed particularly forM = 4; 5. The general trend forM > 1 is increasing performance with sequence order, however note that M = 1 displays the best perfor- mance of all RQDD sequence orders considered. QRUDD performance as a function of p and is characterized by strong dependence on both parameters for allM considered. The smallest sequence order,M = 1, obtains the worse performance, most notably in the regime of small pulse-width (large pulse delay). QRUDD performance forM = 1 is therefore highly dependent on pulse timing in addition to pulse-width and flip-angle error. Small regimes of high fidelity can be found forM > 1 for the lowest values of pulse-widths and flip-angle errors considered implying an emergence of robustness to errors proportional to the pulse delay. These regimes are most prominent in even parity sequence orders for QRUDD as shown in Fig. 4.13(k)-(o). 143 As a final comparison, consider the performance of each variation of robust QDD as a function of the bath strength and system-environment interaction strength J. Such an examination essentially characterizes the performance of each scheme under various system-specific conditions, an analysis that is not attainable solely from the experimental system described in this work. Varyingf;Jg, the average pulse delay is fixed at 30s and pulse width at 10:6s; hence, the parameter regime is that of short pulse-width relative to the pulse delay and the principal source of error is proportional avg andJ avg . All simulations are averaged over 10 random realizations of flip-angle error such that is chosen from a normal distribution with mean zero and a standard deviation of 0:1 (10% flip-angle error). In Fig. 4.15, the performance for each variation of QDD is shown for a range ofJ; values. R4QDD, displayed in panels (a)-(e), exhibits high fidelity regimes which are bounded aroundJ avg 1 for avg < 1. The sudden decline in performance is gener- ated by the fact that the coupling strength, and essentially the decoherence rate, is on the order of 1 avg . DD tends to obtain the highest degree of effectiveness whenJ avg 1 regardless the scheme utilized; therefore, it is an intrinsic property of the protection method and can be observed for all schemes in Fig. 4.15. The performance characteris- tics of R4QDD change for avg 1 with >J due to self-averaging effects produced by pure environment dynamics which dominate the total system-environment time evo- lution dynamics. The qualitative characteristics are consistent across all sequence orders with the primary distinction being a steady attenuation in the regions of high perfor- mance with increasing sequence order. In contrast, the behavior of RQDD [panels (f)-(j)] with increasing sequence order is quite different, with near -independence for M = 1; 2 and fluctuations in perfor- mance with increasingM. While performance most certainly diminishes asJ avg ! 1, 144 the effects of increasing result in reduced performance forN = 3 and increased per- formance for N = 4; 5. Self-averaging effects are first observed for M = 5 where it is suspected that the source of decoherence for RQDD that is proportional to the strength of the environment dynamics contributes more significantly asM increases. In Sec. 4.3.4, the attributes of RQDD are explored further in terms of effective Hamiltonian theory to provide additional insight into the effects of varying sequence order. Such an analysis is further useful for determining the intrinsic differences between each of the schemes since, as a function of J and , the qualitative features of QRUDD [panels (k)-(o)] are nearly identical to R4QDD, except for smaller regions of high performance and advantages in utilizing even parity sequence orders for QRUDD. Most certainly, the justification for their similarity can be attributed to the error terms which dominate the effect Hamiltonian, however the details are indistinguishable from simulation results alone and are further studied in the next section. The numerical comparisons shown here suggest that RQDD is the preferred robust construction of QDD for a variety of pulse-delay timing and pulse imperfection param- eter ranges. Although it is not possible to determine whether RQDD indeed achieves higher order error suppression than the alternative schemes from numerical simulations alone, it is clear that RQDD at least offers improved error reduction over R4QDD and QRUDD. Firstly, this can be seen by the fact that RQDD maintains its superiority in the presence of error accumulation resulting from the application of multiple sequence cycles even though it utilizes the most sequence cycles for allM. In particular this is surprising in the case of fixed total time (see Fig. 4.12) where for allM andT the mini- mum pulse delay for RQDD is never the smallest of the three schemes. In fact, RQDD obtains the largest minimum pulse delay forM = 2; 4; 5 and largest average pulse delay forM = 5 for allT . Sources of error proportional to the inter-pulse delay time must be reduced significantly by the RQDD construction to account for the high performance. 145 Further comparisons pertaining to variations in pulse width and the continual compar- isons with the magnitude of the flip-angle error also exhibit similar results indicating that RQDD is also robust to pulse imperfection errors. The abundant evidence of supe- rior performance leads one to ask whether additional error suppression is attained by RQDD, as opposed to just a higher degree error reduction. In the subsequent section, this question is addressed by direct calculation of the effective Hamiltonian for each scheme and sequence order. 4.3.4 Analytical Comparison The numerical comparisons discussed in the previous section provided insight into the individual performance of each variation of QDD and how their robustness to pulse imperfections is altered by increasing sequence order. Additionally, the comparisons proved to be a useful method for determining which scheme is the more favorable can- didate for robust DD using nonuniform pulse delays. However, inherent properties of the scheme such as order of error suppression achieved by each sequence and an under- standing of the apparent parity variations for RQDD and QRUDD remains unanswered from simulations alone. In this section, AHT is employed to calculate the effective error Hamiltonian and elucidate details of each scheme, which were not addressed in Sec. 4.3.3, using the general dephasing Hamiltonian H 0 =I S B 0 + z B z ; (4.68) whereB , = 0;z are bounded environment operators. The strong pulse assumption, kH C (t)kkH 0 k, is enforced throughout this section in order to describe the dynamics of each sequence in the case that additional errors generated byH 0 during the pulse are reduced. 146 First, consider the case of R4QDD where the effective error Hamiltonians forM = 1; 2;:::; 8 are individually calculated and then summarized by H R4QDD M err p c " X =x;y avg R4 M; + p RQ M; # [B 0 ;B z ] + R4 M p c z B z :(4.69) The complex coefficients f R4 M ; R4 M ; R4 M g are specific to R4QDD and vary with sequence order M. Note that the order of error suppression is not altered by increas- ing sequence order or sequence order parity and that the primary distinction between each value of M is an accumulation of error, observed by an increasing magnitude off R4 M ; R4 M ; R4 M g. This result is consistent with simulations discussed in Figs. 4.12- 4.13 where deviations in sequence performance remained generally uniform for allN values and vary with all three relevant parameters:f; avg ; p g. Self-averaging effects observed in Fig. 4.15 are also captured by Eq. (4.69), particularly by the error terms pro- portional to the commutator [B 0 ;B z ]. Additional characteristics of RQDD include first order error suppression in all three parameters due to the chosen phase symmetrization method enforced on each nested UDD sequence. This method also yields second order error suppression for errors ofO( 2 ). Utilizing an alternative scheme, such as RQDD, introduces further error suppression; however, parity effects emerge as a result of sequence structure. Calculating the effective error Hamiltonian for the first eight sequence orders, the results are summarized by H RQDD N odd err p c avg RQ N + p RQ N x [B 0 ;B z ] (4.70) 147 and H RQDD N even err ~ RQ N p c z B z + p c avg ~ RQ N + p ~ RQ N x [B 0 ;B z ] +O[(J +) 3 3 p ] (4.71) which describe odd and even parity sequence orders, respectively. The apparent bias towards asymmetric QDD observed from simulations is evident from the additionalO( p ) term in Eq. (4.71). Sequence symmetrization prior to the time-symmetrization operation has been shown to affect sequence performance for cer- tain fixed pulse-interval sequence constructions [67]. Here, this effect is shown to exist for nonuniform DD sequence structures where asymmetric QDD-based RQDD yields complete cancellation of second order terms, i.e. quadratic terms composed of products of p , d , and, and symmetric QDD-based RQDD does not. The time-symmetrization technique utilized for QRUDD is similar that of RQDD, with the distinction being the manner in which the time-symmetrization is implemented. In the case of QRUDD, time symmetrization is applied to each nested UDD sequence thereby attempting to reduce the pulse errors generated by each inner sequence. Numer- ical simulations indicate that this technique is not beneficial unless time symmetrization is applied to the entire sequence rather than each constituent sequence. This observation is confirmed analytically where H QRUDD N odd err QR N p c z B z + QR N avg c y B z (4.72) describes the odd order parity sequence orders and H QRUDD N even err ~ QR N p c z B z + ~ QR N 2 p c x [B 0 ;B z ] (4.73) 148 the even orders, for the first eight sequence orders of QRUDD. The dominant error terms areO( p ) for both sequence order types, with additional second order terms ofO( avg ) appearing for odd parity orders. Symmetric UDD sequences yield higher order error suppression, contrary to the results for the alternative time-symmetrization method utilized by RQDD. Consistent with simulations, even parity QRUDD exhibits higher performance in the short pulse delay regime; see Fig. 4.14 and 4.13. Together, simulations and analytical calculations convey that RQDD is indeed the preferred method for constructing robust versions of QDD. While additional schemes like R4QDD attempt to decrease the presence of pulse imperfection errors by phase adjusting each nested UDD sequence, it is found that time-symmetrization generates the highest level of robustness. It is not without specificity, however, as the imple- mentation of time-symmetrization must occur at the conclusion of the QDD sequence rather than