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Towards optimized dynamical error control and algorithms for quantum information processing

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Towards optimized dynamical error control and algorithms for quantum information processing

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Content TOWARDS OPTIMIZED DYNAMICAL ERROR CONTROL AND ALGORITHMS FOR QUANTUM INFORMATION PROCESSING by Wan-Jung Kuo 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) August 2012 Copyright 2012 Wan-Jung Kuo Dedication To my beloved parents, Jung-Fu Kuo and Li-Min Lee; my dearest sister, Chia-Wei; my dearest brother, Tsung-Han. ii Acknowledgments First and foremost, I owe my most profound gratitude to my advisor, Professor Daniel A. Lidar, for his generous support and valuable guidance during my Ph.D. study. This dissertation would not have been possible without his encouragement and help. I am really grateful for his understanding, patience, and his helpful career advice. I am greatly indebted to Alioscia Hamma and Ali Rezakhani for their guidance and advice. Working with them was a valuable experience for me. I am grateful to Professors Todd Brun, Paolo Zarnardi, and Edmond Jonckheere for all that I learned from them. My special thanks to Professors Werner Dappen, Stephan Haas, and Susumu Takahashi, for supervising my qualification and Ph.D. dissertation. I would also like to take the opportunity to give my special thanks to my master thesis advisor at National Taiwan University, Professor Yeong-Chuan Kao, who showed me the beauty of general relativity that drove me to take the path of theoretical physicists. I am blessed with having kind and generous friends and colleagues during my Ph.D. study. Thank to Kristen Pudenz, Soraya Taghavi, Zhihui Wang, Gerardo Paz, Gregory Quiroz, Milad Marvian, Siddharth Muthu Krishnan, Siddhartha Santra, Yueh-Cheng Kuo, Wei-Feng Tsai, Wan-Jen Huang, Alireza Shabani, Kaveh Khodjasteh for their friendship and useful suggestions. Last, but far from least, I owe my deepest gratitude and respect to my family for their sacrifice and their constant support. I am deeply grateful to my husband Bora for iii his love and understanding. Thanks to my lovely kitties, Kappa and Gamma, for making my life cheerful. iv Table of Contents Dedication ii Acknowledgments iii List of Tables vii List of Figures viii Abstract ix Chapter 1: Introduction to Dynamical Decoupling (DD) 1 1.1 General basic setting of dynamical decoupling . . . . . . . . . . . . . . 2 1.1.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Interaction picture . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.3 Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Ideal pulsed-DD-controls for preserving quantum memories . . . . . . . 7 1.2.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.2 Periodic DD (PDD) . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.3 Concatenated DD (CDD) . . . . . . . . . . . . . . . . . . . . . 11 Chapter 2: Nested UDD (NUDD): Universality Proof and Error Analysis 15 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.1 Uhrig DD (UDD) . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.2 Quadratic DD (QDD) . . . . . . . . . . . . . . . . . . . . . . . 17 2.1.3 Nested UDD (NUDD) . . . . . . . . . . . . . . . . . . . . . . 19 2.1.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 NUDD formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.1 NUDD pulse timing . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.2 (r 1 ,r 2 ,...,r ` )-type error . . . . . . . . . . . . . . . . . . . . . 22 2.2.3 (r 1 ,r 2 ,...,r ` )-type error modulation function . . . . . . . . . . 24 2.2.4 Dyson expansion of evolution operator in toggling frame . . . . 26 2.2.5 NUDD coefficients . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3 The performance of NUDD scheme . . . . . . . . . . . . . . . . . . . 28 v 2.3.1 The decoupling order of each error type . . . . . . . . . . . . . 28 2.3.2 The overall performance of NUDD scheme . . . . . . . . . . . 35 2.4 Proof of NUDD Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4.1 Synopsis of the proof . . . . . . . . . . . . . . . . . . . . . . . 37 2.4.2 The outer-layer interval decomposition form ofF (n) ~ r ` . . . . . . 40 2.4.3 The performance of the inner (`−1)-layer NUDD . . . . . . . 42 2.4.4 One more order suppression of(~ r `−1 ,1)-type error . . . . . . . 43 2.4.5 Fourier expansion after linear change of variables . . . . . . . . 45 2.4.6 Integrating out the firstn−1 integrals ofF (n) ~ r ` . . . . . . . . . . 48 2.4.7 The vanishing of the(~ v 0 `−1 ,1)-type error . . . . . . . . . . . . . 51 2.5 Numerical results for 4-layer NUDD sequence . . . . . . . . . . . . . . 53 2.5.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.5.2 ~ r 4 = (r 1 ,r 2 ,r 3 ,r 4 )-type error . . . . . . . . . . . . . . . . . . . 55 2.5.3 Decoupling order of each error type . . . . . . . . . . . . . . . 57 2.5.4 Overall decoupling order . . . . . . . . . . . . . . . . . . . . . 58 2.5.5 Comparison of theoretical predictions with numerical results . . 58 2.5.6 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.7 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.7.1 The outermost UDD interval decomposition . . . . . . . . . . . 65 2.7.2 F (n) ~ r ` in terms ofF (n 0 ) ~ r 0 `−1 . . . . . . . . . . . . . . . . . . . . . . 70 2.7.3 Fourier expansions after linear change of variable . . . . . . . . 75 2.7.4 min[{ ˇ N ~ r `−1 ∈~ v 1 `−1 }] =N k 0 <` o min . . . . . . . . . . . . . . . . . . . . 78 Chapter 3: Adiabatic Quantum Brachistochrone 80 3.1 Introduction of adiabatic quantum computing (AQC) . . . . . . . . . . 80 3.2 Time optimal AQC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.3 Geometrization of AQC . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.3.1 Riemannian geometry of the whole Hamiltonian space . . . . . 86 3.4 Adiabatic quantum search . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.4.1 Adiabatic search on 1-dimensional manifold . . . . . . . . . . . 88 3.4.2 Adiabatic search on 2-dimensional manifold . . . . . . . . . . . 90 3.4.3 Performance comparison between 1-d and 2-d geodesics . . . . 93 3.4.4 Adiabatic search on 4-dimensional manifold . . . . . . . . . . . 97 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Chapter 4: Conclusions 101 References 103 vi List of Tables 2.1 Performance comparison of UDD, PDD, and CDD schemes . . . . . . 16 2.2 Performance comparison of QDD, XY4, CDD, and CUDD schemes . . 18 2.3 Suppression characteristics of QDD . . . . . . . . . . . . . . . . . . . 32 2.4 The function types of error modulation functions . . . . . . . . . . . . 47 2.5 The function types of the resulting integrand ofF (n) ~ r ` . . . . . . . . . . . 49 2.6 16 error types~ r 4 for the 4-layer NUDD scheme. . . . . . . . . . . . . . 55 2.7 Error types for the 4-layer NUDD with the MOOS 1 . . . . . . . . . . 56 2.8 Error types for the 4-layer NUDD with the MOOS 2 . . . . . . . . . . 57 2.9 4-layer NUDD withN 1 = 2,N 2 = 4,N 3 = 6,N 4 = 8 . . . . . . . . . . 59 2.10 4-layer NUDD withN 1 = 2,N 2 = 4,N 3 = 6,N 4 = 3 . . . . . . . . . . 59 2.11 4-layer NUDD withN 1 = 7,N 2 = 5,N 3 = 3,N 4 = 1 . . . . . . . . . . 60 2.12 4-layer NUDD withN 1 = 2,N 2 = 4,N 3 = 1,N 4 = 6 . . . . . . . . . . 60 2.13 4-layer NUDD withN 1 = 1,N 2 = 3,N 3 = 5,N 4 = 7 . . . . . . . . . . 62 2.14 4-layer NUDD withN 1 = 7,N 2 = 3,N 3 = 5,N 4 = 1 . . . . . . . . . . 62 vii List of Figures 3.1 Curvature, RC, 2-d QAB . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.2 Curvatures along RC and 2-d QAB . . . . . . . . . . . . . . . . . . . . 93 3.3 The speedup vs the number of qubits . . . . . . . . . . . . . . . . . . . 94 3.4 Performance between RC and 2-d QAB for3-qubit adiabatic search . . 96 3.5 Performance between RC and 2-d QAB for6-qubit adiabatic search . . 96 3.6 Performance between RC, 2-d, and 4-d QABs for1 qubit case . . . . . 99 viii Abstract Two topics in the field of quantum information processing, optimized dynamical error suppression and quantum algorithms, are considered here. The computational errors induced by the surrounding environment is one of the main obstacles in building a quantum computer. Engineering powerful techniques to combat errors in quantum devices is highly demanding. In the first part of this thesis, I focus on one quantum error correction technique, dynamical decoupling (DD), introduced in Chapter 1. Chapter 2 is dedicated to nested UDD (NUDD), a highly efficient decou- pling scheme that utilizes the decoupling characteristics of UDD by multi-layer nesting. UDD (1-layer NUDD) is an optimal DD method for eliminating single-qubit general dephasing, and QDD (2-layer NUDD) is a near-optimal DD method for eliminating one qubit general decoherence. I present a rigorous analytical proof of the performance and universality of QDD/NUDD, and obtain an explicit formula for the decoupling order of each error type, which elucidates the relationship between the error type and characteris- tics of NUDD. From the explicit formula, a NUDD scheme can be designed accordingly such that optimal efficiency of NUDD is achieved. Moreover, the highly efficient error cancellation mechanism is revealed by the analysis. The proof of QDD has been pub- lished in [31], and the proof of NUDD will be submitted for publication shortly. Chapter 3 is devoted to the Adiabatic Quantum Computation (AQC). In this work (published in [44]), a general time-optimal strategy, which in principle can optimize ix any quantum adiabatic algorithm for which the gap is known or can be estimated, is formulated. In addition, I present a natural differential-geometric framework for AQC. x Chapter 1 Introduction to Dynamical Decoupling (DD) Performing computation and information tasks based on quantum mechanical systems is the fundamental study of quantum computation and quantum information [20, 38, 32]. To build a quantum information processor (QIP), one of the essential requirements is to be able to reliably manipulate quantum systems. However, the inevitable coupling between a quantum system and an uncontrollable environment, or bath, typically results in decoherence [11, 63], deviations from the intended dynamical evolution of the quan- tum system. Decoherence leads to computational errors, reduces the information pro- cessing capabilities, and is a serious obstacle on the path towards building fault tolerant (arbitrarily accurate) quantum computers [21]. Therefore, finding protection against decoherence for open quantum systems is essential in quantum information processing (QIP). The standard method used to this end is quantum error-correcting codes (QECCs) [46, 47, 29, 48, 21]. QECCs encodes the quantum states of the system by adding ancil- lary qubits to carry the redundant information of the original state in such a way that even if noise, namely, decoherence, corrupts some of the information, the errors can be detected by error syndrome measurements, and the state of the original system can be recovered accordingly. However, in order to keep the error rates of logical gates and memory below a certain threshold, the spatial and temporal resource requirements for fault tolerant quantum error correction (FT-QEC) are really daunting [14, 42, 30, 3]. 