the constituent UDD sequence level in order to achieve the greatest advan- tage. Time-symmetrization has been long known to reduce pulse imperfection errors for fixed-interval DD. Here, it is shown that such a technique can also be adopted for nonuniform pulse delay DD constructions as well. 4.3.5 Summary of Results In analyzing various symmetrized versions of QDD, it is found that robustness to finite- width and flip-angle errors is most prominent in time-symmetrized QDD; referred to as RQDD. Additional variations which utilize symmetrization in pulse phases (R4QDD) or time-symmetrization at the each nested UDD level (QRUDD) exhibit inferior perfor- mance due to their inability to address second-order errors in TDPT. While robustness is limited for RQDD, and even reduces for even parity sequence orders, its effectiveness is unsurpassed due to the point where time-symmetrization is utilized. Symmetrization at each nested level offers some benefits, however, subsequent accumulation of these 149 Figure 4.16: Comparison in performance between RQDD and GA-optimized sequences for fixed number of pulses with varying pulse delay and magnitude of flip-angle error. Note that GA-optimized sequences outperform RQDD for the Hamiltonian outlined in Eq. (4.74). errors occurs upon the completion of the outer most sequence. In identifying the most favorable method for constructing robust QDD, an effective bound on robust QDD per- formance is also obtained as the effective error Hamiltonian for RQDD can be shown to scale asO( 2 p ) in the best possible scenario of oddM RQDD. Suppressing decoher- ence to higher orders in TDPT most likely requires more sophisticated methods in the pulse-interval-optimized setting either employing pulse shaping techniques or newly- optimized pulse intervals and sequence structure. 150 4.4 Numerically-Optimized DD vs. Symmetrized QDD Thus far robust sequences have been identified for the fixed pulse-interval and interval- optimized cases. Here the best performers of these scenarios are compared under the conditions of H 0 =I X i<j;j=1 b ij 3 z i z j ~ i ~ j | {z } H B + X X j d j j | {z } H SB (4.74) where the pure bath dipole-dipole Hamiltonian contains the coefficient b ij 2 [1; 1] chosen from a uniform probability distribution and system-environment interaction is essentially a generalization of the NMR system described in Sec. 4.3. The deterministic sequence denoted as RGA (l) 8a =RGA 8a [RGA (l1) 8a ] (4.75) is selected from the finite pulse interval case, while RQDD [Eq. (4.60)] is chosen from the pulse-interval-optimized setting. The number of pulses are set to be commensu- rate with the fourth level of concatenation forRGA (l) 8a ; thus, 8 4 pulses are used for each sequence. The average pulse interval is varied as avg 2 [10s; 120s] and the magnitude of the flip angle error is varied from 1% to 20%. All simulations utilizeJ = 100kHz and = 100Hz, thus, yielding favorable conditions for DD. In addition, the numeri- cal results are average 20 realizations of initial system state and environment operators with the trace-norm distance between the DD-evolved and initial states constituting the measure of performance. In Fig. 4.16(a)-(d), the performance for RQDD is shown for the first four sequence orders: M = 1; 2; 3; 4 such that K cycles are applied to obtain the required pulse count. As expected from previous results, odd parity sequences outperform even parity 151 sequence orders. The overall trend is an increase in performance with increasing num- ber of pulses. Performance is primarily dependent on where small flip-angle errors display the most favorable performance for odd sequences. Even order parity sequences appear to exhibit better performance for higher flip-angle errors, however, this is most likely due to an accumulation of coherent errors. In Fig. 4.16(e)-(h), the first four con- catenation levels of RGA (l) 8a are displayed using K cycles. Performance is primarily dependent on the magnitude of the flip-angle error for the first two levels of concatena- tion, while higher order sequences convey increased performance dependent upon and d as expected from the results shown in Sec. 4.2 when d > p and d 1. Comparing schemes one finds that the numerically-optimized sequences are better suited for general decoherence models when the system-environment interaction sets the relevant time scale. Of course, in the alternative situation, where the pure environment dynamics are dominant, one expects nearly opposite results due to motional narrowing effects which become increasingly prevalent as the DD cycle time grows. The depen- dence of performance on system specifications is an unavoidable feature of DD, and any dynamical control technique for that matter. While robustness to control errors is neces- sary for enhancing DD performance, the results shown here further stress the importance of slow-varying environment dynamics for successful dynamical suppression of deco- herence. 152 Chapter 5 Protecting Adiabatic Quantum Computation via Dynamical Decoupling In adiabatic quantum computation (AQC) a problem is solved by evolving in the ground state manifold of an adiabatic HamiltonianH ad (t), witht2 [0;T ] [3, 118]. The ground state of the beginning Hamiltonian H B = H ad (0) is assumed to be easily preparable, while the ground state of the problem HamiltonianH P = H ad (T ), represents the solu- tion to the computational problem. AQC has been shown to be computationally equiva- lent to the standard circuit model of QC [4, 5, 6, 7, 8], and is being pursued experimen- tally using superconducting flux qubits [33] and nuclear magnetic resonance [119, 120]. However, in spite of evidence of intrinsic robustness [121, 122, 123, 124, 125, 126] and proposals to protect AQC against decoherence [127, 128], AQC still lacks a complete theory of fault-tolerance, unlike the circuit model of QC [129, 130, 131, 132, 133, 134]. In fact, even identifying an acceptable notion of fault-tolerant AQC (FTAQC) is an open problem. 1 To qualify as AQC, at least the computation should remain adiabatic. On 1 In the circuit model a fault-tolerant simulation allows one to generate the output of a given ideal circuit, to arbitrary accuracy, using the faulty components of another circuit [1]. A similar definition for AQC would presumably involve the simulation of an ideal adiabatic evolution using a faulty one, but if “faulty” is simply taken to mean “non-adiabatic”, then one can just use the equivalence proof to argue that the problem is already solved. However, this argument misses the point since the defining feature of AQC—the adiabatic preparation of the ground state ofH P —should be preserved. 153 the other hand, it seems too restrictive to require the techniques used to address decoher- ence and noise to be adiabatic as well. Thus the following characterization of FTAQC is proposed: ‘Given a closed-system AQC specified byH B andH P , and> 0, a fault tol- erant open-system simulation will use adiabatic evolution between two faulty, encoded Hamiltonians H B and H P derived fromH B andH P , so that the final system-only state of the simulation is efficiently decodable to a state that is-close (in fidelity) to the ground state of H P . In addition, the simulation may involve any other faulty non-adiabatic error-correction, suppression, or avoidance operations.’ This characterization is meant to convey that the computation should be adiabatic, and apart from that the error correction can be anything. No attempt is made to rigor- ously quantify the relation between the “ideal” and “faulty” pairsH B ;H P and H B ; H P , the types of allowed error-correction operations, or the nature of the decoding step. Instead, it is demonstrated here that it is possible to approach ideal AQC in an open sys- tem, using ideas guided by the characterization of FTAQC. Specifically, it shall assumed that the underlying computation is indeed adiabatic but encoded into new Hamiltonians, and that the protection is non-adiabatic. Ref. [128] introduced an AQC protection method which fits this FTAQC approach. Protection is carried out by means of dynamical decoupling (DD) [56, 58, 57]. To ensure the compatibility of DD with AQC, all qubits are encoded into a quantum error detecting code [135], which allows DD pulses to be applied that commute with the adiabatic evolution, while at the same time acting to decouple the system from the bath. 2 Ref. [128] relied on first order DD sequences, resulting in a tradeoff between fidelity and DD sequence bandwidth, and conjectured that high-order DD, in particular concatenated DD (CDD) [117], should alleviate this tradeoff. 2 It can be shown [136] that in an appropriate rotating frame the DD protection technique of Ref. [128] is essentially equivalent to the method of energy gap protection of AQC [127]. 154 Here it is demonstrated, using numerical simulations for the random-unitary map model [137], that following the strategy of Ref. [128], but using high-order DD sequences, and in particular CDD, it is possible to dramatically enhance the fidelity of AQC in an open system setting. These results support the idea that FTAQC, in the sense characterized above, is indeed attainable using appropriately chosen DD sequences. 5.1 Closed System Adiabatic Quantum Computation In the limit of closed system evolution, the accuracy of AQC is quantified by the adi- abatic theorem of quantum mechanics. Let H ad (t), where t2 [0;T ], define a time- dependent Hamiltonian which implements AQC on an n-qubit system. The spectrum ofH ad (t) presumably contains a non-degenerate ground statej 0 (t)i and first excited statej 1 (t)i separated by a gap (t) such that min = min t2[0;T ] (t) > 0. Prepar- ing the system initially in the statej (0)i =j 0 (0)i and evolving it according to the Schr¨ odinger equation @ t j (t)i = iH ad (t)j (t)i, the state at the end of the evolu- tion isj (T )i. The adiabatic theorem of quantum mechanics guarantees thatj (t)i will be close to the instantaneous ground statej 0 (t)i for all t with high probability if T poly(k _ H ad k; 1= min ), where the operator normkAk is the largest eigenvalue of p A y A [138]. More precisely, let s := t=T denote the dimensionless time, pick q2 (0; 1), and assume that the total adiabatic evolution time satisfies T & a q (max s kdH ad =dsk) b1 b min : (5.1) Then, according to the adiabatic theorem, D[j (T )i;j 0 (T )i].q a : (5.2) 155 The values of the integer exponents a and b in Eqs. (5.1) and (5.2) depend upon the differentiability and analyticity properties ofH(t), and the boundary conditions satisfied by its derivatives; typically b2f1; 2; 3g [139], while a can be tuned between 1 and arbitrarily large integer values, equal to the number of vanishing derivates ofH(t) at the boundariest = 0 andt =T [140]. AQC directly exploits the adiabatic theorem of quantum mechanics by initializing the quantum system in a HamiltonianH ad (0) = H I that has an easily prepared ground state, usually a complete superposition state between all accessiblen-qubit basis states. The computational problem of interest is then encoded into the problem Hamiltonian H ad (T ) = H P such that ground state ofH P corresponds to the solution. Interpolating betweenH I andH P via H ad (t) =A(t)H I +B(t)H P ; (5.3) whereA(t);B(t) are monotonically decreasing functions, the solution to the computa- tional problem is obtained with an accuracy determined by the adiabatic theorem. 5.2 Adiabatic Quantum Computation in the Presence of Decoherence In realizable systems, the accuracy of AQC is not fully defined by the adiabatic theorem alone as decoherence processes now play a vital role in determining such a quantity. Such processes can occur in a variety of ways all of which are essentially captured by H 0 (t) =H ad (t) +H err (t) (5.4) 156 where, in addition to the adiabatic Hamiltonian H ad (t), the error Hamiltonian H err (t) constitutes any form of decoherence mechanism. When system-environment interac- tions compriseH err (t), relaxation and dephasing times place a constraint on the interpo- lation time of AQC such that the probability of remaining in the ground state throughout the computation reduces if the interpolation time is longer than the associated decoher- ence times of the system. Decoherence may also occur in the form of classical fields that distort the energy spectrum such that the size and location of the minimum gap are modified; thus, increasing the probability of excitation, possibly at numerous locations, throughout the interpolation. The latter case is the focus here, where H err (t) = X 2fx;y;zg n X j=1 j (t) j : (5.5) defines the interaction between the jth qubit and time-dependent stochastic classical field. Each field j (t) is a stationary zero-mean random Gaussian process with spectral densityI jk (!) = jk I(!) [141], where I(!) = 1 p 2 Z 1 1 j (t) j (t +) e i! d = s J 2 exp[ (!=) 2 =2]; (5.6) such thathi denotes a Gaussian ensemble average and I(!) is t-independent due to stationarity. The standard deviation plays the role of the spectral cutoff (1= is the bath correlation time). The correlation function amplitude satisfies j (t) j (t +) / J and consequently j (t)/ p J. Each random realization of H err (t) generates a random unitary U (t), where := f j g [the solution of the Schr¨ odinger equation governed byH 0 (t)], with probabilityp . Applying these unitaries to a fixed initial state 0 is equivalent to the completely-positive random-unitary map 0 7! P A 0 A y , with Kraus operatorsA := p p U [142]. Thus, 157 Eq. (5.5) generates dynamics which can be effectively described in a similar manner to the open system dynamics described in Sec. (1.1.2). While this is not the most general model of decoherence [137], as it does not account for quantum system-environment interactions, it represents an interesting and relevant error model, e.g., due to charge noise in superconducting qubits [143, 144]. 5.3 Applying Dynamical Decoupling to Adiabatic QC The problem one attempts to solve using DD is to perform high fidelity AQC in spite of the presence of H err . The decoupling pulses are introduced through an additional time-dependent control HamiltonianH C (t), which generates a unitary pulse propagator U C (t). Zero-width pulses are considered separated by finite intervals. The inclusion of pulse-width errors is left for a future study focusing on a more complete picture of fault tolerance; CDD is known to be relatively robust against such errors [54, 117, 145]. The HamiltonianH(t) =H C (t) +H 0 (t) generates the complete dynamics of the system in the presence of DD, represented by the unitary evolution operator U(t). To suppress H err (t), while preserving H ad (t), it is required that each term in H err (t) anticommute with some pulse operator comprising H C (t), while [H C (t);H ad (t 0 )] = 08t;t 0 . Upon satisfying these conditions the time evolution operator in theH C (t)-interaction picture becomes ~ U(T ) := U y C (T )U(T ) (5.7) = Te iT R 1 0 H ad (s)ds +O[(k ~ H 0 err kT ) +1 ]; whereT denotes time ordering, and ~ H 0 err is an effective error Hamiltonian, which can be computed using the Magnus series. DD becomes effective provided the “noise strength” 158 k ~ H 0 err kT < 1 [54]. The larger the “decoupling order”, the closer to ideal is the adia- batic evolution. Previously only the case = 1 was analyzed [128]. 5.3.1 Stabilizer decoupling To satisfy the non-interference condition [H C (t);H ad (t 0 )] = 0 we make use of the [[n;n 2; 2]] stabilizer codeC, encoding n 2 logical qubits (n even) into n physi- cal qubits [128, 135, 146]. The stabilizer ofC isS =fI;X;Y;Zg, whereX(Y;Z) = N n j=1 x(y;z) j . The encoded single-qubit operators are x j = x 1 x j+1 and z j = z j+1 z n , wherej = 1; 2;:::;n2. AQC overC is implemented by replacing each Pauli matrix in H ad (t) by its encoded version, yielding an encoded adiabatic Hamiltonian H ad (t) which is fully 2-local. 5.3.2 Concatenated Dynamical Decoupling The general notion of CDD has been outlined in Sec. 2.4.2 for a general decoupling groupG. Here, the decoupling groupG =S and hence G = 4 for the [[n;n2; 2]] stabilizer code. Adapting the method to incorporate the time-dependence ofH(t), the (` + 1)th level of concatenation is given by U (`+1) CDD (T ) = G Y k 1 =1 g k 1 P (`) k 1 g y k 1 ; ` 0: (5.8) Starting fromm = 2,P (`) k 1 is calculated recursively from P (`) k 1 ;:::;k m1 := G Y km=1 g km P (`1) k 1 ;:::;km g y km ; l 1 (5.9) 159 with DD-free evolution segments P (0) k 1 ;:::;k l :=U 0 t `1 +T k ` G ` ;t `1 +T k ` 1 G ` t ` :=T ` X j=1 k j G j ; t 0 = 0; ` 1; (5.10) where the DD-free evolution operator is U 0 (t a ;t b ) =T exp i R ta t b dtH 0 (t) . These definitions ensure that at concatenation levell the symmetrization procedure is applied to DD-free evolution segments of duration ` = T=G ` , using a total ofG ` pulses. For piecewise constant Hamiltonians it has been shown that CDD achieves = lth order decoupling [54, 76, 145, 147, 148]. 5.3.3 Quadratic Dynamical Decoupling The QDD scheme is used in addition to CDD to analyze DD performance and compare two fundamentally distinct decoupling protocols. In the original construction discussed in Sec. 2.4.3 the time intervals between successive pulses were given in terms of the minimum pulse delay. Expressing the QDD scheme instead in terms of the total AQC interpolation timeT in order to account for the time-dependent ofH(t) can be done in a straightforward by replacing the normalized time intervals by the actual time intervals: U (M 1 ;M 2 ) QDD;f 1 ; 2 g (T ) =U (M 2 ) UDD; 2 U (M 1 ) UDD; 1 U 0 (T ); (5.11) where U (M) UDD; U 0 (T ) = M+1 M+1 Y k=1 U 0 (t (M) k ;t (M) k1 ); (5.12) t (M) k = T sin 2 [k=(2M + 2)]. The pulse operators are chosen as the generators of the stabilizerS, 1 6= 2 2fX;Zg. 160 5.4 Performance of Deterministic Decoupling Schemes For each algorithm (Grover, 2SAT)n = 4 physical qubits are used to encode two log- ical qubits in the codeC, and study both CDD- and QDD-protected AQC. In these simulations the pulse intervals decrease as the total number of pulses increases with the concatenation or QDD sequence order, for each given value of the total time T . Starting for both algorithms from the uniform superposition state as the initial encoded statej (0)i,j (T )i =U xDD; (T )j (0)i is computed, where x=C or Q [Eqs. (5.8) and (5.11)], for a given Gaussian noise realization. Performance is assessed using the trace- norm distance D Tr; (T ) := D Tr [j (T )i;j 0 (T )i] = p 1F 2 (T ) , where F (T ) = jh (T )j 0 (T )ij is the fidelity, to quantify the difference between the encoded, xDD- protected statej (T )i and the desired encoded final statej 0 (T )i, i.e., the ground state of H ad (T ). All plots exhibit the average distance D Tr (T ) = P p D Tr; (T ), and it is easily verified that 1D Tr (T ) lower-bounds the output fidelity of the random-unitary map with Kraus operators p p U xDD; (T ). In the ideal case of noise- and DD-free evolution j ad (t)i =T exp Z t 0 H ad (t 0 )dt 0 j (0)i; (5.13) the adiabatic theorem [138] guaranteesD Tr [j ad (T )i;j 0 (T )i] 1 providedT is suf- ficiently large [see Sec. 5.1]. In the simulations shown below a finite range ofT ’s was used, and as can be seen from Figs. 5.1-5.5, D Tr (T ) does indeed tend to zero for the ideal case asT is increased, though not monotonically in the Grover case. 3 The main effect ofH err (t) is to causeD Tr (T ) to diverge away from zero asT is increased, so that 3 Ideal case oscillations are consistent with the adiabatic theorem, which predictsD! 0 only in the limitT!1. For an analytical expression predicting such oscillations in the case of Grover’s algorithm see Ref. [149], Eq. (174). 161 there is an optimal evolution time. 4 The main role of DD protection, then, is to keep the fidelity of the AQC process as close as possible to the ideal and, in particular, to prevent min T2[0;Tmax] D Tr (T ) from growing larger than some tolerance> 0 away from min T2[0;Tmax] D Tr [j ad (T )i;j 0 (T )i]. This is the sense in which DD-protected AQC approaches the ideal of FTAQC described in the introduction. 