1 Another powerful technique for decoherence control in QIP, adapted from nuclear magnetic resonance (NMR) refocusing techniques [24, 23], is dynamical decoupling (DD) [60]. In the conventional setting of DD, the unwanted system-bath interaction is mitigated through the application of a predetermined sequence of short and strong pulses (unitary operations), acting purely on the system [54, 53]. In more general terms, any purely Hamiltonian (open-loop) control engineered for enforcing active error cancella- tion is referred to as a DD scheme. The advantage of DD is that it bypasses the need for measurement, feedback, or encoding overhead implicit in closed-loop QEC. Therefore, DD is a substantially less resource intensive method for decoherence suppression. 1.1 General basic setting of dynamical decoupling In this section, we shall lay out the general formula for incorporating dynamical Hamil- tonian control into arbitrary gates. 1.1.1 Formulation DenoteS as the target quantum system which could consist of one qubit or many qubits and B as its quantum bath, the noise environment. Suppose H S is the Hamiltonian that implements the desired quantum gateU S (T) at timeT on the target system in the absence of the noise. The presence of the system-bath couplingH SB (t) which could be time-dependent induces additional unwanted non-trivial actions on the system, resulting in decoherence. To counteract the damage caused byH SB (t), we apply a DD scheme realized via a time-dependent system-only HamiltonianH c (t), designed for refocusing 2 the evolution towards the ideal intended one. The overall Hamiltonian with DD that governs the joint evolution of the system and the bath is given by H(t) =H S +H B (t)+H SB (t)+H c (t) (1.1) whereH B (t) acts purely on the bath. Note that every term in the above equation is so far general and no details have been specified yet. The unitary evolution operator or the propagator which describes how the quantum state of the joint system changes with time satisfies the following Schrodinger equation with initial condition, dU(t) dt =−iH(t)U(t), U(0) =I S ⊗I B , (1.2) where we assume units such that the reduced Planck constant~ = 1. If one can engineer the DD controlH c (t) such that the overall unitary evolution is equivalent to the gate operation up to an error of orderT N+1 , i.e., U(T) =U S (T)⊗B 0 +O(T N+1 ) (1.3) whereB 0 is an arbitrary bath operator, then we say that this DD control decouples the system from the environment to theN th order, namely, it achievesN th order decoupling. 3 1.1.2 Interaction picture To analyze the noise suppression performance of external DD controlH c (t), it is usually convenient to work in the interaction picture with respect to the joint HamiltonianH S + H c (t) which generatesU S+c (t). The propagator in this interaction picture e U(t)≡U S+c (t) † U(t) (1.4) satisfies the same linear homogeneous ordinary differential equation of the first order as Eq. (1.2) subjecting to the same boundary condition, d e U(t) dt =−i e H(t) e U(t), e U(0) =I S ⊗I B , (1.5) with the new Hamiltonian e H(t) ≡ U S+c (t) † (H B (t)+H SB (t))U S+c (t) (1.6) = H B (t)+ e H SB (t) (1.7) where the property[H S +H c (t),H B (t)] = 0 is being used in the second equality. If H c (t) is devised such that U S+c (T) coincides with the intended gate operation U S (T) at the final time, namely, U S+c (T) =U S (T), (1.8) then we have U(T) =U S (T) e U(T) (1.9) and thereby the effectiveness of DD for noise suppression is reflected by how close e U(T) is toI S up to an arbitrary bath operator. 4 1.1.3 Tools In order to evaluate how good the performance of a given DD sequence is, we need to analyze the distance between e U(T) andI S up to an arbitrary bath operator. Hence, the solution of e U(T) is required. Two known perturbative power series used for approxi- mating e U(T) are explained next with some remarks about their relative advantages and disadvantages. Dyson expansion The usual treatment for solving quantum evolution e U(T) is to solve the following inte- gral equation which is equivalent to Eq. (1.5), e U(T) =I−i Z T 0 e H(t) e U(t)dt. (1.10) A formal solution of e U(T) can be built up by iteration of Eq. (1.10) which leads to the known Dyson series [39], e U(T) =I + ∞ X n=0 (−iT) n Z 1 0 dη n Z ηn 0 dη n−1 ... Z η 2 0 dη 1 e H(η n )... e H(η 1 ) (1.11) with normalized time variableη≡ t T . Formally e U(T) is expressed as the time-ordered exponential e U(T) =T exp[−i Z T 0 e H(t)dt] (1.12) whereT denotes time-ordering. Note that in terms of the normalized time variableη, it is clear that e U(T) in Eq. (1.11) is a power series in the total timeT , where each term is a time-ordered product. One advantage of the Dyson expansion is that the convergence of Dyson’s series is guaranteed for any finite time interval if e H(t) is a bounded operator. 5 Magnus expansion The solution of e U(T) can be also expressed in a matrix exponential representation e U(T) =e −iH eff (T) (1.13) where the matrix in the exponent has a series expansion referred to as the Magnus expan- sion [10] H eff (T) = ∞ X n=0 ¯ H (n) (T) (1.14) where ¯ H (n) (T) are given as multiple integrals of nested commutators. For illustration, the three first terms of that series: ¯ H (1) (T) = Z T 0 dt 1 e H(t 1 ) (1.15) ¯ H (2) (T) = −i 2 Z T 0 dt 2 Z t 2 0 dt 1 [ e H(t 2 ), e H(t 1 )] (1.16) ¯ H (3) (T) = −1 6 Z T 0 dt 3 Z t 2 0 dt 2 Z t 2 0 dt 1 ([ e H(t 3 ),[ e H(t 2 ), e H(t 1 )]]+[ e H(t 1 ),[ e H(t 2 ), e H(t 3 )]]). (1.17) For higher orders, the computation becomes increasingly more complicated. Note that the superscriptn in ¯ H (n) indicates that ¯ H (n) consists of an nested integrals and therefore is of the order ofT n . The most appealing characteristic of the Magnus expansion is that the approxima- tion of e U(T) given by truncating at any order of the Magnus series preserves its uni- tary property (probability conservation), which is not shared by the truncation of the Dyson expansion. However, our purpose is to estimate the performance of a DD scheme 6 through the Dyson or Magnus expansions of e U(T) and does not intend to really approxi- mate e U(T) by those perturbative expansions. Therefore, preserving the unitary character of e U(T) is not necessary in that regard. There are two disadvantages of the Magnus expansion compared to the Dyson series. One is the difficulty in constructing explicit higher order terms and the other is that the validity, i.e. existence and convergence, of the Magnus expansion is subject to the condition [10] Z T 0 k e H(t)kdt≤ 1.086869 μ (1.18) where0<μ≤ 1 andk.k is an operator norm, which we require to satisfy the so-called submultiplicative property,kABk≤kAkkBk. 1.2 Ideal pulsed-DD-controls for preserving quantum memories Following the formulation in the previous section, we focus on a specific category of DD schemes which consists of a sequence of infinitely strong and instantaneous pulses (Bang-Bang control) for long-time storage of quantum memories. In practice, a real pulse is not infinitely strong and not instantaneous, i.e., it can only have a finite ampli- tude and finite pulse width. However, for the purpose of demonstrating how DD control averages out the undesired system-bath couplings, analysis with ideal pulses is easier, and can give a good approximation as long as the effect of the non-ideal (finite amplitude and finite pulse width) pulses is negligible. 7 1.2.1 Formulation The goal on which we focus here is to prolong the coherence time of the target quantum system which is essential in QIP. From the language of the previous section, the intended gate we would like to apply reliably and coherently is the NOOP gate,U S (T) =I S with H S = 0, which realizes the no-operation on the quantum system. The pure bath term H B (t) and the system-bath couplingH SB (t) remains as general as Eq. (1.1). However, the type of DD schemes we consider to apply is a sequence of ideal pulses, i.e. infinitely strong and instantaneous pulses, such as H c (t) = N+1 X j=1 δ(t−t j )H p j . (1.19) From Eq. (1.19), the control Hamiltonian is infinitely strong at instants t j and zero elsewhere. The overall unitary evolution under the ideal pulsed DD control can be decomposed by using the composition property of the unitary propagator as follows, U(T,0) =P N+1 U 0 (T,t N )P N ···P 2 U 0 (t 2 ,t 1 )P 1 U 0 (t 1 ,0), (1.20) whereU 0 is the free evolution generated byH B (t)+H SB (t) and the evolution at instant t j is P j ≡e i Rt j +0 + t j−0 + dtδ(t−t j )Hp j =e iHp j (1.21) whereH B (t)+H SB (t) is negligible compare to the dominant termδ(t−t j )H p j . 8 The joint interaction pictureU S+c (t) in Sec. 1.1.2 for the case of the NOOP gate, U S =I, reduces to the interaction picture with respect to the control HamiltonianH c (t) only, namelyU c (t) known as the toggling frame. Explicitly,U c (t) is expressed as U c (t) =P j ...P 1 P 0 t∈ (t j+1 ,t j ] (1.22) whereP 0 ≡I,t 0 ≡ 0, andt N+1 ≡T . Following Eq. (1.8),H c (t) is to be designed to meet the following constraint at the final timeT , U c (T) =P N+1 ...P 1 P 0 =U S (T) =I, (1.23) so that the toggling-frame time evolution operator coincides with the overall propagator at timeT , e U(T) =U c (T) † U(T) =U(T). (1.24) The toggling-frame Hamiltonian which generates e U(t) reads e H(t) = H B (t)+U c (t) † H SB (t)U c (t) = H B (t)+(Π j k=0 P k ) † H SB (t)(Π j k=0 P k ). t∈ (t j+1 ,t j ]. (1.25) Once the explicit form of e H(t) is obtained, the distance between e U(t) and I S can be evaluated by Dyson’s or Magnus’ expansions of e U(t). For example, the first order term for both Dyson’s and Magnus’ expansions of e U(t) Eqs. (1.11) and (1.17) happens to be the same, Z T 0 dt e H(t) =H B T + N+1 X j=1 τ j (Π j−1 k=0 P k ) † H SB (t)(Π j−1 k=0 P k ) (1.26) 9 where τ j = t j − t j−1 are pulse intervals. If P N+1 j=1 τ j (Π j−1 k=0 P k ) † H SB (t)(Π j−1 k=0 P k ) is proven to be zero, then the first order of e U(t) is a pure bath term H B T which has no effect on our target quantum system. Then we say that the DD pulse sequence,{P j } N+1 j=1 or Eq. (1.19), decouple the noiseH SB (t) at least to the first order. The performance of two typical DD sequences, PDD and CDD, shall be demon- strated as examples as in the following. 