5.4.1 Grover’s Algorithm The problem solved by Grover’s algorithm is the identification of a marked element in an unsorted list ofN elements, using the minimum number of oracle queries [151]. This can be done inO( p N) queries, which is a quadratic improvement over the best possible classical algorithm [152]. Recast in the language of AQC [153, 154], Grover’s algorithm is defined by thenqubit Hamiltonian H G ad (s) = [1f(s)](Ijuihuj) +f(s)(Ijmihmj); (5.14) wherejui denotes the uniform superposition over allN = 2 n computational basis states, jmi is the marked state, and I is the identity operator. The minimum spectral gap G min = O(1= p N), and the total run timeT O( p N) is found from Eq. (5.1) with b = 1 [139, 140] provided the optimized interpolation functionf(s) is used [153, 154]: f(s) 1 2 1 2 p N 1 tan[(1 2s) arccos(1= p N)]: (5.15) 162 á á á á á á á á á á á á á áá á á á áá á á á á á á á á á á á á á áá á á á á á á á á á á á á á á á á á á á á á á á áá á á á áá á á á á á á æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æææ æ æ æ æ æ æ ææ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ìì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ò ò ò ò ò ò ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò ò ò òòò ò ò ò òò ò ò òò òò ò ò òòòò ò ò ò ò ò ò ò ò òò òò ò ò ò à à à à à à à à à à à à à àà à à à à à à à à à à àà à à à à à à à à à àà à à à à à à à à à àà à à à à àà à à à àà à à à à à à à à à àà 0 5 10 15 20 25 30 35 0.0 0.2 0.4 0.6 0.8 T Hunits of 1D min G L D Tr HTL æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à 15 20 25 30 35 0.01 0.02 0.05 0.10 0.20 à CDD 4 ò CDD 3 ì CDD 2 æ CDD 1 á Faulty Ideal Figure 5.1: Averaged trace-norm distance D Tr (T ) between the CDD-protected final state and the desired ground state as a function of total run time T for Grover’s algo- rithm, in units of the inverse minimum gap. Cutoff frequency: = min =5. The ideal (solid black) and faulty (empty squares) evolutions are included for reference. Insert shows a close-up for largeT , using a log scale for the vertical axis. Performance improves monotonically with concatenation level l, with the corresponding sequence denoted CDD l . Error bars are due to averaging over 30 random realizations ofH err (t). Concatenated Dynamical Decoupling CDD results for Grover’s problem are shown in Fig. 5.1, for increasing concatenation levels, at = min =5. The faulty Grover evolution [generated by H G ad (t) +H err (t)] reaches a minimum deviation D Tr (T ) 0:13 at T 4:5= G min and then diverges from the ideal evolution [generated by H G ad (t)]. In contrast, CDD-protected evolution becomes remarkably close to the ideal evolution as the level of concatenation increases. Nearly ideal evolution is maintained essentially over the entire range ofT values simu- lated for a concatenation level ofl = 4. 4 This effect of open-system AQC, due to a competition between the benefit of increasingT for adia- baticity and the simultaneous increasing damage due to system-bath coupling, has been previously pointed out experimentally [119] and theoretically [122, 150]. 163 à à à à à à à à à à à à à àà à à à àà à à à à à à à à à à à à à àà à à à à à à à à à à à à à à à à à à à à à à à àà à à à àà à à à à à à æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æææ æ æ æ æ æ æ ææ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ ì ì ì ì ì ì ì ì ì ì ì ì ì ìì ì ì ì ì ì ì ì ì ì ì ìì ì ì ì ì ì ì ì ì ì ìì ì ì ìì ìì ì ì ìì ì ì ì ì ìì ì ì ìì ì ì ì ì ì ìì ì ì ì ì ìì ò ò ò ò ò ò ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò ò ò òòò ò ò ò ò ò ò ò ò ò ò òòò òò òò òòò ò ò ò ò ò ò ò òòò òò á á á á á á á á á á á á á áá á á á á á á á á á á áá á á á á áá á á á áá á á á á á á á á á áá á á á á áá á á á ááá ááá á áá á ááá ç ç ç ç ç ç ç ç ç ç ç ç ç çç ç ç ç ç ç ç ç ç ç ç çç ç ç ç ç ç ç ç ç ç ççç ç ç ç ç ç ç ç ç ççç ç ç ç çç ç ç ç çç ç ç ç ç çç ç ç ç çç í í í í í í í í í í í í í íí í í í í í í í í í í íí í í í í í í í í í ííí í í í í í íí í ííí í í ííí í í í í íí íí í í í ííí í í ó ó ó ó ó ó ó ó ó ó ó ó ó óó ó ó ó ó ó ó ó ó ó ó óó ó ó ó ó ó ó ó ó ó óó ó ó ó óó ó ó ó ó ó óó ó ó ó óó ó ó ó óó ó ó ó ó ó óó ó ó óó 0 5 10 15 20 25 30 35 0.0 0.2 0.4 0.6 0.8 T Hunits of 1D min G L D Tr HTL ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò òòò ò ò ò á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç í í í í í í í í í í í í í í í í í í í í í í í í í í í í í í í í í í í í í í í í í í ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó óó ó 15 20 25 30 35 0.100 0.050 0.020 0.030 0.015 0.150 0.070 ó QDD 15 í QDD 14 ç QDD 7 á QDD 6 ò QDD 4 ì QDD 3 æ QDD 1 à Faulty Ideal Figure 5.2: Trace-norm distance between the QDD-protected final state (T ) and the desired ground state M , as a function of total run time T for Grover’s algorithm, in units of the inverse minimum gap. Specific sequence orders are considered: M = 1; 3; 4; 6; 7; 14; 15. Odd parity sequence orders contain an equal number of pulses to those of the CDD sequences shown in Fig. 1 (main text). In contrast to CDD, QDD- protected evolution does not more closely resemble closed-system evolution at largeT as the number of pulses grows. Quadratic Dynamical Decoupling In Fig. 5.2, the results for QDD-protected Grover are shown for = min =5. Closed- system-like behavior is observed forM = 1; 3; 6; 7 andM = 14; 15 up toT 4:5= G min andT 11:5= G min , respectively, where the minimum deviation min T D Tr (T ) generally decreases with increasing sequence order. QDD does not achieve a higher performance than CDD for an equivalent number of pulses, as CDD minimum deviations range from 10% to 15% smaller than QDD values at equivalent “optimal” values ofT . QDD is also distinct from CDD in its behavior at largeT , with respect to the simulated range, where QDD-protected evolution begins to divergence from the ideal closed-system evolution as the number of pulses is increased. Although the performance of each scheme, desig- nated by the minimum deviation, is well-characterized in the smallT regime for Grover, 164 this characteristic of QDD could ultimately lead to a bound on QDD effectiveness for algorithms where DD extends ideal-like behavior to larger values ofT . 5.4.2 2-bit Satisfiability The second algorithm considered is the 2-bit satisfiability (2-SAT) problem on a ring. This problem is not associated with a quantum speedup, but is instructive nonetheless. Given a set of clausesfC k g n k=1 associated withn bits, such that each clause acts only on adjacent bitsk andk + 1, the 2-SAT Hamiltonian acquires minimum energy when there are satisfying assignments for all clauses. In the case that the clauses define agreement between adjacent bits (00 or 11, but not 01 or 10), the 2-SAT Hamiltonian is that of the transverse-field Ising model [3] H 2SAT ad (t) = 1 t T n X j=1 (I x j ) + t T n X j=1 1 2 (I z j z j+1 ); (5.16) where periodic boundary conditions z n+1 z 1 are imposed. Note that the symmetric ground state ofH P ,j 0 (T )i = (j0::: 0i +j1::: 1i)= p 2, satisfies the clause agreement condition. The minimum gap occurs near t = 2T=3 and min = O(1=n); it can be found using the standard Jordan-Wigner-Fourier transform method [3, 155]. The linear interpolation given in Eq. (5.16) is not optimal; see Ref. [156] for the optimal path. Concatenated Dynamical Decoupling Figure 5.3 shows our CDD results for the 2-SAT problem. CDD-protected evolution is essentially indistinguishable from the ideal at high enough concatenation level. The improved performance in the 2-SAT case, relative to the Grover case, are consistent with earlier observations that algorithms associated with second order quantum phase 165 à à à à à à à à à à à à à à à à à à à à à àà àà à à à à à à à à à à ààà à àà à à à à à à à à àà à à à à à à à à à à à à àà à àà à à à ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç çç ç ç ç çç çç ç ç çç çç ç ç çç çççç ç çç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç çç ç ç çç í í í í í í í í í í í í í í í í í í í í í í í í í í íííííííí ííí íí í í í í íí í íí íííííí í íí ííí í ííí í í íí í í í ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó óóóóóóóóóóóóóóóóóóóóóóóóóóó óóó óóóóó óóóó ó óóóó ó á á á á á á á á á á á á á á á á á á á á á á á á á á ááááááááááááááááááááááááááááááááááááááááááááá 0 5 10 15 20 25 30 35 0.0 0.2 0.4 0.6 0.8 T Hunits of 1D min 2 SAT L D Tr HTL ç çç ç ç çç ç ç ç ç çç çççç ç ç ç ç í íííí í íí í í í í í í íí í í í ííí í íí í í í ííí í í í í í í í í í í í ó ó ó ó ó ó ó óóó ó ó ó ó ó ó ó óó ó ó ó óó ó ó ó ó ó ó ó ó ó ó óó ó ó ó ó ó ó á á á áá áá á áá á áá á á ááááá á áááá ááááá á á ááá áá á áá á á 15 20 25 30 35 0.100 0.050 0.020 0.030 0.070 á CDD 4 ó CDD 3 í CDD 2 ç CDD 1 à Faulty Ideal Figure 5.3: As in Fig. 5.1, for the 2-SAT on a ring problem. The curves for ideal evolution and CDD 4 overlap to within our numerical accuracy up to T = 25= 2SAT min . Note the minimum atT 12 2SAT min for the faulty evolution, suggesting the existence of an optimal open system evolution time. This optimal time increases with concatenation level, until it disappears in the ideal case and for CDD 4 . CDD boosts the deviation by a factor of 10 fromD Tr 0:2 (atT 12:5= 2SAT min ) in the faulty case toD Tr 0:02 (at T 34:5= 2SAT min ) for CDD 4 . transitions are more amenable to AQC than those for first order transitions [125, 157, 158]. Quadratic Dynamical Decoupling Interestingly, 2-SAT on a ring is one such case where the large T regime is relevant and QDD performance diminishes at a critical value of sequence order. As with Grover, a general trend of decreasing min T D Tr (T ), with increasing sequence order exists for all M 14 considered here. However, the appealing relationship between minimum deviation and M disappears for 2-SAT at M = 15, where the minimum deviation is nearly 15% larger than that ofM = 14. [See Fig. 5.4 for numerical results.] The values ofT associated with the minimum deviation, arg min T D Tr (T ), extend far into the large 166 à à à à à à à à à à à à à à à à à à à à à àà àà à à à à à à à à à à ààà à àà à à à à à à à à àà à à à à à à à à à à à à àà à àà à à à æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ ææ ææ æ æ ææ ææ æ æ ææ ææææ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ ææ ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ììììììììì ìì ìììììì ììììì ììì ìì ìììì ì ì ìì ì ì ìì ì ì ì ì ìì ì ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò òòòòòòòòòòòòòò òòòòò òòò òò òòòòòò òòòòò ò ò ò ò òò ò òòò á á á á á á á á á á á á á á á á á á á á á á á á á á ááááááááááááááááááááááááááááááááá áááá áá á á ááá á ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç çççççççççççççççççççççççççççç ç çççççç ççç ççççç ç í í í í í í í í í í í í í í í í í í í í í í í í í í íííííííííí íí í í í í íííííí ííí ííííí í íííííí í í ííí íí í ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó óóóóóóóó óóóóó óóó ó óóó óóó ó óóó ó óó óó óó ó ó óóó óóóóó ó 0 5 10 15 20 25 30 35 0.0 0.2 0.4 0.6 0.8 T Hunits of 1D min 2 SAT L D Tr HTL ò ò ò ò ò ò ò ò òòò ò ò òò ò òò ò ò ò ò òòò òò ò ò ò òò ò ò ò ò ò ò ò ò òò á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç íí í í íí í í í í í í í í íí íí í íí í íí í í í í í í í í í í í í í í í í í í óó ó ó ó óó óó ó óóó ó ó ó ó ó óó ó óó ó ó ó ó óó ó ó ó ó ó ó ó ó ó ó ó ó ó 15 20 25 30 35 0.100 0.050 0.020 0.030 0.070 ó QDD 15 í QDD 14 ç QDD 7 á QDD 6 ò QDD 4 ì QDD 3 æ QDD 1 à Faulty Ideal Figure 5.4: Same as in Fig. 5.2, for the 2-SAT on a ring problem. QDD protection extends ideal evolution as sequence order increases, ultimately saturating atM = 14. T regime and reach a maximum of T 35:5= 2SAT min for M = 14; 15. In the case of CDD at ` = 4 the maximum T value is nearly equivalent [T 34:5= 2SAT min ], yet CDD-protected evolution does not exhibit a divergence within the simulated range and obtains a minimum deviation of approximately half that of M = 14; 15. The bound on QDD performance is indicative of a preference to concatenation rather than nesting, perhaps most notably for algorithms with second order quantum phase transitions, due to the recursive suppression of the system-bath interaction throughout the entire evolution provided by the former. 