1.2.2 Periodic DD (PDD) The simplest possible DD protocol, Periodic DD (PDD) [54, 6, 16, 62, 53], has been shown to suppress one qubit general static dephasing due to coupling to an arbitrary quantum bath,σ z ⊗B z , up to the first order in the Magnus or Dyson expansions with respect to the total sequence durationT by using equidistant idealX-typeπ-pulses. Specifically, the overall evolution propagator under PDD control is U(T,0) =XU 0 (T, T 2 )XU 0 ( T 2 ,0), (1.27) where U 0 is the free evolution generated by the static pure bath and dephasing, I S ⊗ B 0 +σ z ⊗B z . Recalling Eq. (1.21), X = e iπ σx 2 suddenly rotates the state about the x-axis byπ angles at the pulse timings, and is therefore called an idealX-typeπ-pulse. From Eq. (1.22) for this case, the control propagatorU c (t) isI andX in the first and second pulse interval, respectively, and satisfies the constraintU c (T) = XXI = I at final timeT . Hence, the first order term for both Dyson’s and Magnus’ expansions of the propagator in the toggling frame e U(t), (1.26), follows Z T 0 dt e H(t) = (I S ⊗B 0 )T + T 2 σ z ⊗B z + T 2 Xσ z X⊗B z = (I S ⊗B 0 )T + T 2 σ z ⊗B z − T 2 σ z ⊗B z 10 which gives rise to only a pure bath term (I S ⊗B 0 )T , which has no effect on the target system. Therefore,U(T) = e U(T) =I +(I S ⊗B 0 )T +O(T 2 ) indicates that the PDD sequence achieves first order decoupling from general static dephasing noise. As a matter of fact, PDD is not only applicable to the one qubit dephasing model but also to the multi-qubit linear dephasing model. Suppose the system consists of` qubits and suffers linear dephasing P ` i=1 σ (i) Z ⊗B (i) Z . It is easy to check that PDD with pulses X (1) ⊗X (2) ⊗...X (`) also decouples the linear dephasing to the first order. 1.2.3 Concatenated DD (CDD) Concatenated DD (CDD) proposed in [27, 28] recursively embeds a given basic DD sequence into itself, providing a systematic way to build up DD schemes to achieve better performance. For an example, concatenating the X-type PDD with the Z-type PDD as follows, U(T) = XU z ( T 2 )XU z ( T 2 ), (1.28) U z ( T 2 ) = ZU 0 ( T 4 )ZU 0 ( T 4 ), (1.29) gives rise to the universal decoupling sequence, XY 4 , U(T) =YU 0 ( T 4 )XU 0 ( T 4 )YU 0 ( T 4 )XU 0 ( T 4 ), (1.30) which can eliminate arbitrary single-qubit decoherence, modeled as H =J 0 I⊗B 0 +J x σ x ⊗B x +J y σ y ⊗B y +J z σ z ⊗B z , (1.31) to the first order. The first order decoupling of XY 4 shall be shown explicitly as follows. 11 From Eq. (1.22), the control evolution operator for this XY 4 sequence reads U c (t) = I t∈ ( T 4 ,0], U c (t) = XI =X t∈ ( T 2 , T 4 ], U c (t) = YXI =Z t∈ ( 3T 4 , T 2 ], U c (t) = XYXI =Y t∈ (T, 3T 4 ], U c (T) = YXYXI =I (1.32) up to irrelevant±1 coefficients. Substituting Eq. (1.32) into Eq. (1.25), the toggling- frame Hamiltonian for XY 4 , which generates e U(t) Eq. (1.24), becomes e H 1 ≡ e H(t) = J 0 I⊗B 0 +J x σ x ⊗B x +J y σ y ⊗B y +J z σ z ⊗B z t∈ ( T 4 ,0], e H 2 ≡ e H(t) = J 0 I⊗B 0 +J x σ x ⊗B x −J y σ y ⊗B y −J z σ z ⊗B z t∈ ( T 2 , T 4 ], e H 3 ≡ e H(t) = J 0 I⊗B 0 −J x σ x ⊗B x −J y σ y ⊗B y +J z σ z ⊗B z t∈ ( 3T 4 , T 2 ], e H 4 ≡ e H(t) = J 0 I⊗B 0 −J x σ x ⊗B x +J y σ y ⊗B y −J z σ z ⊗B z t∈ (T, 3T 4 ]. (1.33) Therefore, the first order term of (Dyson’s or Magnus ) expansions of e U(t), Z T 0 dt e H(t) = T 4 4 X i=1 e H i =J 0 I⊗B 0 T, (1.34) results in a pure bath term, which indicates that the decoherence is suppressed at least to the first order. Moreover, one can construct a CDD scheme which employs XY 4 as a building block sequence. Denote the propagator under one layer of XY 4 control Eq. (1.30) as p 1 . 12 Then the propagator under CDD N control denoted as p N (T), where T is the duration of the whole CDD N cycle, containsN nested layers of XY 4 , and is constructed in the following recursive way, p N (T) =Y p N−1 ( T 4 )Xp N−1 ( T 4 )Y p N−1 ( T 4 )Xp N−1 ( T 4 ). (1.35) wherep N−1 ( T 4 ) contains the firstN−1 nested layers of XY 4 with the duration T 4 . As a matter of fact, forp N (T), the sequence duration of its innerp i , which contains the first i XY 4 layers, is T 4 N−i . It turns out that CDD N with XY 4 as the recursive unit is capable of achievingN th -order decoupling for general single-qubit decoherence, i.e., complete error suppression up to an error of orderT N+1 , as long asT is sufficiently small such that the condition for the convergence of Magnus expansion Eq. (1.18) is satisfied. This can be easily proven by induction. Suppose CDD N−1 with XY 4 as the recursive unit achieves (N − 1) th -order decoupling. Accordingly, for the Magnus expansion of p N−1 (τ) whereτ, the total duration of CDD N−1 , satisfied the convergence condition Eq. (1.18), its leading order for each non-trivial error isN. Hence, the effective Hamiltonian ofp N−1 (τ) can be expressed in the form, H eff p N−1 (τ) =τJ 0 0 I⊗B 0 0 (τ)+τ N [J 0 x σ x ⊗B 0 x (τ)+J 0 y σ y ⊗B 0 y (τ)+J 0 z σ z ⊗B 0 z (τ)] (1.36) whereB 0 λ (τ) withλ = 0,x,y,z is a polynomial ofτ. Choose the total duration of CDD N ,T , to satisfy Eq. (1.18) such that the Magnus expansion ofp N (T) is converged. Accordingly, the Magnus expansion of its innerp N−1 , CDD N−1 , also converges in that its duration τ = T 4 , which is smaller than T , also satisfies Eq. (1.18). Note that since XY 4 is an equidistant DD sequence, all p N−1 in the intervals of the outermost layer of XY 4 Eq. (1.35) are the same, due to the same durationτ = T 4 . Hence, in the toggling frame with respect to the outer-most XY 4 layer 13 Eq. (1.32), the effective Hamiltonians from the first to the four pulse interval have the same sign changes as Eq. (1.33), i.e., e H eff 1 = J 0 0 I⊗B 0 0 (τ)+τ N [J 0 x σ x ⊗B 0 x (τ)+J 0 y σ y ⊗B 0 y (τ)+J 0 z σ z ⊗B 0 z (τ)], e H eff 2 = J 0 0 I⊗B 0 0 (τ)+τ N [J 0 x σ x ⊗B 0 x (τ)−J 0 y σ y ⊗B 0 y (τ)−J 0 z σ z ⊗B 0 z (τ)] e H eff 3 = J 0 0 I⊗B 0 0 (τ)+τ N [−J 0 x σ x ⊗B 0 x (τ)−J 0 y σ y ⊗B 0 y (τ)+J 0 z σ z ⊗B 0 z (τ)] e H eff 4 = J 0 0 I⊗B 0 0 (τ)+τ N [−J 0 x σ x ⊗B 0 x (τ)+J 0 y σ y ⊗B 0 y (τ)−J 0 z σ z ⊗B 0 z (τ)]. withτ = T 4 . Therefore, despite the unknown polynomialsB 0 λ (τ) withλ = 0,x,y,z, the first Magnus order ofp N (T) still leaves only the pure bath term, i.e., H eff p N (T) = 4 X i=1 e H eff i = 4J 0 0 I⊗B 0 0 ( T 4 ) (1.37) where all the errors with orderT N vanish. The second or higher Magnus order terms of p N (T) based on Eq. (1.17) turn out to be multiple products of e H eff i , which implies that the leading order is higher thanT N . Therefore, we have proven that CDD N with XY 4 as the recursive unit can suppress general single-qubit decoherence completely up to an error of orderT N+1 . CDD has been amply tested in recent experimental studies [41, 5, 1, 57, 7] and demonstrated to be fairly robust against pulse imperfections. However, the number of pulses required by CDD grows exponentially with decoupling order: 4 N pulses to attain Nth order error suppression, which is not efficient enough to implement scalable quan- tum computing whenN is large. 14 Chapter 2 Nested UDD (NUDD): Universality Proof and Error Analysis 2.1 Introduction For scalable quantum computing to be possible, it is desirable to design a DD sequence which is both accurate (high decoupling order) and efficient (small number of pulses). In this chapter, I present a rigorous comprehensive analysis of the performance and universality of nested UDD (NUDD) scheme, which is the most efficient DD scheme for suppression of general decoherence model on multi-qubit or multi-level system. Since both UDD and QDD sequences belong to NUDD schemes, this NUDD proof also gives the detailed study on the performance of UDD and QDD. UDD, QDD, and NUDD sequences are reviewed in the introduction. 2.1.1 Uhrig DD (UDD) For the single-qubit pure dephasing spin-boson model, Uhrig discovered a DD sequence (UDD) which is optimal in the sense that it achieves N th -order decoupling with the smallest possible number of pulses, N orN + 1, depending on whetherN is even or odd [51, 49]. The key difference compared to other DD schemes is that the X-type π-pulses of UDD are applied at non-uniform pulse timing, t j =T sin 2 jπ 2(N +1) . (2.1) 15 DD scheme Decoupling order Number of pulses PDD 1 2 CDD N N O(2 N ) UDD N N N orN +1 Table 2.1: Performance comparison of UDD, PDD, and CDD schemes for the general single-qubit dephasing suppression. withj = 1,2,...,N whereN =N ifN even andN =N +1 ifN odd. The additional pulse, applied at the end of the sequence whenN is odd, is required in order to make the total number ofX-type pulse seven, so that the overall effect of theX-type pulses at the final time T will be U c (T) = I, leaving the qubit state unchanged. Note that N in Eq.(2.1) is referred to as the sequence order, and should not be confused with decoupling order. Hence, we used different notation ˇ N for decoupling order. In UDD case, decoupling order is equal to the UDD sequence order, namely, ˇ N =N. Table 2.1 compares the performance of UDD with other standard DD schemes, PDD and CDD which recursively embeds PDD into itself, as mentioned in the Secs. 1.2.2 and 1.2.3, for suppressing the general single-qubit dephasing. The number of pulses required by UDD grows polynomially with the degree of accuracy, N, which is an exponential improvement compared with PDD and CDD. The performance of the UDD sequence was the subject of a wide range of recent numerical and experimental studies [13, 33, 12, 8, 9, 8, 15, 5, 25, 7, 56, 4, 61]. Due to analyticity of the UDD pulse timings, UDD can be proven rigorously to be not only optimal but also universal for the generic dephasing bath, i.e,J 0 I S ⊗B 0 +J z σ z ⊗ B z withB 0 andB z being general [59, 52]. This indicates that UDD has a broad range of applicability, because its optimality is insensitive to the details of the decohering environment. Moreover, it has also been proven to be applicable to analytically time- dependent Hamiltonians [40]. 