5.4.3 Dependence of DD Performance on Cutoff Frequency One contributor to the better performance of CDD is the Gaussian spectral density of the noise process j (s); the performance of UDD-based schemes is adversely affected if the spectral cutoff is not sufficiently sharp at high frequencies [102, 159]. Transforming the error Hamiltonian into the toggling frame and analyzing the resultant DD modulation 167 Figure 5.5: CDD 4 vs QDD 15 for the Grover problem as a function of the normalized bath correlation time 1=(T ) and normalized total run timeT= G min . Increasing the bath correlation time 1= at fixed T generally results in improved performance for CDD 4 . whose performance is significantly better than QDD 15 in in the large T regime. All results were averaged over 30 realizations ofH err (t). functions in the frequency domain, one finds that DD can be interpreted as a high- pass filter whose cut-off frequency increases as the inter-pulse free evolution period decreases [144]. UDD-based schemes tend to have the flattest filter functions [102, 144], and hence their cutoff frequencies are much smaller than concatenation-based schemes. Here, we examine the effects of the spectral cutoff on QDD and CDD-protected AQC for both Grover’s search problem and 2-SAT on a ring. First, consider the performance of CDD and QDD as a function of the total adiabatic run timeT and normalized correlation time (T ) 1 . The focus will be on the sequences which exhibit the smallest minimum deviation between DD-protected and ideal evolu- tion for both schemes: CDD 4 and QDD 15 for Grover and CDD 4 and QDD 14 for 2-SAT. In Figs. 5.5 and 5.6, the deviation of DD-protected evolution from ideal AQC evolution is shown for both sequences in the case of Grover’s search algorithm and 2-SAT on a ring, respectively. The normalized correlation time (T ) 1 2 [10 3 ; 10 3 ] and the total run time is varied fromT = 0 toT = 36= min . We average all results over 30 random realizations of j (s). 168 Figure 5.6: Comparison between CDD 4 and QDD 14 performance for 2-SAT on a ring as a function of the normalized correlation time (T ) 1 and total run time T . As in Fig. 3 (main text), sequence performance is nearly the same for short correlation times. Increases in correlation time result in smaller deviations between DD-protected and ideal evolution for both schemes. In the case of Grover’s search algorithm, CDD 4 dominates for short correlation times while QDD 15 performance increases substantially at large (T ) 1 and attains minimum deviations approximately equal to CDD 4 . CDD 4 -protected and closed-system evolu- tion nearly coincide up toT values ranging fromT 4:5= G min at (T ) 1 = 10 3 to T 16:5= G min at (T ) 1 = 10 3 , where the minimum deviation reduces by about 36% over the entire range of (T ) 1 . QDD 15 obtains similar values ofT , however, minimum deviation reduces by about 60% over the range of cutoff frequencies. The distinction between the DD schemes is most notable in the largeT regime, where QDD 15 -protected evolution diverges from the ideal evolution more dramatically than CDD 4 as the spec- tral cutoff is reduced. However, this effect is inconsequential for QDD 15 performance since minimum deviation occurs predominately in the shortT regime. Additional arti- facts, such as self-averaging due to rapid fluctuations of j (s) at short correlation times (“motional narrowing”) are also present for both schemes. The results for the 2-SAT problem differ from Grover’s algorithm in that CDD 4 maintains its superiority over QDD 14 for all . CDD-protected evolution reaches 169 æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô 0.001 0.01 0.1 1 10 100 0.01 0.02 0.05 0.10 0.20 0.50 t Hunits of 1bL D Tr HtL ô b=200D min G à b=20D min G ò b=2D min G ì b=0.2D min G æ b=0.02D min G Figure 5.7: Performance of CDD 4 for Grover’s search algorithm as a function of the pulse interval for various values of the frequency cutoff . Short correlation times ( = 200 G min ) demand a large pulse interval to obtain minimum deviation from ideal evolution. As is decreased, reduced minimum deviation is achieved at pulse intervals much smaller than the correlation time (1=). Results are averaged over 30 realizations ofH err . minimum deviation at T 28= 2SAT min and T 34:5= 2SAT min for (T ) 1 = 10 3 and (T ) 1 = 10 3 , respectively, where the value of min T D Tr (T ) reduces by 56% over the entire range of (T ) 1 considered. Similar values of T corresponding to arg min T D Tr (T ) are also obtained for QDD, however, the dramatic reduction in mini- mum deviation with decreasing is not observed. In fact, even for the largest normal- ized correlation time (T ) 1 = 10 3 QDD 14 minimum deviation remains approximately 50% larger than CDD 4 . As discussed in Sec. 5.4.2, increasing the sequence orders of QDD does not appear to be beneficial for the 2-SAT problem due to the divergent behav- ior of QDD-protected evolution from the ideal case at large T . Here, it is shown that similar results are found for a wide range of cutoff frequencies. It is suspected that sim- ilar results would be obtained for any algorithm where DD-protection extends ideal-like behavior to largeT values. 170 æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô 10 -6 10 -5 10 -4 0.001 0.01 0.1 1 0.01 0.02 0.05 0.10 0.20 0.50 t Hunits of 1bL D Tr HtL ô b=200D min G à b=20D min G ò b=2D min G ì b=0.2D min G æ b=0.02D min G Figure 5.8: As in Fig. 5.7, for QDD 15 . The pulse interval represents the minimum delay between pulses for QDD. In contrast to CDD 4 , minimum deviation between DD- protected and ideal evolution does not increase with correlation time for QDD. Note also that the minimum deviations are approximately twice as large as those obtained for CDD and require a minimum pulse interval 10 times smaller. The expectation of spectral cutoff-dependent variations in DD performance is clearly evident for both Grover’s algorithm and 2-SAT. The minimum deviations for CDD 4 and QDD 15 decrease with increasing correlation time for the Grover case, with QDD 15 obtaining the more significant reductions over the range of considered. Closed- system-like evolution is observed for extended time durations, dependent upon corre- lation time, in the case of 2-SAT for both CDD 4 and QDD 14 , with CDD 4 maintaining the more favorable minimum deviations. The attributes of DD-protected AQC appear to be dependent upon the order of the quantum phase transition, however it is necessary to consider additional algorithms to truly validate this observation. Whether or not such a conclusion can be drawn, the algorithm-dependent variations in DD performance are still quite intriguing, and perhaps unexpected, results. 171 æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô 10 -4 0.001 0.01 0.1 1 10 100 0.02 0.05 0.10 0.20 0.50 t Hunits of 1bL D Tr HtL ô b=200D min 2 SAT à b=20D min 2 SAT ò b=2D min 2 SAT ì b=0.2D min 2 SAT æ b=0.02D min 2 SAT Figure 5.9: Performance of CDD 4 -protected evolution for the 2SAT problem, as a func- tion of pulse interval , for different values of the frequency cutoff ( in units of 1= to separate the curves). Total timeT = 4 4 . The minimum in each fixed curve corresponds to an optimal pulse interval opt (), and correspondingT opt (). Peak per- formanceD Tr (T opt ()) improves as is decreased, except at = 2SAT min =0:005 where self-averaging effects result from rapid fluctuations in j (t) (“motional narrowing”). Results are averaged over 30 realizations ofH err (t). As an additional analysis, consider DD performance with respect to the minimum delay between successive pulses for various values of the spectral cutoff. The min- imum pulse delay is a canonical parameter that represents a physical constraint in stan- dard DD studies, where the primary objective is to apply the control pulses such that 1 in order to generate the desired effective averaging ofH err to some order in (orT ). This condition on presents a contradictory situation for AQC, where extended total evolution time is demanded to obtain higher computational accuracy. Although pre- vious work has addressed this issue for the case of PDD, a general understanding of the relationship between and for AQC evolution is still lacking for more sophisticated DD schemes. By varying the spectral cutoff, one seeks to gain insight into the connec- tion between and the value of where minimum deviation between DD-protected and ideal evolution occurs, opt = arg min D Tr (), specifically for CDD and QDD. 172 æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô 10 -5 0.001 0.1 10 0.10 0.50 0.20 0.30 0.15 0.70 t Hunits of 1bL D Tr HtL ô b=200D min 2 SAT à b=20D min 2 SAT ò b=2D min 2 SAT ì b=0.2D min 2 SAT æ b=0.02D min 2 SAT Figure 5.10: As in Fig. 5.9, for QDD 14 . Here, the pulse interval represents the mini- mum delay between pulses. Minimum deviations from ideal evolution are approximately twice as large as those obtained for CDD 4 . Peak performance appears insensitive to the value of. First, consider Grover’s search problem for CDD 4 and QDD 14 as a function of shown in Figs. 5.7 and 5.8, respectively. CDD 4 -protected evolution clearly exhibits a direct relationship between the minimum deviation from ideal evolution and, generally reducing as the spectral cutoff reduces. Large spectral cutoffs ( = 200 G min ) require a relatively large pulse delay, opt 10=, to generate an evolution long enough to reach minimum deviation. Consequently, the 1 condition is violated and DD-protected evolution deviates more significantly from ideal evolution, D Tr 0:02 at the lowest point. Short spectral cutoffs ( = 0:02 G min ) result in a opt approximately 1,000 times smaller than 1= and a considerable reduction in the minimum deviation,D Tr 0:005. The-dependence of QDD 15 is clearly noticable when transitioning into the 1, yet the reduction in minimum deviation is unsubstantial thereafter. Peak performance occurs for = 0:02 G min , where opt is about 10 5 times smaller than 1=. In general, minimum deviation is reached at minimum pulse intervals of approximately 10 times smaller than those obtained for CDD 4 . 173 The results are qualitatively equivalent for the 2-SAT problem, as displayed in Figs. 5.9 and 5.10, for CDD 4 and QDD 14 , respectively. The minimum deviation and opt decrease with decreasing spectral cutoff for CDD 4 , while appreciable changes in the minimumD are not observed for QDD 14 . Again, large spectral cutoffs require mini- mum pulse intervals that are longer than 1= ( opt 10= for = 200 2SAT min ) to acquire the total time necessary to reach minimum deviation. Self-averaging effects are more noticeable for 2-SAT and contribute to the improvement in CDD 4 performance seen in Fig. 5.9 for = 200 2SAT min since opt > 1. In contrast to the Grover case, the values of opt are comparable for both sequences, which is consistent with the extended ideal- like AQC evolution observed for increasing correlation time in Fig. 5.6. Furthermore, a distinction can be made between Grover and 2-SAT with respect to their susceptibility to variations in. CDD 4 -protected AQC significantly improves with increasing correla- tion time for 2-SAT, whereas the performance variations are less dramatic for the Grover case. 5.4.4 Effect ofH ad (s)-induced Noise An additional contributor to the favorable performance of CDD is the scheme’s effec- tiveness in suppressing decoherence associated indirectly with the time-dependent inter- polation functionf(s). AlthoughH ad (s) is, of course, by design not an error-generating term, at second order in the Magnus expansion it couples toH err via a double commuta- tor of the form [[H ad (s);H err (s)];H err (s)]. Focusing on the case where the error model is time-independent, j (s) , the interplay between DD andf(s) for Grover’s problem is analyzed, where the interpolation function is non-linear. In Figs. 5.11 and 5.12, the performance of CDD and QDD-protected AQC is shown, respectively, for = 10 2 . The faulty evolution only coincides with the ideal case up to T 4= G min , thereafter diverging rapidly. Protecting AQC by CDD does not appear to 174 á á á á á á á á á á á á á á á á á áá á á á á á á á á á á á á á á á á á á á áááááááááááááááááááááááááááááááááá æ æ æ æ æ æ æ ææ æ æ æ æ ææ æ æ æ æ æ ææ æ æ æ ææ æ æ æææ æ æ æ ææææææææææææææ æ æ æ æ æææ æ æ æ æ æ ææææ æ æ æ æ æ æ ì ì ì ì ì ì ì ì ì ì ì ì ì ìì ì ì ì ì ì ì ì ì ì ì ì ìì ììì ì ì ì ì ì ìì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ìì ì ì ì ì ì ì ì ì ì ò ò ò ò ò ò ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò ò ò òòò ò ò ò ò òòòòòòòòò ò ò ò ò òòòòòòòò ò ò ò ò òòò à à à à à à à à à à à à à àà à à à à à à à à à à àà à à à à à à à à à àà à à à à à à à à à àà à à à à àà à à à ààà à à à àà à à ààà 0 5 10 15 20 25 30 35 0.0 0.2 0.4 0.6 0.8 1.0 T Hunits of 1D min G L D Tr HTL à CDD 4 ò CDD 3 ì CDD 2 æ CDD 1 á Faulty Ideal Figure 5.11: Performance of CDD for a constant error Hamiltonian [ j (s) 10 2 8;j in Eq. (2) (main text)]. The curves for ideal evolution, faulty evolution, and CDD 4 overlap up to T 4= G min , after which CDD 4 continues to track the ideal evolution essentially perfectly throughout the simulated range. be beneficial untill = 4, where CDD 4 -protected evolution closely coincides with ideal evolution throughout the simulated range of T . The results for QDD-protected AQC are quite different in that increasing the number of pulses does not necessarily prolong ideal evolution, nor reduce the minimum deviation. Significant deviations from ideal evolution are observed for allM6= 15 whenT 25= G min , whileM = 6 appears to be optimal forT 25= G min . Interestingly, the evolution generated byM = 15 produces the longest closed-system-like evolution and the most considerable divergence from the ideal case. The fact that the optimal interpolation function for Grover’s search is a non-linear function most likely dictates the bias in DD scheme. The functionf(s) essentially plays the role of a time-dependent noise process that increases rapidly throughout the AQC 175 à à à à à à à à à à à à à à à à à àà à à à à à à à à à à à à à à à à à à à àààààààààààààààààààààààààààààààààà æ æ æ æ æ æ æ ææ æ æ æ æ ææ æ æ æ æ æ ææ æ æ æ ææ æ æ æææ æ æ æ ææææææææææææææ æ æ æ æ æææ æ æ æ æ æ ææææ æ æ æ æ æ æ ì ì ì ì ì ì ì ì ì ì ì ì ì ìì ì ì ì ì ì ìì ì ì ìì ì ì ì ì ìì ì ì ì ì ì ìììììì ì ì ììì ì ì ì ì ìì ì ì ì ì ì ìì ì ì ì ì ìì ì ì ì ì ò ò ò ò ò ò ò ò ò ò ò ò ò òò ò ò ò ò òò ò ò ò òò ò ò ò ò ò ò ò ò ò ò ò òò ò ò ò òò ò ò ò ò òòòòòòòòòòò ò ò ò ò ò ò òò ò ò ò ò á á á á á á á á á á á á á áá á á á á á á á á á á á ááááááááá á á á á á á á áá á á á ááá á á á á áá á á á ááá á á á áá á á á á ç ç ç ç ç ç ç ç ç ç ç ç ç çç ç ç ç ç ç ç ç ç ç ç ç ççççç ç ç ç ç ç çç ç ç ç ç ç ç ç ççç ç ç çççççç ç ç ç ç ç ççççççççç ç í í í í í í í í í í í í í íí í í í í í í í í í í íí í í í í í í í í í íí í í í í íí í í í í í í í í í í í í í í í í í í í í íí í í í í í ó ó ó ó ó ó ó ó ó ó ó ó ó óó ó ó ó ó ó ó ó ó ó ó óó ó ó ó ó ó ó ó ó ó óó ó ó ó ó ó ó ó ó ó óóó óóó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó óó ó 0 5 10 15 20 25 30 35 0.0 0.2 0.4 0.6 0.8 1.0 T Hunits of 1D min G L D Tr HTL ó QDD 15 í QDD 14 ç QDD 7 á QDD 6 ò QDD 4 ì QDD 3 æ QDD 1 à Faulty Ideal Figure 5.12: As in Fig. 5.11, for QDD withM =f1; 3; 4; 6; 7; 14; 15g. QDD-protected evolution deviates minimally from ideal evolution for M = 15 up to T 25= G min , thereafterM = 6; 7 are the optimal sequence orders. Unlike the CDD case, increasing the number of pulses does not result in closed-system-like behavior over the entire range ofT considered. M = 14; 15 evolution closely coincides with ideal evolution more so than any otherM values, however also diverges faster than all order sequence orders at criticalT values. evolution, except near the minimum energy gap. As with the time-dependent j (t) func- tions, CDD is better equipped to deal withf(s)-induced decoherence due to the recur- sive error suppression provided by each sub-level of concatenation. In contrast, UDD protocols do not complete the error averaging process until the last pulse is applied: not all error channels are addressed by each nested sequence. The procedure appears to be most detrimental for increasing sequence orders at relatively large values of T , which implies that the non-linearity off(s) is generating effective Hamiltonians at each nested sub-level that cannot be averaged out completely. This result is quite similar to the spectral cutoff analysis shown in Fig. 5.8 for QDD 15 , where minimum deviation grew considerably for short correlation times, most notably at = 20 G min , due to rapid 176 fluctuations in j (s) that lead to additional unsuppressed decoherence at the end of the QDD evolution. 5.5 Summary of Results Here a high-order DD-based strategy for protected open-system AQC was introduced and shown using numerical simulations to be capable of achieving high fidelities for a random-unitary map model. At high enough concatenation level, the CDD-based pro- tection strategy achieves fidelities which are essentially indistinguishable from closed- system adiabatic evolution, for two algorithms associated with first and second order quantum phase transitions. CDD outperforms QDD in these simulations, an effect which can be attributed to CDD’s ability to address all error types throughout the evolution, as opposed to each nested sequence combating only a subset of errors and only reaching full decoupling potential at the end of the sequence as in the case of QDD. 177 Chapter 6 Conclusions and Future Work Various aspects of quantum error suppression via dynamical decoupling have been dis- cussed in this work in order to facilitate, not only a theoretical understanding of the technique, but also pitfalls which may arise and hinder the methods ability to effectively suppress decoherence in physically realizable systems. Furthermore, dynamical decou- pling has been examined in the context of adiabatic quantum computation to exhibit how one could conceivably protect continuous quantum evolution from decoherence; thus, displaying the versatility of the approach when utilized in conjunction with quantum error correction. In concluding this work, a summary of the key results and future work is given below. 6.1 Analysis of Quadratic DD Quadratic DD (QDD) is an attractive approach for combating general single-qubit deco- herence due to its favorable decoupling efficiency and it’s ability to address non-uniform decoherence. While much is known about the minimum decoupling order of this method, full characterization of the error suppression capabilities of QDD were not fully understood prior to this study. By varying the number of pulses in each nested UDD sequence and extracting the errors associated with each system basis operator the relationship between the number of pulses and decoupling order was effectively determined. The results shown here clearly suggest that the method discussed can be extended to more general NUDD settings to characterize the error suppression properties 178 as a function of the number of pulses in each nested UDD sequence; hence, providing a complete understanding of non-uniform UDD-based schemes. 6.2 Robust Dynamical Decoupling Decoherence due to system-environment interactions presents one of the most challeng- ing hurdles to overcome in order to achieve high-fidelity quantum computation. While dynamical decoupling seeks to address this issue by suppressing such interactions, addi- tional errors introduced by intrinsic control pulse imperfections are inevitable. Physical implementation and realization of dynamical decoupling as a viable method for quan- tum protection therefore requires simultaneous suppression of decoherence generated by faulty pulses in addition to unwanted interactions. Unfortunately, many methods which exhibit favorable error suppression for system-environment interactions tend to lack robustness to pulse imperfections. Here it is shown that such issues can be dealt with by employing genetic algorithms and symmetrization techniques to construct robust DD sequences. The success of the genetic algorithm approach in locating sequences which suggest arbitrary order error suppression in the presence of rotation angle errors, specifically, raises questions regarding what forms of additional errors may be effec- tively addressed by the optimization algorithm. While already considering such errors as finite amplitude pulse errors in this study, future work would include rotation axis errors which are prevalent in QC systems such as Nitrogen-Vacancy (NV) centers in diamond. Furthermore, one can build upon the results shown here by considering addi- tional error models. Dynamical decoupling tends to be less effective when the spectral density of the environment contains a significant high frequency contribution. Locating optimized DD sequences in this regime would be truly beneficial for systems which are 179 susceptible to such environmental conditions, e.g., NV centers. Consideration of vari- ous error models and control error specifications are essential towards understanding the limitations of DD both from the perspective of candidate physical systems and reduction in pulse imperfection propagation. 