16 2.1.2 Quadratic DD (QDD) Unlike the one qubit dephasing model, universal optimal DD sequences are still absent for the single-qubit system subjected to general decoherence, which is modeled via the following Hamiltonian: H =J 0 I⊗B 0 +J x σ x ⊗B x +J y σ y ⊗B y +J z σ z ⊗B z , (2.2) whereB λ ,λ∈{0,X,Y,Z}, are arbitrary bath-operators withkB λ k = 1 (the norm is the largest singular value), the Pauli matrices,σ λ ,λ ∈ {X,Y,Z}, are the unwanted errors acting on the system qubit, andJ λ ,λ∈{0,X,Y,Z}, are bounded coupling coefficients between the qubit and the bath. Nevertheless, attempting to exploit the decoupling efficiency of UDD, West et al. introduced a near-optimal pulse sequence called Quadratic DD (QDD) which nests aZ- type UDD N 1 sequence, designed to eliminate the longitudinal relaxation errorsσ z ⊗B z andσ y ⊗B y by usingN 1 orN 1 +1Z pulses, with anX-type UDD N 2 sequence, designed to eliminate the pure dephasing errorσ Z ⊗B Z by usingN 2 orN 2 +1X pulses [58]. The X-type UDD N 2 pulses comprising the outer layer of the QDD N 1 N 2 sequence are applied at the original UDD N 2 timing Eq. (2.1) withN replaced byN 2 , while theZ-type pulses of the inner level UDD N 1 , applied fromt j−1 tot j , are executed at times t j,k =t j−1 +τ j sin 2 kπ 2(N 1 +1) (2.3) withN 1 +1 pulse intervals τ j,k ≡t j,k −t j,k−1 . (2.4) 17 DD scheme Decoupling order Number of pulses XY4 1 4 CDD N N O(4 N ) CUDD N N O(N2 N ) QDD N N O(N 2 ) Table 2.2: Performance comparison of QDD, XY4, CDD, and CUDD schemes for the single-qubit general decoherence suppression. Hence, the evolution operator at the final time T , at the completion of the QDD N 1 N 2 sequence, is U(T) = X N 2 U Z (T,t N 2 )X···XU Z (t 2 ,t 1 )XU Z (t 1 ,0) (2.5) U Z (t j ,t j−1 ) = Z N 1 U f (t j,N 1 +1 ,t j,N 1 )Z···ZU f (t j,2 ,t j,1 )ZU f (t j,1 ,t j,0 ) (2.6) where U Z (t j ,t j−1 ) is the UDD N 1 sequence evolution on the inner level, and U f is the pulse-free evolution generated byH Eq. (2.2). Table 2.2 compares the performance of QDD with other DD schemes, XY4 (in Sec. 1.2.3), CDD (in Sec. 1.2.3) and CUDD. For suppressing the single-qubit general decoherence up to an order N, CDD requiresN layer’s concatenation of XY 4 , which costsO(4 N ) pulses scaling exponentially withN. CUDD is a DD sequence combining orthogonal single-axis CDD and UDD sequences [50]. The number of pulses required for CUDD isO((N +1)2 N ), reducing the number of control pulses required compared to two-axis CDD but still grows exponentially with N. In contrast, QDD N,N (where the inner and outer UDD sequences have the same decoupling order) was conjectured to suppress arbitrary qubit-bath coupling to orderN by usingO(N 2 ) pulses, based on numerical studies [58, 43], which saves exponential number of pulses compared with CDD, CUDD, and all other known DD schemes. 18 2.1.3 Nested UDD (NUDD) To protect a set of qubits or multilevel systems against arbitrary system-bath interac- tions, nested UDD (NUDD) scheme, the generalization of QDD, was proposed [36, 55]. A `-layer NUDD scheme with a sequence order set {N 1 N 2 ...N ` } is constructed by concatenating ` levels of UDD sequences, where N i is the sequence order of UDD N i sequence on thei th level of NUDD. In order to address the dominant sources of error in any particular implementation more efficient, the sequence orders of different UDD layers can be different values. In [55], the control pulses operator set{Ω 1 ,Ω 2 ,...,Ω ` }, with the subscript being the layer index, is generalized from a set of single-qubit Pauli matrices to the mutually orthogonal operation set (MOOS), which consists of indepen- dent unitary Hermitian system operators mutually commuting or anti-commuting with each other, i.e. Ω 2 i = Ω † i Ω i =I (2.7) Ω i Ω j =±Ω i Ω j . (2.8) And each Ω i is required not to commute with the total Hamiltonian, otherwise, the control pulse type does not perform any effect on the noise. Suppose the number of the qubits in the target system isk. Then 2k-layers NUDD scheme was conjectured to suppress arbitrary qubit-bath coupling to orderN by using O(N 2k ) pulses, again an exponential improvement over CDD (kN layers ofXY 4 ) with O(4 kN ) pulses. Therefore, NUDD is much more efficient, though far from optimal, to deal with general multi-qubit decoherence than other known DD schemes. 19 2.1.4 Motivation Since NUDD sequences (including QDD and UDD) are most efficient DD schemes by far for preserving the large size of quantum memories, a detailed rigorous analysis of the performance of NUDD scheme can help to understand how the high noise suppres- sion mechanism actually works. Moreover, a more powerful DD sequence could be constructed by extracting and modifying the key elements of NUDD noise suppression mechanism. Therefore, it is worthy to give a thorough study on the performance of NUDD schemes. It turns out that proving the actual performance of QDD and NUDD analytically were non-trivial and challenging because the pulse intervals of NUDD’s recursive unit, UDD, are not equidistant, unlike the case of CDD withXY 4 as the recur- sive unit. The earlier work in [55] proved the validity of NUDD for only the cases with even sequence orders, and another work in Ref.[26] showed only the overall perfor- mance of NUDD which only considers Pauli matrices as control pulses types. Moreover, none of [55] and [26] addressed the decoupling order for each individual error. I have successfully provided a rigorous comprehensive analysis with a compact for- mulation for the university and performance of NUDD with arbitrary sequence order set and generalized control pulses set, MOOS [31]. This proof not only fills the gaps left in Ref. [55] but also goes beyond the overall performance proof in [26]. The con- cept of error types is generalized and more abstract in the sense that they are classified according to the chosen control MOOS set. Most importantly, we obtain the explicit decoupling order formula, a function of a given error type, the parities and magnitudes of all the sequence orders of NUDD, which shows explicitly how each UDD layer con- tributes to suppress a given type of error. It immediately follows that the overall sup- pression order of NUDD with MOOS as the control pulse set is the minimum among all the sequence orders of NUDD. Moreover, our analysis identifies the condition under 20 which the suppression ability of a given UDD layer is being hindered, totally ineffec- tive or enhanced by other UDD layers with odd sequence orders. Thereby, one can design a NUDD scheme accordingly such that the full power of each UDD layer is fully exploited. Note that our analysis shows that the performance of NUDD scheme with generalized control pulse types and arbitrary sequence orders is also universal as other UDD-like schemes with Pauli matrices as control pulses [59, 52, 55, 26, 31], i.e. the performance of general NUDD remains the same for multi-qubit or multi-level systems coupling to arbitrary bounded environments. Our theoretical results for the decoupling order of each error type are supported by the numerical simulations for QDD schemes on one qubit and four-layer NUDD schemes on 2-qubit system. In Sec. 2.2, a general NUDD scheme with MOOS as the control pulse set is formu- lated. In Sec. 2.3, the results of the performance of NUDD are presented. Specifically, NUDD Theorem 1 gives the decoupling order formula for each error type and its corol- lary 1 gives the overall performance of NUDD with its proof provided. The detailed complete proof of NUDD Theorem 1 is presented in Sec. 2.4. Numerical results for 4-layer NUDD scheme on 2 qubit system are demonstrated in 2.5 in support of our the- oretical analysis. We conclude in Sec. 2.6. The appendixes provide additional technical details. 2.2 NUDD formulation 2.2.1 NUDD pulse timing Normalized`-layer NUDD pulse timingη j ` ,j `−1 ,...,j 1 is defined as the actual NUDD tim- ing divided by total evolution timeT , wherej i is called thei th layer UDD pulse timing index and is from1 toN i +1. With fixed{j k } ` k=i+1 and{j k =N k +1} i−1 k=1 ,η j ` ,j `−1 ,...,j 1 21 withj i running from1 toN i +1 constitute one cycle of UDD N i timing with total duration s j ` s j `−1 ...s j i+1 , i.e., η j ` ,...,j i+1 ,j i = η j ` ,...,j i+1 −1 (2.9) + s j ` s j `−1 ...s j i+1 sin 2 j i π 2(N i +1) , (2.10) where η j ` ,...,j i ≡ η j ` ,...,j i+1 ,j i ,N i−1 +1,...,N 1 +1 (2.11) ≡ η j ` ,...,j i+1 ,j i +1,0,...,0 (2.12) withη j ` =0 ≡ 0, and s j k = sin π 2(N k +1) sin (2j k −1)π 2(N k +1) (2.13) is the normalized UDD N k pulse interval. Accordingly, the i th level Ω i pulses are applied at the η j ` ,...,j i , where {j k = 1,...,N k +1} ` k=i+1 , andj i = 1,2,...,N i whereN i =N i ifN i even andN i =N i +1 if N i odd. The additional pulse, applied at the end of the sequence when N i is odd, is required in order to make the total number of Ω i pulses even, so that the constraint U c (T) =I (1.23) is satisfied. 2.2.2 (r 1 ,r 2 ,...,r ` )-type error Each control operatorΩ i can divide the total Hamiltonian into two parts: one commutes with Ω i and the other anti-commutes with Ω i . For example, starting from the control pulses operatorΩ 1 on the first layer, the Hamiltonian is divided in two parts as follows, H = X r 1 =0,1 H (r 1 ) , (2.14) 22 where H (r 1 ) ≡ H +(−1) r 1 Ω 1 HΩ 1 2 . (2.15) It is straightforward to verify thatH (0) commutes withΩ 1 andH (1) anti-commutes with Ω 1 . For each H (r 1 ) with r 1 = 0 or1, it can be divided further by the second layer of control pulses operatorΩ 2 with the same procedure as Eq. (2.15), i.e., H (r 1 ,r 2 ) ≡ H (r 1 ) +(−1) r 2 Ω 2 H (r 1 ) Ω 2 2 . (2.16) Therefore, by using Eq. (2.15) iteratively as H (r 1 ,...,r ` ) ≡ [H (r 1 ,...,r `−1 ) +(−1) r ` Ω ` H (r 1 ,...,r `−1 ) Ω ` ]/2, (2.17) the total Hamiltonian can be written as a sum of independent 2 ` pieces accordingly by the given MOOS{Ω 1 ,Ω 2 ,...,Ω ` } as H = X {r i =0,1} ` i=1 H (r 1 ,r 2 ,...,r ` ) (2.18) where        {Ω i ,H (r 1 ,r 2 ,...,r ` ) } = 0 ifr i = 1 [Ω i ,H (r 1 ,r 2 ,...,r ` ) ] = 0 ifr i = 0 (2.19) withi∈ 1,2,...` and X {r i =0,1} ` i=1 ≡ X r 1 =0,1 X r 2 =0,1 ··· X r ` =0,1 . (2.20) H (r 1 ,r 2 ,...,r ` ) is classified as a (r 1 ,r 2 ,...,r ` )-type error, whose definition includes all the operators that have the same commuting or anti-commuting relation (2.19) 23 as H (r 1 ,r 2 ,...,r ` ) with respect to a given MOOS {Ω 1 ,Ω 2 ,...,Ω ` }. In particular, ~ 0 ` ≡ (0,0,...,0)-type error, which commutes with all control pulses and then is not sup- pressed by NUDD sequence, is called trivial error while the rest types as non-trivial errors. For example, suppose the first and second pulse types used in QDD (namely, 2- layers NUDD) areZ andX type pulses, respectively. Then this MOOS{Z,X} divides the Hamiltonian of a single qubit general decoherence model (2.2) into four pieces, H (0,0) =J 0 I⊗B 0 ,H (1,0) =J x σ x ⊗B x ,H (0,1) =J z σ z ⊗B z , andH (1,1) =J y σ y ⊗B y . 2.2.3 (r 1 ,r 2 ,...,r ` )-type error modulation function Due to the discreteness of control pulses, it is easier to the whole analysis in the toggling frame. Up to±1, the normalized (T = 1) control evolution operator is U c (η) = Ω j ` −1 ` ...Ω j 2 −1 2 Ω j 1 −1 1 (2.21) whenη∈ [η j ` ,j `−1 ,...,j 1 −1 ,η j ` ,j `−1 ,...,j 1 ) andU c (1) =I. Note that commuting and anti-commuting relation, Eq. (2.19), can be reformulated equivalently as follows, Ω i H {r 1 ,r 2 ,...,r ` } Ω i = (−1) r i H (r 1 ,r 2 ,...,r ` ) , (2.22) fori from 1 to`, by using the unitary Hermitian propertyΩ 2 i =I. 24 Hence, with Eqs. (2.21) and (2.22), we obtain the normalized Hamiltonian in the toggling frame, ˜ H(η) = U c (η) † HU c (η) = X {r i =0,1} ` i=1 (f 1 ) r 1 ...(f ` ) r ` H (r 1 ,r 2 ,...,r ` ) (2.23) where f i (η) = (−1) j i −1 η∈ [η j ` ,...,j i −1 ,η j ` ,...,j i ) (2.24) is called normalizedi th -layer modulation function for`-layer NUDD scheme and it only switches sign when thei th layer UDD N i pulse index,j i , changes. Q ` i=1 f i (η) r i in front of(r 1 ,r 2 ,...,r ` )-type errorH (r 1 ,r 2 ,...,r ` ) in Eq. (2.23) is called(r 1 ,r 2 ,...,r ` )-type error modulation function. Note that since n 0 Y p=1 f i (η) r (p) i =f i (η) P n 0 p=1 r (p) i =f i (η) ⊕ n 0 p=1 r (p) i (2.25) where ⊕ is the binary addition defined as the ordinary integer addition followed by modulo 2 operation, {f r i =0 i ,f r i =1 i } under ordinary multiplication forms a Z 2 group. Likewise, for Q ` i=1 f i (η) r i , n 0 Y p=1 ` Y i=1 (f i (θ)) r (p) i = ` Y i=1 (f i (θ)) P n 0 p=1 r (p) i (2.26) = ` Y i=1 (f i (θ)) ⊕ n 0 p=1 r (p) i (2.27) indicates that the set of the(r 1 ,r 2 ,...,r ` )-type error modulation functions constitutes a Z ⊗` 2 group. 