6.3 DD-Protection for Adiabatic Quantum Computa- tion Utilizing a hybrid technique involving dynamical decoupling and quantum error cor- rection codes it was shown that deterministic DD sequences can be utilized to generate closed system-like AQC evolution. The particular error model considered displayed preference toward CDD due to its concatenated structure which effectively addressed all terms in the error Hamiltonian in each level of concatenation. Combating decoher- ence via QDD did not appear beneficial due to its nested structure which combats only a subset of error-producing terms at each nested UDD level. Coupled with the fact the QDD does not reach its full error suppression capabilities until the end of the total cycle time, as discussed in Chapter 3, increasing the number of pulses results in an overall accumulation of error. Although the pulse-interval optimized scheme did not produce favorable results here it is likely that one may be able to recover high efficiency for such schemes if the optimization accounts for the location of the minimum energy gap between the ground and first excited state of the total Hamiltonian. Since this location is where a high probability of excitation can be expected, both in the closed and open sys- tem case, one would expect that a majority of the DD pulses should be centered around the location of the minimum energy gap. A possible attractive feature of this approach is a reduced overhead in the number of pulses required to achieve a particular accuracy for the AQC algorithm. In consideration of more realistic physical settings where the exact 180 spectrum of the Hamiltonian may not be tractable due to large Hilbert space dimension or incomplete noise characterization, the method may be extended by introducing feed- back measurements. Recently, it has been shown that weak measurements of the ground state energy curvature for superconducting flux qubit systems are achievable. Noting that the curvature is maximum at the minimum energy gap, one may exploit this mea- surement and utilize it to determine where a majority of the DD pulses are to be applied; essentially a feedback-assisted DD protection approach to open system AQC. 181 Appendix A Numerical Techniques A.1 Genetic Algorithm for Optimized DD Genetic Algorithms represent an approach to optimization problems based on the prop- erties of natural evolution. Given an initial population and a definition of fitness, the algorithm simulates the processes of selection, reproduction, and mutation in an attempt to locate the member in the population with the highest probability of survival. In regards to DD, the population can be thought of as a subset of all possible sequence configurations, where a configuration is specified by the order and types of pulses, for a given sequence lengthK. The member with the highest probability of survival is the sequence which maximally suppresses system-bath interactions with respect to a partic- ular distance measure. In the following subsections we outline the representation of the population and discuss how selection, reproduction, and mutations are implemented in the setting of DD optimization. A.1.1 Chromosome structure The canonical approach to GAs is to define a member of the population by a set of genes, loosely referred to as a chromosome. Each gene can be thought of as a parameter in the optimization problem which contributes in some way to the fitness, and therefore 182 the probability of selection, of the member. Defining a member in the population as a DD sequence, Eq. (4.7) is translated directly into its corresponding chromosome C () j =fP 1 ;P 2 ;:::;P K1 ;P K g; (A.1) representing thejth member in theth generation. The genes are given by the pulses in the sequence, therefore the number of genes increases with increasing sequence length. In general, a sequence and its corresponding chromosome do not have to be structurally equivalent. Later we will elaborate on why the naive translation of Eq. (A.1) is not favorable for DD optimization and discuss how it can be refined; however, for now Eq. (A.1) is adequate to describe each aspect of the algorithm outlined in the subsequent subsections. The population is given by the set of chromosomesfC () j g Q j=1 , where each C () j corresponds to a sequence U () j ( c ) and Q is the population size. The total number of possible sequence configurations,N (K), is determined by both the length of the sequence and the number of pulse types inD. The size of the sequence space grows exponentially with the length of the sequence,N (K) =jDj K , wherejDj is the number elements inD. The search space can be reduced by imposing the cyclic DD conditionU C ( c ) =I S , which is applicable for our focus on quantum memory preservation. The condition can be recast in the context of the search problem as K Y j=1 P ideal j /I S (A.2) on allC () j , where only the ideal, zero-width version of the pulse is used when finite- width or flip-angle error pulse profiles define V (t). Applying Eq. (A.2), the search 183 space is reduced toN R (K) =jDj K1 , where onlyjDj 1 of the original search space accounts for viable DD sequences. The initial population is chosen at random from the reduced search space, such that QN R (K). In general, the size of Q is somewhat arbitrary and expected to vary depending on the number of degrees of freedom specified by the problem. In the context of DD optimization, the size of the initial population will ultimately end up fixed for all sequence lengths due to the structure of the initial chromosomes; see Section A.1.5 for additional details. A.1.2 Selection Associated with each chromosome C () j is a selection probability p () j . This quantity defines the probability of being selected for reproduction in generation and is given by p () j = q () j P i q () i ; (A.3) where q () j = log 10 D (;j) K represents the performance, or fitness, of the jth sequence. Here, the cyclic DD condition is imposed as well, G = I S , and denote D (;j) K D (;j) K (U( c );I S ). The logarithm is included in the definition of the fitness due to com- plications with the selection probability that are attributed to the extreme sensitivity of Eq. (2.72), and any distance measure for that matter, to sequence variations. The exchange of a single pulse in a sequence with any other member of the decoupling set can result in a change in performance up to many orders of magnitude. Since the reduced search space does not eliminate all poorly performing sequences, the fitness can vary greatly in any generation. As a result, there is a reduced contribution of high performance sequences in the selection probability distribution. The logarithm counter- acts this issue by increasing the resolution of the selection probability. 184 A.1.3 Crossover In each generation 2Q offspring are produced from the current population. Members of the population are chosen for reproduction based on their probability of selection. The selection process is constrained such that the crossover procedure only occurs between two distinct members of the population. Members with a high probability of selection not only possess a higher likelihood of reproduction, but also have a higher probability of reproducing with multiple members in a single generation since each crossover is an independent event. Reproduction is implemented by a crossover between two members in the popula- tion, yielding two offspring. To best illustrate the crossover, consider the two chromo- somes C () j = fP 1 ;:::;P i ;:::;P k g; (A.4) C () j 0 = fR 1 ;:::;R i ;:::;R k g; (A.5) whereP i ;R i 2D. The offspring are created by splicing the parent chromosomes at a location chosen at random, where each pulse location has an equal probability of being chosen. Taking the splice point to be theith pulse site, the resulting offspring are ~ C () j = fP 1 ;:::;P i ;R i+1 ;:::;R k g; (A.6) ~ C () j 0 = fR 1 ;:::;R i ;P i+1 ;:::;P k g: (A.7) It is essential that the offspring still satisfy Eq. (A.2), however it is not necessarily true that each is guarenteed to do so. If the DD condition is not satisfied, the pulse located at 185 the splice point is manipulated until the condition is satisfied. For example, if ~ C () j does not fulfill the DD condition then it is transformed to ~ ~ C () j =fP 1 ;:::;P 0 i ;R i+1 ;:::;R k g; (A.8) where it now is in agreement with Eq. (A.2) and P 0 i 2D. In the situation that ~ ~ C () j cannot be found, the splice point is chosen again and the process is repeated until the proper offspring are created. The condition set forth by Eq. (A.2) restricts the crossover process and in some cases does not allow it at all. By permitting the manipulation of the pulse at the splice point, it is ensured that only the probability of selection dictates reproduction. It is always possible to construct offspring from the above process, since there is no constraint on yielding offspring which are identical to the parent chromosomes. Thus, every set of parent chromosomes is guaranteed to produce some form of offspring. Upon producing the 2Q offspring, the bestQ=4 parents and 3Q=4 offspring are taken to be the new population. The partitioning was chosen based on what appeared to be the most beneficial to the convergence of the algorithm. No duplicate sequences are allowed in the new population, however if the updated population size is less thanQ then new members are generated at random from within the reduced search space. A.1.4 Mutation After reproduction, the new population composed ofQ=4 parents and 3Q=4 offspring is used to create 2Q mutated sequences,Q single-site andQ double-site. Every sequence in the population participates in both mutation processes, however only a portion of the mutated sequences is retained for the succeeding generation. 