25 2.2.4 Dyson expansion of evolution operator in toggling frame With shorthand notations, ~ r ` ≡ (r 1 ,r 2 ,...,r ` )∈{0,1} ⊗` (2.28) and X ~ r ` ≡ X {r i =0,1} ` i=1 (2.29) where the subscript` indicates the vector~ r ` has` components, the Dyson’s series expan- sion of the evolution propagator in the toggling frame, e U(T) = b T exp[−iT R 1 0 e H(η)dη], reads e U(T) = ∞ X n=0 X {~ r (p) ` } n p=1 (−iT) n n Y p=1 H ~ r (p) ` F ⊕ n p=1 ~ r (p) ` (2.30) where n Y p=1 H ~ r (p) ` ≡H ~ r (n) ` H ~ r (n−1) ` ...H ~ r (1) ` , (2.31) and F ⊕ n p=1 ~ r (p) ` ≡ n Y p=1 Z η (p+1) 0 ` Y i=1 f i (η (p) ) r (p) i dη (p) (2.32) withη (n+1) = 1 andr (p) i is thei th component of~ r (p) ` . Since Ω i n Y p=1 H ~ r (p) ` Ω i = n Y p=1 Ω i H ~ r (p) ` Ω i (2.33) = n Y p=1 (−1) r (p) i H ~ r (p) ` (2.34) = (−1) ⊕ n p 0 =1 r (p 0 ) i n Y p=1 H ~ r (p) ` (2.35) 26 where the first equality is obtained by inserting Ω 2 i = I between any adjacent H ~ r (p) ` and the second equality is obtained by using Eq. (2.22), Q n p=1 H ~ r (p) ` either commutes (⊕ n p=1 r (p) i = 0) or anti-commutes (⊕ n p=1 r (p) i = 1) with each control pulses operator Ω i . In other words, Q n p=1 H ~ r (p) ` belongs to~ r ` -type error with componentr i =⊕ n p=1 r (p) i . We would like to denote Q n p=1 H ~ r (p) ` as eitherH ⊕ n p=1 ~ r (p) ` =~ r ` orH (n) ~ r ` if we only care about the resulting error type~ r ` . Moreover, Eq. (2.35) also indicates that all 2 ` (r 1 ,r 2 ,...,r ` )- type errors under multiplication constitute aZ ⊗` 2 group. 2.2.5 NUDD coefficients From Eq. (2.32), F ⊕ n p=1 ~ r (p) ` is a normalizedn-nested integral with total duration being 1, and ~ r (p) ` in its subscript indicates that the pth integrand of F (n) ⊕ n p=1 ~ r (p) ` is ~ r (p) ` = (r (p) 1 ,r (p) 2 ,...,r (p) ` )-type error modulation function Q ` i=1 f i (η (p) ) r (p) i . According to F ⊕ n p=1 ~ r (p) ` ’s associated error typeH (n) ⊕ n p=1 ~ r (p) ` =~ r ` (orH (n) ~ r ` in shorthand), we can also denote F ⊕ n p=1 ~ r (p) ` to be F ⊕ n p=1 ~ r (p) ` =~ r ` orF (n) ~ r ` (2.36) and name it then th order normalized`-layers NUDD~ r ` -type error coefficient. Some- time we would just callF ⊕ n p=1 ~ r (p) ` =~ r ` as NUDD coefficient in short if it causes no confu- sion. From Eq. (2.32), the form of 1-layer NUDD(~ r 1 = 1)-type error coefficients F ⊕ n p=1 ~ r (p) 1 =1 = n Y p=1 Z η (p+1) 0 f 1 (η (p) ) r (p) 1 dη (p) (2.37) is exactly the same as the UDD coefficients appeared in [52] for the pure dephasing model where the dephasing termσ z ⊗B z is our(~ r 1 = 1)-type error and Pauli matrixσ x is our control pulse operator. 27 Moreover, the form of 2-layers NUDD~ r 2 = (r 1 ,r 2 )-type error coefficients F ⊕ n p=1 ~ r (p) 2 =~ r 2 = n Y p=1 Z η (p+1) 0 f 1 (η (p) ) r (p) 1 f 2 (η (p) ) r (p) 2 dη (p) (2.38) is exactly the same as the QDD coefficients defined in our earlier paper [31] where σ x ⊗B x ,σ z ⊗B z , andσ y ⊗B y are our (1,0)-, (0,1)- and (1,1)-type errors, and Pauli matricesσ z andσ x are our 1-layer and 2-layer control pulses. Note that from Eq. (2.32), NUDD coefficients are actually the same no matter what the control pulse operators are as long as they are independent and constitute a MOOS set. Therefore, the proofs for the performance of UDD and QDD sequence in [52] and [31], which used single qubit Pauli matrices as control pulses, can be immediately applied to the proofs for 1-layer NUDD and 2-layer NUDD scheme with general control pulses. 2.3 The performance of NUDD scheme 2.3.1 The decoupling order of each error type Since the summands in the Dyson’s series expansion of the evolution propagator Eq. (2.30) are all classified as one of2 ` types of errors, for a given~ r ` -type error, if all its first ˇ N ~ r ` orders NUDD coefficients vanish, then the `-layer NUDD eliminates the ~ r ` -type error to the order ˇ N ~ r ` , i.e. ˇ N ~ r ` is the decoupling order of the~ r ` -type error. We shall prove the following theorem in Sec. 2.4, 28 NUDD Theorem 1. A `-layers NUDD scheme with a given sequence order set {N 1 ,N 2 ,...,N ` } where the o th 1 layer is the first UDD layer with odd sequence order N o 1 eliminates~ r ` = (r 1 ,r 2 ,...,r ` )-type errors to the order ˇ N ~ r ` , i.e. F (n) ~ r ` = 0, ∀n≤ ˇ N ~ r ` , (2.39) with ˇ N ~ r ` ≡ max[{r i ((⊕ i−1 k=1 r k [N k ] 2 ) ⊕1) e N i + ` X k=i+1 r k [N k ] 2 } ` i=1 ] (2.40) where [.] 2 modulos the value inside the brackets by 2, and e N i =        N i wheni≤o 1 min[N i ,N k 0 <i o min +1] wheni>o 1 , (2.41) where N k 0 <i o min ≡ min 6=0 [{[N k 0] 2 N k 0} i−1 k 0 =1 ] (2.42) where min 6=0 takes the “nonzero” minimum value from the set{[N k ] 2 N k } i−1 k=1 . Note that the value of [N i ] 2 indicates the parity of the sequence orderN i of thei th UDD layer,        [N i ] 2 = 0 whenN i is even [N i ] 2 = 1 whenN i is odd. (2.43) Hence, with Eq. (2.43), some quantities appear in Eq. (2.40) having the following inter- pretations: 1. ⊕ i−1 k=1 r k [N k ] 2 , whose value is either 0 or 1, gives the parity of the total number of UDD layers, each of which has odd sequence order and control pulses that anti-commute with the~ r ` -type error, in the firsti−1 levels of NUDD. 29 2. P ` k=i+1 r k [N k ] 2 counts the total number of those UDD layers after thei th UDD layer, with odd sequence orders and control pulses anti-commuting with the~ r ` - type error. 3. N k 0 <i o min is the minimum value among all “odd” sequence orders of the first i− 1 UDD layers of NUDD. In contrast with the sequence order N i of the i th UDD layer, e N i is interpreted as the suppression order of thei th UDD layer. One can understand this by examining the error which anti-commutes only with the control pulses of the i th UDD layer. This error is expected to be suppressed by only thei th UDD layer and not affected by other UDD layers. Therefore, the decoupling order of this error, which is e N i according to Eq. (2.40), should reflect the suppression ability of thei th UDD layer. The suppression order e N i , by definition Eq. (2.41), is always either equal or smaller than the sequence orderN i , namely, e N i ≤N i . (2.44) When e N i < N i occurs, we have e N i = N k 0 <i o min +1, which suggests that the suppression order of thei th UDD layer is partially hindered by the UDD layer with the smallest odd sequence order, which is nested inside thei th UDD layer. The coefficient in front of e N i in Eq. (2.40) is 1 only when both r i = 1 and ⊕ i−1 k=1 r k [N k ] 2 = 0 are satisfied, otherwise it is 0. Accordingly, there are two require- ments for the i th UDD layer to be effective on a given~ r ` -type error. First, the error anti-commutes with the control pulses of this UDD layer (r i = 1). Second, the error anti-commutes with total even number of odd order UDD among the first (i−1) UDD layers (⊕ i−1 k=1 r k [N k ] 2 = 0). 30 In contrast, if an error anti-commutes with total odd number of odd order UDD among the first(i−1) UDD layers (⊕ i−1 k=1 r k [N k ] 2 = 1) , then the suppression ability of thei th -layer UDD sequence is totally ineffective on this error type even though if this error also anti-commutes with the control pulses of thei th UDD layer. Note that for the trivial error type, ~ 0 ` -type error, we haveN ~ 0 ` = 0, which suggests thatF (n) ~ 0 ` 6= 0. However, more precisely speaking,F (n) ~ 0 ` = 0 forn ≤ 0 in our context indicates that the vanishing of the ~ 0 ` -type errors cannot be concluded in our proof. In the following, we shall explicitly discuss the decoupling order formula Eq. (2.40) for some particular examples of NUDD schemes. UDD (1-layer NUDD) Since 1-layer NUDD has only one UDD layer, for the decoupling order of (~ r 1 = 1)- type error, Eq. (2.40) gives rise to ˇ N ~ r 1 =1 = e N 1 . By definition of e N 1 Eq. (2.41), the suppression order of the first UDD layer e N 1 is always equal to its sequence orderN 1 , i.e. e N 1 =N 1 , no matter ifN 1 is even or odd parity. Therefore, we have ˇ N ~ r 1 =1 =N 1 (2.45) which shows that the non-trivial error is eliminated up to the UDD’s sequence order, the same result in [59, 52]. QDD (2-layers NUDD) According to Eq. (2.40), the decoupling order formula for QDD (2-layer NUDD) with the MOOS{Z,X} and sequence order set{N 1 ,N 2 } is simplified as ˇ N ~ r 2 = max[r 1 e N 1 +r 2 [N 2 ] 2 ,r 2 (r 1 [N 1 ] 2 ⊕1) e N 2 ] (2.46) 31 Single-axis Inner sequence order Outer sequence order Decoupling order error type N 1 N 2 ˇ N (1,0)-type error :σ x arbitrary arbitrary N 1 (1,1)-type error :σ y even even max[N 1 ,N 2 ] even odd max[N 1 +1,N 2 ] odd even N 1 odd odd N 1 +1 (0,1)-type error :σ z even arbitrary N 2 odd arbitrary min[N 1 +1,N 2 ] Table 2.3: Summary of single-axis error suppression for QDD with all possible sequence order set{N 1 ,N 2 }. Substitutingr 1 = 1 andr 2 = 0 to Eq. (2.46) gives rise to the decoupling order of the(1,0)-type error (σ x error), ˇ N (1,0) = e N 1 =N 1 . (2.47) where the last equality is always true from the definition of e N i Eq. (2.41) for the innermost UDD N1 layer no matter if N 1 is even or odd. Eq. (2.47) indicates that the inner Z-type UDD N1 sequence always suppressesσ x error to the sequence orderN 1 . The outer X-type UDD N2 sequence has no effect at all on theσ x error which makes sense because (1,0)-type error anticommutes with the first UDD layer only and commutes with the second UDD layer. Since the (1,0)-type error (σ x error) is supposed to be suppressed by the first layer of UDD N1 sequence only, the decoupling order of the(1,0)- type error reflects the suppression ability of the first UDD layer. Therefore, e N 1 can be viewed as the suppression ability of the first UDD layer. The decoupling order of the(0,1)-type error (σ z error) reads ˇ N (0,1) = max[[N 2 ] 2 ,(0⊕1) e N 2 ] = e N 2 (2.48) 32 Again, the decoupling order of the(0,1)-type error is supposed to reflect the suppression ability of the second UDD layer because the (1,0)-type error anticommutes with the second UDD N2 . Therefore, e N 2 can be viewed as the suppression ability of the second UDD layer. From the definition of e N i Eq. (2.41), the suppression order of the second UDD layer, e N 2 , is equal to its corresponding sequence order N 2 if N 1 is even, and is equal tomin[N 1 +1,N 2 ] ifN 1 is odd. In other words, if the inner sequence orderN 1 is even or ifN 1 is odd andN 1 +1≥N 2 , the(0,1)-type error is eliminated to the sequence orderN 2 . However, ifN 1 is odd andN 1 +1 < N 2 , the (0,1)-type error is eliminated to N 1 + 1 which is smaller than the sequence order N 2 . This indicates that the inner UDD sequences with odd and smallN 1 can hinder the suppression ability of the outer sequence. The(1,1)-type error (σ y error) which anticommutes with the pulses of both the inner and outer sequences is expected to be suppressed by both sequences. However, from ˇ N (1,1) = max[ e N 1 +[N 2 ] 2 ,([N 1 ] 2 ⊕1) e N 2 ], (2.49) one can see that if the inner sequence order is odd, then [N 1 ] 2 ⊕1 = 0 makes the sup- pression of the second UDD layer become ineffective. In contrast, if the outer sequence order is odd, it can help the inner UDD sequences to suppress one more order. The results of Eq. (2.46) for QDD case with all possible{N 1 ,N 2 } is summarized in Table 2.3. Arbitrary layers NUDD with all inner sequence orders even For `-layer NUDD with all inner sequence orders even with any `, [N i ] 2 = 0 for all i < ` while the outermost sequence order is not specified, i.e. N ` could be even or 33 odd. In these cases, the suppression ability e N i of each i th UDD layer is equal to its corresponding sequence orderN i , e N i =N i . If [N ` ] 2 = 0, i.e. the cases with all sequence orders to be even, then the decoupling order formula reads ˇ N ~ r ` = max[{r i N i } ` i=1 ], (2.50) which indicates that the suppression ability of the UDD sequence in each layer remains the same as the one without nesting other UDD sequences and each UDD layer does not interfere with each other. Note that Eq. (2.50) coincides with the naive expectation that each UDD layer can always suppress to its sequence order, which is apparently not correct from the above QDD case and the following cases. If [N ` ] 2 = 1, then we have ˇ N ~ r ` = max[{r i N i ⊕r ` } `−1 i=1 ,r ` N ` ]. (2.51) which shows that the outermost layer of UDD sequence can help inner UDD layers to suppress the error which anti-commutes with the outermost UDD control pulses one more order. Arbitrary layers NUDD with all sequence orders odd For any NUDD with all sequence orders, in general, e N i =N k 0 <i o min +1<N i could occur which suggests that the suppression ability of thei th UDD layer is hindered by one of the odd order UDD layer inside thei th layer. Nevertheless, for the case of the`-layer NUDD with a[N i ] 2 = 1∀i andN 1 >N 2 > ···>N ` , e N i =N i Eq. (2.41) is guaranteed . Moreover, it is easy to show that N i + ` X k=i+1 r k >N j + ` X k=j+1 r k (2.52) 34 for anyi<j. Accordingly, due to Eq. (2.52), it turns out that the maximum value in Eq. (2.40) occurs at the inner-most UDD layer that the error anti-commutes with. Suppose the first non-zero component of a~ r ` -type error isr M = 1. Then it follows that theM th UDD layer has the maximum suppression on this error, i.e. ˇ N ~ r ` =N M + ` X k=M+1 r k . (2.53) The second term of the above equation implies that the outer-odd-UDD suppression effects contributed by UDD layers with odd sequence orders outside theM th layer are all added up. In other words, the outer-odd-UDD suppression effect is cumulative. 2.3.2 The overall performance of NUDD scheme The overall performance of a `-layer NUDD scheme is quantified by the minimum among the decoupling orders of all error types, i.e., ˇ N min ≡ min[{ ˇ N ~ r ` }] (2.54) Then we call ˇ N min as the overall decoupling order. The following corollary of the NUDD theorem 1 states the relationship between the overall decoupling order ˇ N min and given sequence orders{N i }: Corollary 1. A `-layer NUDD scheme with a given sequence order set {N 1 ,N 2 ,...,N ` }, its overall decoupling order ˇ N min is ˇ N min = min[{N 1 ,N 2 ,...,N ` }] (2.55) 35 Proof of Corollary 1. For a given error type~ r ` 6= ~ 0 ` , suppose the inner-most non-zero component isr i = 1, which also impliesr k<i = 0, leading to⊕ i−1 k=1 r k [N k ] 2 = 0. Due to r i ((⊕ i−1 k=1 r k [N k ] 2 )⊕1) = 1 and ˇ N ~ r ` taking the maximum value over the set displayed in Eq. (2.40), it follows that ˇ N ~ r ` ≥r i (p ⊕ (1,i−1)⊕1) e N i = e N i (2.56) Among~ r ` 6= ~ 0 ` error types, there are error types which anti-commute with only one control pulse type denoted as~ e i withr i = 1 and allr j6=i = 0. According to Eq. (2.40), the decoupling order of~ e i -type error is ˇ N ~ e i = e N i . (2.57) Owing to Eq. (2.56) and (2.57), ˇ N min , the minimum among decoupling orders of all non-trivial error types [Eq. (2.54)], occurs among the decoupling orders of all~ e i -type errors, i.e., ˇ N min ≡ min[{ ˇ N ~ r ` }] = min[ ˇ N ~ e 1 , ˇ N ~ e 2 ,..., ˇ N ~ e ` ] = min[ e N 1 , e N 2 ,..., e N ` ] (2.58) Suppose among ` layers, the o th 1 , o th 2 , ... , and o th b UDD layers where 0 < o 1 < o 2 < ··· < o b ≤ ` are the layers with odd sequence orders, namely, [N o j ] 2 = 1 for j = 1,2,...,b. By the definition of e N i [Eq. (2.41)], we have e N i =        N i fori≤o 1 min[N o 1 +1,N o 2 +1,...,N o j−1 +1,N i ] foro j−1 <i≤o j (2.59) 36 withj ∈ 2,...,b. Substituting the above e N i Eq.(2.59) into Eq. (2.58) and applying the equality such asmin[A,min[B,C]] = min[A,B,C], it follows that min[ e N 1 , e N 2 ,..., e N ` ] = min[{N 1 ,N 2 ,...,N ` }] (2.60) which proves Eq. (2.55). 2.4 Proof of NUDD Theorem 1 2.4.1 Synopsis of the proof The proof is done by induction. As discussed in Sec. 2.2.5, the proofs of the vanishing of UDD coefficients in [52] and QDD coefficients in our paper [31], which used single qubit Pauli matrices as control pulses, are also valid for the 1-layer and the 2-layer NUDD schemes with more general control pulses set (MOOS). From [52] and [31], the decoupling orders of each error type for UDD and QDD match exactly with the formula Eq. (2.40). Accordingly, NUDD theorem 1 is proven to be true for the 1-layer and the 2-layer NUDD sequence. Suppose that NUDD Theorem 1 holds for (`− 1)-layer NUDD with an arbitrary integer`, i.e. F (n) ~ r `−1 = 0, ∀n≤ ˇ N ~ r `−1 . (2.61) Then the remaining task is to show that NUDD Theorem 1 also holds for`-layer NUDD, Eqs. (2.39) and (2.40). The procedure is the following. F (n) ~ r ` can be re-expressed two different forms independently by two different approaches adapted from our QDD proof [31], outer-most-layer interval decomposi- tion and the nested integral analysis with certain function types. Each method is able to show the vanishing ofF (n) ~ r ` for some error types. 37 In Sec. 2.4.2, by the first approach, outer-most-layer interval decomposition, the`- layer NUDD coefficientF (n) ~ r ` is expressed in terms of the`−1-layer NUDD coefficients. Then in Sec. 2.4.3, it follows that it is the vanishing of the (`−1)-layer NUDD coef- ficients that make the first ˇ N ~ r `−1 orders of the `-layer NUDD coefficient F (n) ~ r ` vanish, i.e., F (n) ~ r ` =(~ r `−1 ,r ` ) = 0, ∀n≤ ˇ N ~ r `−1 (2.62) due to the assumption Eq. (2.61). Moreover, in Sec. 2.4.4, from the outer-layer interval decomposition form ofF (n) ~ r ` , we shall show that for the (~ r `−1 ,1)-type error which anti-commutes with the control pulses of the ` th UDD layer, if the ` th sequence order is odd ([N ` ] 2 = 1), due to the anti-symmetry of the` th UDD layer, it helps the first`−1 UDD layers to suppress the error by one more order, namely, F ( ˇ N ~ r `−1 +1) (~ r `−1 ,1) = 0. (2.63) . Combining Eq. (2.63) with Eq. (2.62), the first approach gives rise to F (n) ~ r ` = 0, ∀n≤ ˇ N ~ r `−1 +r ` [N ` ] 2 . (2.64) The decoupling order of the (~ v 0 `−1 ,1)-type error which by Definition 1 includes the ( ~ 0 `−1 ,1)-type error, shall be derived independently by the second approach presented in Sec. 2.4.5- 2.4.7. Definition 1. ~ v 0 `−1 :~ r `−1 with⊕ `−1 k=1 r k [N k ] 2 = 0 ~ v 1 `−1 :~ r `−1 with⊕ `−1 k=1 r k [N k ] 2 = 1 38 The second part of the NUDD proof is summarized as follows. In Sec. 2.4.5, we apply a piecewise linear change of variables toF (n) ~ r ` , such that the Fourier expansions of all its possible integrands, Q ` i=1 f i (η) r i , belong to some specific function types. In particular, for NUDD with all even inner sequence orders, there are only two function types, {c N ` ,0 even ,ζ N ` ,0 odd }; for NUDD with at least one odd sequence order, there are four types,{c N ` ,0 even ,ζ N ` ,0 odd ,ζ N ` ,0 even ,c N ` ,0 odd }, defined as follows, Definition 2. k,q∈Z with|q|≤n;d k1 , d k1 , d k1 , d k1 ∈R are arbitrary. : the product-to-sum trigonometric function operation c N ` ,n odd : X k d k1 cos[(2k+1)(N ` +1)θ+qθ] c N ` ,n even : X k d k2 cos[2k(N ` +1)θ+qθ] ζ N ` ,n even : X k d k3 sin[2k(N ` +1)θ+qθ] ζ N ` ,n odd : X k d k4 sin[(2k+1)(N ` +1)θ+qθ] By utilizing the group properties of {c N ` ,0 even ,ζ N ` ,0 odd } (or {c N ` ,0 even ,ζ N ` ,0 odd ,ζ N ` ,0 even ,c N ` ,0 odd } ), after integrating out the firstn−1 integrals ofF (n) ~ r ` , we obtain the second expression of F (n) ~ r ` , F (n) (~ v 0 `−1 ,1) = Z π 0 c N ` ,n odd dθ = X k d k sin[(2k+1)(N ` +1)θ+qθ]| π 0 = 0. (2.65) for the (~ v 0 `−1 ,1)-type error, where n ≤ N ` for NUDD with all even inner sequence orders, and n ≤ min[N k 0 <` o min + 1,N ` ] for NUDD with at least one odd inner sequence 39 order. Moreover, the vanishing orders of the other types of errors cannot be concluded from this method, denoted as F (n) ~ r ` / ∈(~ v 0 `−1 ,1) = 0, ∀n ≤ 0. In conclusion, the second approach gives rise to F (n) ~ r ` = 0, ∀n≤r ` ((⊕ `−1 k=1 r k [N k ] 2 ) ⊕1) e N ` . (2.66) where the definition of e N ` is Eq. (2.41) withi replaced by`. Combining Eq. (2.64) with Eq. (2.66), we haveF (n) ~ r ` = 0 for alln smaller or equal to Max[ ˇ N ~ r `−1 +r ` [N ` ] 2 ,r ` ((⊕ `−1 k=1 r k [N k ] 2 ) ⊕1) e N ` ]. (2.67) The above equation is equivalent to the decoupling order formula Eq. (2.40) shown in NUDD Theorem 1 by substituting the explicit expression of ˇ N ~ r `−1 , i.e., Eq. (2.40) for (`−1)-layer NUDD, in Eq. (2.67) and using the property such as Max[Max[A,B],C] =Max[A,B,C]. (2.68) Therefore, NUDD Theorem 1 also holds for the`-layer NUDD scheme with arbitrary sequence orders. The induction method also implies that NUDD Theorem 1 holds for any number of nested layers of NUDD. 2.4.2 The outer-layer interval decomposition form ofF (n) ~ r ` It is expected that the(~ r `−1 6= ~ 0 `−1 ,r ` )-type error which anti-commutes one or more of the first(`−1) inner-layer control operators is mainly suppressed by the inner(`−1)- layer NUDD sequences. In order to extract the action of the inner (`−1)-layer NUDD scheme and factor out the action of the outer-most(` th )-layer UDD sequence, we employ 40 the outer-layer interval decomposition which splits each integral ofF (n) ~ r ` Eq. (2.32) into a sum of sub-integrals over the` th -layer UDD N ` pulse intervalss j ` Eq. (2.13). The derivation ofF (n) ~ r ` given in Appendix 2.7.1 shows that each`-layer NUDD coef- ficient can be expressed in terms of the (`−1)-layer NUDD coefficients as follows, F (n) ~ r ` =(~ r `−1 ,r ` ) ≡F ⊕ n p=1 (~ r (p) `−1 ,r (p) ` ) = n X m=1 X { P m a=1 na=n} ( m Y a=1 F (na) ~ r <a> `−1 )( m Y a=1 j (a+1) ` −1 X j (a) ` =a (−1) (j (a) ` −1)r <a> ` s na j (a) ` ) (2.69) where j (m+1) ` − 1 ≡ N ` + 1, and for given m, P { P m a=1 na=n} sums over all possible combinations of{n a } m a=1 under the condition m X a=1 n a =n (2.70) with 1 ≤ n a ≤ n. From the derivation of Eq. (2.69) in Appendix 2.7.1, one can see that each segment with a specific configuration{m,{n a } m a=1 } defines a unique way to separate n vectors {~ r (p) ` } n p=1 into m clusters. The a th cluster contains n a number of vectors, {~ r (p) ` } P a−1 k=1 n k +na p= P a−1 k=1 n k +1 with n 0 ≡ 0, and ~ r <a> ` is the resulting vector of the a th cluster, i.e., ~ r <a> ` ≡⊕ P a−1 k=1 n k +na p= P a−1 k=1 n k +1 ~ r (p) ` , (2.71) which implies⊕ m a=1 ~ r <a> ` =~ r ` , i.e., ⊕ m a=1 ~ r <a> `−1 =~ r `−1 (2.72) ⊕ m a=1 r <a> ` =r ` . (2.73) 41 Moreover, eacha th cluster associatesF (na) ~ r <a> `−1 , an th a order (`− 1)-layer~ r <a> ` -type error NUDD coefficient, and P j (a+1) ` −1 j (a) ` =a (−1) (j (a) ` −1)r <a> ` s na j (a) ` which contains the outer-most (` th ) UDD layer’s information. 