186 Single-site mutations are performed by choosing a pulse site at random and altering the pulse until Eq. (A.2) is again satisfied. If the DD condition is unsatisfiable then the original pulse is replaced and a different pulse site is chosen. It is possible that only the original configuration satisfies Eq. (A.2). In this situation the mutated member is simply a duplicate sequence, therefore it is discarded. Double-site mutations correspond to linked single-site mutations. The process begins in a similar manner by choosing a pulse site at random, say theith site with pulse P i . An additional pulse site is now chosen at random from the set of pulse sites which have pulse types equivalent toP i , e.g., thejth site. BothP i andP j are updated simul- taneously until the DD condition is again satisfied. If an additional pulse site does not exist, then the initial site is re-selected and the double-site mutation process is repeated. As in the case of the single-site mutation, if the DD condition cannot be satisfied then the mutated sequence is accepted as the original configuration and discarded. At the conclusion of the two mutations, a portion of the parent, offspring, and mutated sequences will comprise the new population. Only the sequences that have the highest fitness with respect to Eq. (2.72) are desired from each division of the popu- lation. It is found that the bestQ=8 parent, 5Q=8 offspring,Q=8 single-site mutated, and Q=8 double-site mutated sequences comprise a favorable distribution for the new pop- ulation. Other distributions were considered such as taking the bestQ=4 of all mutated sequences, as well as different proportions of the offspring. However, no distribution appeared to yield a higher probability of optimal sequence convergence. A.1.5 Necessary Convergence Accelerators As noted above, single-site perturbations may result in large deviations in sequence performance. Hence, the logarithm was introduced to decrease the performance gap between poor- and well-performing sequences, thereby increasing the resolution of the 187 selection probability. However, this adjustment only proves to aid in optimal conver- gence for sequences comprised ofK < 16 pulses. This is evident from a simple com- parison between CDD and numerically located sequences atK = 16; 64; 256, where the numerically “optimal” sequences perform far worse than CDD. It is suspected that the local minima convergence is ultimately attributed to sig- nificant deviations in sequence performance that result in relatively large local min- ima traps. This complication is alleviated by introducing two convergence accelerators which act to reduce the size and presence of large traps, thereby smoothening what we refer to as the fitness landscape. Both accelerators are crucial for the algorithm to converge on global optima as the number of pulses increases beyondK = 16. Reducing Local Traps via Complexity Although the size ofN (K) is decreased by imposing the cyclic DD condition, the result- ing reduced search space,N R (K), maintains its exponential scaling in the number of control pulses. Hence, there is still a high probability of the subspace containing low- performance sequences that lead to large local traps. This issue is resolved by reducing the search space further and systematically increasing its size as the algorithm iterates, such that the search space at the termination of the algorithm isN R (K). The addi- tional reduction is achieved by constraining the complexity of the chromosome, thereby moderating the possible sequence configurations. Initially, each chromosome is chosen to represent the most elementary two- dimensional sequence C (l=0;=0) j =fP s 1 ;P s 2 g; (A.9) wherel=0 is the initial complexity index. The notationP s j denotes a pulseP j applied at the locations specified by the sets j . We chooses 1 ands 2 to contain only the odd and even pulse sites, respectively, for the initial population. This is a relevant construction 188 Figure A.1: Complexity-reduction protocol forK = 16. The upper (blue) curves denote the linking between odd pulse sites and the lower (red) curves denote even pulse site linking. The process begins with all odd and even pulses linked, then continues by removing links between every other even pulse site. Links are removed until all even sites are uncorrelated, thereafter the odd sites undergo the same process. since all known deterministic DD schemes utilizing fixed free intervals contain the same pulse at either every even or odd site [85, 76]. Moreover, it conveniently reduces the space to onlyjGj 2 sequence configurations for allK. For a general complexity indexl, the chromosome is defined as C (l;) j =fP s 1 ;P s 2 ;:::;P s ~ K(l) g; (A.10) where we require that ~ K(l) [ i=1 s i = all sites and ~ K(l) \ i=1 s i =; (A.11) 189 be satisfied so that only one control pulse is applied at each pulse site. The number of setsfs j g is determined by ~ K(l) = 8 > > > < > > > : 3 2 (l + 4 3 ) : l even 3 2 (l + 1) : l odd (A.12) for l = 0; 1; 2;:::;l max . At maximum complexity, l max , the most general sequence withinN R (K) is permitted. Hence, eachs j is a single element set containing only the jth pulse site. An example of the complexity-increase procedure is illustrated in Figure A.1 for K = 16. Note that at each level of complexity-increase we have chosen to remove constraints only pertaining to odd or even sites. The constraint between every other even site is removed until each even pulse site is independent, after which the same is performed on the odd sites. In contrast to Eq. (A.1), the number of elements in Eq. (A.10) increases as the algo- rithm iterates. It is important to note that this aspect does not imply an increase in the number of pulses, rather an increase in the permissible search space. This is an attrac- tive feature since it not only diminishes the presence of local traps, but also yields an initial set of sequence configurations which only scales quadratically injDj. Choosing sequences for the initial population is obviously much more favorable here since the space is drastically smaller thanN R (K). In principle, it may even be possible to choose the entire set as the initial population. For a single-qubit system subjected to ideal- pulses, we find that the complete initial set of configurations is indeed computationally convenient, consisting of only 16 possible configurations. Other pulse profiles lead to larger initial sets, but, remarkably, optimal sequence convergence is possible for initial populations of only 16 sequences. 190 Fitness annealing Substantial differences in sequence fitness manifest local traps in the fitness landscape. By decreasing the complexity of the chromosome only the probability of generating local traps is diminished. In order to control the relative differences between high- and low-performance sequences, we introduce an annealing process into the selection probability. Adopted from Ref. [160], the selection probability is redefined as p (l;) j (T ) = ~ q (l;) j (T ) P i ~ q (l;) i (T ) ; (A.13) [compare with Eq. (A.3)] such that ~ q (l;) j (T ) = exp q () j q () best T () ! : (A.14) The performance of the most fit member in theth generation is denoted byq () best and the temperature function is given by T () =T 0 T f T 0 =c 1 sin c : (A.15) The temperature function utilized here is a modified version of the one introduced in Ref. [160], where we have included the sinusoidal function to reduce the probability of local minima convergence asT () decreases from the initial temperatureT 0 to the final temperatureT f . The number of generations between these temperatures is dictated by the cutoff generation c , which is chosen based on the value of K. For large K, we pick c to be large as well since the annealing process is to accelerate global mini- mum convergence while reducing the probability of local minimum convergence. The 191 remaining parameters and are related to the amplitude and frequency of the oscilla- tions, respectively. Upon increasing the complexity indexl, the annealing process resets with an initial temperatureT 0 chosen so that all sequences in the current population have an equal likelihood of being chosen for reproduction. A.2 Extracting effective error Hamiltonian scaling numerically In Sec. 4.2.1, the scaling of the distance measureD K for each optimal sequence in the ideal pulse limit was discussed without direct calculation of the effective Hamiltonian. This scaling is obtained by assuming thatD K has the form D Kos O(J n J n N+1 d ); (A.16) where N is the decoupling order of the sequence and n J +n = N + 1. First, the decoupling order is determined by examining log 10 D K as a function of d , as this quan- tity scales linearly in d with a slope ofN + 1. The scaling of d is only dependent upon the decoupling order N, and therefore is independent of the relative magnitudes of J and. 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Abstract (if available)

Abstract Quantum computation (QC) relies on the ability to implement high-fidelity quantum gate operations and successfully preserve quantum state coherence. One of the most challenging obstacles for reliable QC is overcoming the inevitable interaction between a quantum system and its environment. Unwanted interactions result in decoherence processes that cause quantum states to deviate from a desired evolution, consequently leading to computational errors and loss of coherence. Dynamical decoupling (DD) is one such method, which seeks to attenuate the effects of decoherence by applying strong and expeditious control pulses solely to the system. Provided the pulses are applied over a time duration sufficiently shorter than the correlation time associated with the environment dynamics, DD effectively averages out undesirable interactions and preserves quantum states with a low probability of error, or fidelity loss. In this study various aspects of this approach are studied from sequence construction to applications of DD to protecting QC. First, a comprehensive examination of the error suppression properties of a near-optimal DD approach is given to understand the relationship between error suppression capabilities and the number of required DD control pulses in the case of ideal, instantaneous pulses. While such considerations are instructive for examining DD efficiency, i.e., performance vs the number of control pulses, high-fidelity DD in realizable systems is difficult to achieve due to intrinsic pulse imperfections which further contribute to decoherence. As a second consideration, it is shown how one can overcome this hurdle and achieve robustness and recover high-fidelity DD in the presence of faulty control pulses using Genetic Algorithm optimization and sequence symmetrization. Thirdly, to illustrate the implementation of DD in conjunction with QC, the utilization of DD and quantum error correction codes (QECCs) as a protection method for adiabatic quantum computing (AQC) is discussed. A performance comparison between two deterministic DD schemes is given, where preference towards one particular method is found due to sequence structure and procedure from which higher accuracy sequences are generated.

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Creator Quiroz, Gregory (author)

Core Title Towards robust dynamical decoupling and high fidelity adiabatic quantum computation

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Physics

Publication Date 12/09/2013

Defense Date 10/18/2013

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag adiabatic quantum computation,adiabatic quantum computing,decoherence,decoherence control,dynamical decoupling,noise mitigation,OAI-PMH Harvest,quantum computation,quantum control

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Advisor Lidar, Daniel A. (committee chair), Brun, Todd A. (committee member), Reichardt, Benjamin (committee member), Takahashi, Susumu (committee member), Zanardi, Paolo (committee member)

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