2.4.3 The performance of the inner (`−1)-layer NUDD We shall show the vanishing of the inner (`− 1)-layer NUDD coefficients, F (na) ~ r <a> `−1 = 0,∀n a ≤ ˇ N ~ r <a> `−1 , makes the first ˇ N ~ r `−1 orders ofF (n) ~ r ` =(~ r `−1 ,r ` ) vanish. First, we have the following lemma with the proof provided in Appendix 2.7.2: Lemma 1. ˇ N ~ r `−1 ≤ P m a=1 ˇ N ~ r <a> `−1 where~ r `−1 =⊕ m a=1 ~ r <a> `−1 With Lemma 1, forF (n) ~ r ` =(~ r `−1 ,r ` ) withn≤ ˇ N ~ r `−1 , each Q m a=1 F (na) ~ r <a> `−1 with⊕ m a=1 ~ r <a> `−1 = ~ r `−1 in Eq. (2.69) satisfies m X a=1 n a =n≤ ˇ N ~ r `−1 ≤ m X a=1 ˇ N ~ r <a> `−1 . (2.74) From Eq. (2.74), it follows that there must exist at least onea from 1 tom such that n a ≤ ˇ N ~ r <a> `−1 , because ifn a > ˇ N ~ r <a> `−1 for alla, it leads to m X a=1 n a > m X a=1 ˇ N ~ r <a> `−1 (2.75) which contradicts the assumption Eq. (2.74). Accordingly, each Q m a=1 F (na) ~ r <a> `−1 term of F (n) ~ r ` =(~ r `−1 ,r ` ) withn≤ ˇ N ~ r `−1 contains at least oneF (na) ~ r <a> `−1 withn a ≤ ˇ N ~ r <a> `−1 , which is zero due to the assumption Eq. (2.61) for (`− 1) NUDD coefficients. Therefore, the first stepF (n) ~ r ` =(~ r `−1 ,r ` ) = 0 for alln≤ ˇ N ~ r `−1 Eq. (2.62) is proven. From the proof, one can see that the effect of the outer`-layer UDD sequence which is entirely contained in the outer part Q m a=1 P j (a+1) ` −1 j (a) ` =a (−1) (j (a) ` −1)r <a> ` s na j (a) ` in Eq. (2.69) does not interfere with the elimination ability of the inner(`−1)-layer NUDD sequence. 42 Note that for~ r ` = ( ~ 0 `−1 ,1)-type error,N ~ 0 `−1 = 0 by the formula Eq. (2.40) gives rise to F (n) ( ~ 0 `−1 ,1) = 0 for all n ≤ N ~ 0 `−1 = 0 Eq. (2.62). However, it does not mean thatF (n) ( ~ 0 `−1 ,1) 6= 0 in our context but that the vanishing ofF (n) ( ~ 0 `−1 ,1) cannot be concluded by the method just mentioned, which only considers the contribution of the first(`−1) UDD layers. Thus~ r ` = ( ~ 0 `−1 ,1)-type error, which commutes with all the control pulses of the inner (`− 1) UDD layers, is supposed to be suppressed by the ` th -layer UDD sequence. 2.4.4 One more order suppression of(~ r `−1 ,1)-type error In the previous section, we proved thatF (n) ~ r ` =(~ r `−1 ,r ` ) = 0 for alln≤ ˇ N ~ r `−1 Eq. (2.62). Now we shall show that for(~ r `−1 ,1)-type error, there is an additional order suppression, i.e.,F ( ˇ N ~ r `−1 +1) (~ r `−1 ,1) = 0 Eq. (2.63), if[N ` ] 2 = 1. We show the following lemma, Lemma 2. For P m a=1 n a = n = ˇ N ~ r `−1 +1 withm≥ 2, there must exist at least onea such thatn a ≤ ˇ N ~ r <a> `−1 Proof of Lemma 2. With Lemma 1, we have m X a=1 n a ≤ m X a=1 ˇ N ~ r <a> `−1 +1. (2.76) Form≥ 2, randomly pickn a 0 among{n a } m a=1 . Ifn a 0 ≤ ˇ N ~ r <a 0 > `−1 , then Lemma 2 is true. On the other hand, ifn a 0 > ˇ N ~ r <a 0 > `−1 which is equal ton a 0 ≥ ˇ N ~ r <a 0 > `−1 +1, subtractingn a 0 from Eq. (2.76) leads to X a6=a 0 n a ≤ m X a=1 ˇ N ~ r <a> `−1 +1−n a ≤ X a6=a 0 ˇ N ~ r <a> `−1 . (2.77) 43 Accordingly, by the same counterargument as mentioned in Sec. 2.4.3, there must exist at least onea 00 6= a 0 such thatn a 00 ≤ ˇ N ~ r <a 00 > `−1 , otherwise we would obtain P a6=a 0 n a ≥ P a6=a 0 ˇ N ~ r <a> `−1 which contradicts Eq. (2.77). With Lemma 2, for the outer-layer decomposition form ofF ( ˇ N ~ r `−1 +1) (~ r `−1 ,1) Eq. (2.69), in each Q m a=1 F (na) ~ r <a> `−1 withm≥ 2 and P m a=1 n a = n = ˇ N ~ r `−1 +1, there must exist at least oneF (na) ~ r <a> `−1 withn a ≤ ˇ N ~ r <a> `−1 . Due to the assumption Eq. (2.61), all Q m a=1 F (na) ~ r <a> `−1 with m ≥ 2 in Eq. (2.69) vanish. Therefore, only the{m = 1,n a = n} term,F ( ˇ N ~ r `−1 +1) ~ r `−1 , remains in the outer-layer decomposition form ofF ( ˇ N ~ r `−1 +1) ~ r ` . In particular, for(~ r `−1 ,1)- type error, we have F ( ˇ N ~ r `−1 +1) (~ r `−1 ,1) =F ( ˇ N ~ r `−1 +1) ~ r `−1 N ` +1 X j ` =1 (−1) (j ` −1) s n j ` (2.78) At this point, we have no information of F ( ˇ N ~ r `−1 +1) ~ r `−1 and indeed, it may be non- vanishing. It turns out that it is the outer part P N ` +1 j ` =1 (−1) j ` −1 (s j ` ) ˇ N ~ r `−1 +1 that vanishes. To prove it, first note that due to the identitysinθ = sin[π−θ],sin[ (2j k −1)π 2(N k +1) ] ins j k (2.13) satisfies sin (2j k −1)π 2(N k +1) = sin[π− (2j k −1)π 2(N k +1) ] = sin[ (2N k +2−2j k +1)π 2(N k +1) ] = sin[ (2(N k +2−j k )−1)π 2(N k +1) ] It follows that the normalized UDD pulse intervals of each layer of NUDD are time- symmetric, i.e. s j k =s N k +2−j k (2.79) 44 withk ∈ {1,2,...,`}. Second,j ` andN ` +2−j ` have opposite parities whenN ` is odd. Accordingly, we have (−1) j ` −1 (s j ` ) ˇ N ~ r `−1 +1 = (−1) j ` −1 (s N ` +2−j ) ˇ N ~ r `−1 +1 (2.80) = −(−1) N ` +2−j ` −1 (s N ` +2−j ) ˇ N ~ r `−1 +1 . where Eq. (2.79) is used in Eq. (2.80). Then it follows that whenN ` is odd, the outer part P N ` +1 j ` =1 (−1) j ` −1 (s j ` ) ˇ N ~ r `−1 +1 vanishes due to the mutual cancellation of terms with equal magnitude but opposite sign. The proof shows the anti-symmetry of the ` th outer-most UDD layer with odd sequence order can help the inner(`−1) NUDD sequences to suppress the(~ r `−1 ,1)-type error by one more order. Therefore, F ( ˇ N ~ r `−1 +1) (~ r `−1 ,1) = 0 Eq. (2.63) is proven. Combining Eq. (2.63) with Eq. (2.62) which proved in the previous section leads toF (n) ~ r ` = 0 forn≤ ˇ N ~ r `−1 +r ` [N ` ] 2 Eq. (2.64). 2.4.5 Fourier expansion after linear change of variables We shall complete the proof of NUDD Theorem 1 by another approach which analyzes the nested integral with certain Fourier function types. First apply a piecewise linear transformation, θ = π N ` +1 ( η−η j ` −1 s j ` )+ (j ` −1)π N ` +1 η∈ [η j ` −1 ,η j ` ) (2.81) 45 with j ` from 1 to N ` , to the n nested integral F (n) ~ r ` . With dη = G 1 (θ)dθ, F (n) ~ r ` = F ⊕ n p=1 ~ r (p) ` Eq. (2.32) is reexppressed as F (n) ~ r ` = n Y p=1 Z θ (p+1) 0 G 1 (θ (p) ) ` Y i=1 f i (θ (p) ) r (p) i dθ (p) (2.82) with θ (n+1) = π, (2.83) f i (θ) = (−1) j i −1 θ∈ [θ j ` ,...,j i −1 ,θ j ` ,...,j i ), (2.84) G 1 (θ) = N ` +1 π s j ` θ∈ [θ j ` −1 ,θ j ` ) (2.85) whereθ j ` ,...,j i is the new pulse timing. The advantage of using this piecewise linear transformation Eq. (2.81) is that all the modulation functions become periodic functions. Consequently, as explained in the Appendix 2.7.3, the Fourier expansion of each functions appearing inF ⊕ n p=1 ~ r (p) ` belongs to one of the function types defined in the Definition 2, Ψ(f ` (θ)) = ζ N ` ,0 odd , (2.86) Ψ(f i<` (θ)) =      c N ` ,0 even whenN i even ζ N ` ,0 even whenN i odd (2.87) Ψ(G 1 (θ)) = ζ N ` ,1 even , (2.88) where Ψ maps a function to the function type of its Fourier expansion up to the unim- portant coefficients. 46 Note that the sets {c N ` ,0 even ,ζ N ` ,0 odd } and {c N ` ,0 even ,ζ N ` ,0 odd ,ζ N ` ,0 even ,c N ` ,0 odd } constitute Z 2 and Z 2 × Z 2 groups with c N ` ,0 even as identity under the binary operation (the product-to- sum trigonometric formula). Then one can obtain the function types of the~ r ` -type error modulation functions by employing the group algebra ofZ 2 andZ 2 ×Z 2 to Ψ( ` Y i=1 f i (θ) r i ) = ` i=1 Ψ(f i (θ) r i ). (2.89) where Ψ(f i (θ) 0 = 1) =c N ` ,0 even for alli. For example, for the~ v 1 `−1 -type error (see Defini- tion 1), its `−1 i=1 Ψ(f i (θ) r i ) would result inζ N ` ,0 even , because the condition⊕ `−1 k=1 r k [N k ] 2 = 1, that the components of ~ v 1 `−1 satisfy, implies that there is a total odd number of r i = 1 associated with [N i ] 2 = 1 such that Ψ(f i (θ) r i ) = ζ N ` ,0 even . It turns out that Ψ( Q ` i=1 f i (θ) r i ) is determined by only two values,⊕ `−1 k=1 r k [N k ] 2 andr ` , (see Table 2.4). error type~ r ` Ψ( Q ` i=1 f i (θ) r i ) (1)(~ v 0 `−1 ,0): ⊕ `−1 k=1 r k [N k ] 2 = 0, r ` = 0 c N ` ,0 even (2)(~ v 0 `−1 ,1): ⊕ `−1 k=1 r k [N k ] 2 = 0, r ` = 1 ζ N ` ,0 odd (3)(~ v 1 `−1 ,0): ⊕ `−1 k=1 r k [N k ] 2 = 1, r ` = 0 ζ N ` ,0 even (4)(~ v 1 `−1 ,1): ⊕ `−1 k=1 r k [N k ] 2 = 1, r ` = 1 c N ` ,0 odd Table 2.4: The first column classifies 2 ` ~ r ` -type errors into four groups by two values, ⊕ `−1 k=1 r k [N k ] 2 andr ` , where~ v 0 `−1 and~ v 1 `−1 are defined in Definition 1. The second col- umn shows the function types of the Fourier expansion of the~ r ` -type error modulation functions. Note that for the `-layer NUDD with [N i ] 2 = 0∀i ≤ `− 1,⊕ `−1 k=1 r k [N k ] 2 is zero for all 2 ` ~ r ` -type errors. Therefore, Ψ( Q ` i=1 f i (θ) r i ) for all error types in this case are eitherζ N ` ,0 odd orc N ` ,0 even , the first two rows in Table 2.4. On the other hand, for the`-layer NUDD with at least one UDD layer with odd sequence order, there are four functions types as shown in Table 2.4. 47 The following expression of F (n) ~ r ` focuses on the function types of the integrands where the coefficients in their Fourier expansion are not important in the proof, F (n) ~ r ` = n p=1 Z θ (p+1) 0 dθ (p) Ψ( ` Y i=1 f i (θ (p) ) r (p) i ) Ψ(G 1 (θ (p) ) (2.90) As a matter of fact, due to Ψ( ` Y i=1 f i (θ) r (1) i )Ψ( ` Y i=1 f i (θ) r (2) i ) (2.91) = Ψ( ` Y i=1 f i (θ) r (1) i ∗ ` Y i=1 f i (θ) r (2) i ) = Ψ( ` Y i=1 f i (θ) ⊕ 2 p=1 r (p) i ), Ψ is a homomorphism mapping from the Z ⊗` 2 group { Q ` i=1 f i (θ) r i } to either the Z 2 group{c N ` ,0 even ,ζ N ` ,0 odd } or theZ 2 ×Z 2 group{c N ` ,0 even ,ζ N ` ,0 odd ,ζ N ` ,0 even ,c N ` ,0 odd }. 2.4.6 Integrating out the firstn−1 integrals ofF (n) ~ r ` We shall prove the following lemma, Lemma 3. For alln≤ Λ+1, the form ofF (n) ~ r ` aftern−1 integrations becomes F (n) ~ r ` = Z π 0 dθ (n) Ψ( ` Y i=1 f i (θ (n) ) r i ) Ψ(G n (θ (n) )) (2.92) up to different irrelevant coefficients in the Fourier expansion where Ψ(G n (θ (n) )) ≡ ζ N ` ,n even . With Ψ( Q ` i=1 f i (θ (n) ) r i ) given explicitly in Table 2.4, all the possible function types of the resulting integrands of F (n) ~ r ` Ψ( Q ` i=1 f i (θ (n) ) r i ) Ψ(G n (θ (n) )) in Eq. (2.92) are listed in Table 2.5: 48 error type~ r ` the resulting integrand ofF (n) ~ r ` Λ ~ r ` (1)(~ v 0 `−1 ,0) : ζ N ` ,n even ∞ (2)(~ v 0 `−1 ,1): c N ` ,n odd N ` (3)(~ v 1 `−1 ,0): c N ` ,n even ˇ N ~ r `−1 (4)(~ v 1 `−1 ,1): ζ N ` ,n odd ∞ Table 2.5: The second column shows the function type of the resulting integrand ofF (n) ~ r ` aftern−1 integrations. The third column shows the maximum order Λ ~ r ` up to which the function type of the resulting integrand ofF (n) ~ r ` does not contain a constant term. Λ ~ r ` is defined as the maximum order such that the function type of the resulting integrand ofF (n) ~ r ` does not contain any constant term and Λ≡ min[{Λ ~ r ` }]. (2.93) The proof of Lemma 3 is done by induction. It is obvious that Lemma 3 is true for the first orderF (1) ~ r ` based on the form of Eq. (2.90). Suppose Lemma 3 holds for all the (n−1) th order NUDD coefficients wheren−1≤ Λ, i.e., F (n−1) ~ r ` = Z π 0 dθ (n−1) Ψ( ` Y i=1 f i (θ (n−1) ) r i ) Ψ(G n−1 (θ (n−1) )). (2.94) where Ψ( Q ` i=1 f i (θ (n−1) ) r i ) Ψ(G n−1 (θ (n−1) )) belongs to one of {c N ` ,n−1 even ,ζ N ` ,n−1 odd ,ζ N ` ,n−1 even ,c N ` ,n−1 odd }. To proceed to the next order, first compare the forms of Eq. (2.90) between the n th order and the (n− 1) th order NUDD coefficients. One can see that then th order 49 NUDD coefficients can actually be viewed as one integral nested with one order lower [(n−1) th -order] NUDD coefficients, i.e., F (n) ~ r ` = Z π 0 dθ (n) Ψ( ` Y i=1 f i (θ (n) ) r (n) i )Ψ(G 1 (θ (n) ))F (n−1),θ (n) ~ r 0 ` , (2.95) where~ r ` = ~ r (n) ` ⊕~ r 0 ` implies that~ r 0 ` could be a different vector from~ r ` , and the extra superscriptθ (n) ofF (n−1) ~ r 0 ` indicates that the upper integration limit,π, of the last integral of F (n−1) ~ r 0 ` is replaced by θ (n) . Therefore, we can substitute the results of F (n−1) ~ r 0 ` Eq. (2.94) directly to Eq. (2.95) to evaluateF (n) ~ r ` . From the definition of Λ, the resulting integrands of F (n−1) ~ r 0 ` for all possible~ r 0 ` are guaranteed to contain no constant term forn−1≤ Λ. Accordingly, the operation Ψ(G 1 (θ (n) ) Z θ (n) 0 dθ (n−1) (2.96) maps{c N ` ,n−1 even ,ζ N ` ,n−1 odd ,ζ N ` ,n−1 even ,c N ` ,n−1 odd } to its corresponding{c N ` ,n even ,ζ N ` ,n odd ,ζ N ` ,n even ,c N ` ,n odd } regardless of the unimportant change of the coefficients in the linear combination. Therefore, up to different coefficients in the linear combination, the operation Eq. (2.96) gives rise to the following mapping, Ψ( ` Y i=1 f i (θ (n−1) ) r 0 i ) Ψ(G n−1 (θ (n−1) ))− → Ψ( ` Y i=1 f i (θ (n) ) r 0 i ) Ψ(G n (θ (n) )). (2.97) By substituting Eq. (2.97) to Eq (2.95), the resulting integrand ofF (n) ~ r ` becomes Ψ( ` Y i=1 f i (θ (n) ) r (n) i )Ψ( ` Y i=1 f i (θ (n) ) r 0 i ) Ψ(G n (θ (n) )) (2.98) 50 Applying the homomorphism property Eq. (2.91) to the above equation, we obtain Eq. (2.92) for then th order wheren≤ Λ+1. For the order n = Λ + 1, the resulting integrands of F (n) ~ r ` for some of the errors begin to contain a constant term. Then the operationΨ(G 1 (θ (n+1) ) R θ (n+1) 0 dθ (n) will map them to different functions other than a purely cosine series or a purely sine series. Hence, it follows that Eq. (2.92) or the column of Table 2.5 are no longer true for the Λ+2 or higher order NUDD coefficients. Now let’s prove the values of Λ ~ r ` shown in the last column of Table 2.5. First, by definition 2,c N ` ,n odd , which is the function type of the resulting integrand of the (~ v 0 `−1 ,1)- type error NUDD coefficients in Table 2.5, does not contain cosine function with zero argument (a constant 1 term) whenn ≤ N ` . Therefore, Λ (~ v 0 `−1 ,1) = N ` . Second, both ζ N ` ,n even and ζ N ` ,n odd have no constant terms under all circumstances which indicates that Λ (~ v 0 `−1 ,0) = Λ (~ v 1 `−1 ,1) = ∞. Finally, for the (~ v 1 `−1 ,0)-type error, c N ` ,n even are in general by definition allowed to have cosine function with zero argument, a constant term. How- ever, from Sec. 2.4.3, we’ve already shown thatF (n) ~ r ` = 0 for alln≤ ˇ N ~ r `−1 Eq. (2.64). Hence, with Eq. (2.64), F (n) ~ r ` ∈(~ v 1 `−1 ,0) = 0 = Z π 0 c N ` ,n even dθ ∀n≤ ˇ N ~ r `−1 ∈~ v 1 `−1 (2.99) suggests that c N ` ,n even in Eq. (2.99) has no constant term for the first ˇ N ~ r `−1 ∈~ v 1 `−1 orders. Accordingly, Λ ~ r ` ∈(~ v 1 `−1 ,0) = ˇ N ~ r `−1 ∈~ v 1 `−1 . (2.100) 2.4.7 The vanishing of the(~ v 0 `−1 ,1)-type error According to Lemma 3, for the(~ v 0 `−1 ,1)-type error,F (n) (~ v 0 `−1 ,1) = R π 0 c N ` ,n odd dθ, wherec N ` ,n odd does not contain any constant term, for alln≤ min[Λ+1,N ` ], wheren≤ Λ+1 ensures 51 that the function type of the resulting integrand ofF (n) (~ v 0 `−1 ,1) isc N ` ,n odd , andn≤ Λ (~ v 0 `−1 ,1) = N ` ensures that there is no constant term. Then as shown in Eq. (2.65), it follows that F (n) (~ v 0 `−1 ,1) = 0 for n≤ min[Λ+1,N ` ]. (2.101) where the value ofΛ expressed explicitly in terms of the sequence orders shall be deter- mined as follows. For the NUDD with [N i ] 2 = 0 for alli < `,⊕ `−1 k=1 r k [N k ] 2 = 0 is always true for all errors, so there are only two error types,(~ v 0 `−1 ,0)- and(~ v 0 `−1 ,1)-type errors in this case. Therefore, from the definition ofΛ Eq. (2.93), we have Λ = min[Λ (~ v 0 `−1 ,0) ,Λ (~ v 0 `−1 ,1) ] = min[∞,N ` ] =N ` . (2.102) Substituting Eq. (2.102) into Eq. (2.101) leads to min[N ` +1,N ` ] = N ` . Therefore, we prove that F (n) (~ v 0 `−1 ,1) = 0 ∀n≤N ` (2.103) for the NUDD with all even inner sequence orders. For the NUDD with at least one odd sequence order in the first`−1 UDD layers, from the definition ofΛ Eq. (2.93), Λ = min[Λ (~ v 0 `−1 ,0) ,Λ (~ v 0 `−1 ,1) ,Λ ~ r ` ∈(~ v 1 `−1 ,0) ,Λ (~ v 1 `−1 ,1) ] = min[∞,N ` ,{ ˇ N ~ r `−1 ∈~ v 1 `−1 },∞] = min[N ` ,{ ˇ N ~ r `−1 ∈~ v 1 `−1 }] (2.104) Due to Lemma 4. min[{ ˇ N ~ r `−1 ∈~ v 1 `−1 }] =N k 0 <` o min 52 with proof provided in Appendix 2.7.4, Eq. (2.104) reads Λ = min[N ` ,N k 0 <` o min ]. (2.105) Substituting the aboveΛ into Eq. (2.101), we have min[Λ+1,N ` ] = min[min[N ` ,N k 0 <` o min ]+1,N ` ] (2.106) = min[N k 0 <` o min +1,N ` ]. (2.107) Therefore, we prove that F (n) (~ v 0 `−1 ,1) = 0 ∀n≤ min[N k 0 <` o min +1,N ` ] (2.108) for the NUDD with at least one odd inner sequence order. Combining Eq. (2.103), and Eq. (2.108), with the fact that the vanishing of the other error types can not be concluded by this approach due to their function types shown in Table 2.5, we obtain F (n) ~ r ` = 0 for all n ≤ r ` ((⊕ `−1 k=1 r k [N k ] 2 ) ⊕ 1) e N ` Eq. (2.66), where e N ` = N ` for NUDD with all even inner sequence orders, and e N ` = min[N k 0 <` o min + 1,N ` ] for NUDD with at least one odd inner sequence order, which is equivalent to the definition defined in Eq. (2.41). Therefore, the last step of the proof for NUDD Theorem 1 is completed. 2.5 Numerical results for 4-layer NUDD sequence All possible variations of MOOS with the same number of pulses for each layer of NUDD sequence are expected to have the same decoupling order formula Eq. (2.40) and preserve the same minimum order scaling. In this section, numerical simulations 53 are employed to examine the performance of 4-layers NUDD scheme for two contrast- ing MOOS as our control pulses. Given the sequence order set, {N 1 ,N 2 ,N 3 ,N 4 }, as conveyed above, this 4-layer NUDD sequence ensures the minimum suppression of all non-trivial error types to at least ˇ N min = min[N 1 ,N 2 ,N 3 ,N 4 ], leading to U(T) ∼ O(T ˇ N min +1 ) as long as the control pulses are independent and constitute as a MOOS. Further suppressions of2 4 error types are given by the formula Eq. (2.40). 2.5.1 Model We consider 2-qubit system coupling to general quantum baths which can be fully pro- tected by general 4-layer NUDD scheme. The total Hamiltonian for 2-qubit generic decoherence model can be written as H = X λ 1 =0,x,y,z X λ 2 =0,x,y,z J λ 1 λ 2 σ λ 1 ⊗σ λ 2 ⊗B λ 1 λ 2 (2.109) whereσ λ j are the standard Pauli matrices withσ 0 ≡ I for the first (j = 1) and second (j = 2) system qubits,B λ 1 λ 2 are arbitrary bath-operators withkB λ 1 λ 2 k = 1 (the norm is the largest singular value), andJ λ 1 λ 2 are bounded coupling coefficients between the qubits and the bath. Modeling the environment as a spin bath consisting of four spin-1/2 particles with randomized couplings between them, the operatorB λ 1 λ 2 which couples to system oper- atorσ λ 1 σ λ 2 is given by B λ 1 λ 2 = X λ 3 ,λ 4 ,λ 5 ,λ 6 c λ 3 λ 4 λ 5 λ 6 λ 1 λ 2 σ λ 3 ⊗σ λ 4 ⊗σ λ 5 ⊗σ λ 6 , (2.110) 54 where the interactions are non-restrictive, i.e. 1- to 4-local interactions are permitted. The indexλ j = 0,x,y,z wherej = 3,4,5,6 stands for the bath qubit and the randomly chosen coefficientc λ 3 λ 4 λ 5 λ 6 λ 1 λ 2 ∈ [0,1]. In our simulations, we consider the case that the strengths of all system-bath inter- actionsJ λ 1 λ 2 are the same except one of the pure environment dynamicsJ 00 . Denote J 00 =β, J λ 1 λ 2 =J. (2.111) For all simulations, we takeβ = 1kHz andJ = 1MHz, the regimeβ J where the environment dynamics are effectively static with respect to system-environment inter- actions. 2.5.2 ~ r 4 = (r 1 ,r 2 ,r 3 ,r 4 )-type error Given a MOOS {Ω 1 ,Ω 2 ,Ω 3 ,Ω 4 }, the 2-qubit general Hamiltonian Eq. (2.109) can be divided into 2 4 types of errors H ~ r 4 = H (r 1 ,r 2 ,r 3 ,r 4 ) by the procedure given in (2.17). There is anther way to obtain H (r 1 ,r 2 ,r 3 ,r 4 ) . First, find the following 4 errors, H (1,0,0,0) ,H (0,1,0,0) , H (0,0,1,0) , andH (0,0,0,1) , which are the generators of theZ ⊗4 2 group {H (r 1 ,r 2 ,r 3 ,r 4 ) }. Then according to H ~ r (1) 4 H ~ r (2) 4 = H ~ r (1) 4 ⊕~ r (2) 4 the other error types can be obtained by all possible product of those four error generators. We display 16 error types in the following way, * H (0,0,0,0) H (0,0,1,0) H (0,0,0,1) H (0,0,1,1) H (0,0,0,0) H (0,0,0,0) H (0,0,1,0) H (0,0,0,1) H (0,0,1,1) H (1,0,0,0) H (1,0,0,0) H (1,0,1,0) H (1,0,0,1) H (1,0,1,1) H (0,1,0,0) H (0,1,0,0) H (0,1,1,0) H (0,1,0,1) H (0,1,1,1) H (1,1,0,0) H (1,1,0,0) H (1,1,1,0) H (1,1,0,1) H (1,1,1,1) Table 2.6: 16 error types~ r 4 for the 4-layer NUDD scheme. 55 For demonstration, we choose two contrasting MOOS as our control pulses sets. One is {I⊗σ z ,I⊗σ x ,σ z ⊗I,σ x ⊗I} (2.112) which consists of single-qubit Pauli matrices. It’s corresponding error generator are H (1,0,0,0) =I⊗σ x , H (0,1,0,0) =I⊗σ z , H (0,0,1,0) =σ x ⊗I, H (0,0,0,1) =σ z ⊗I. (2.113) up to arbitrary baths which are omitted in the above equations. Replacing Eq. (2.113) to Table 2.6, we can get the explicit forms for all the error types as shown in the following table. * I⊗I σ x ⊗I σ z ⊗I σ y ⊗I I⊗I I⊗I σ x ⊗I σ z ⊗I σ y ⊗I I⊗σ x I⊗σ x σ x ⊗σ x σ z ⊗σ x σ y ⊗σ x I⊗σ z I⊗σ z σ x ⊗σ z σ z ⊗σ z σ y ⊗σ z I⊗σ y I⊗σ y σ x ⊗σ y σ z ⊗σ y σ y ⊗σ y Table 2.7: 16 error types~ r 4 for the 4-layer NUDD scheme with the MOOS{I⊗σ z ,I⊗ σ x ,σ z ⊗I,σ x ⊗I}. The other MOOS is chosen as {σ z ⊗σ z ,I⊗σ x ,σ z ⊗I,σ x ⊗I} (2.114) with error generators, H (1,0,0,0) =I⊗σ x , H (0,1,0,0) =I⊗σ z , H (0,0,1,0) =σ x ⊗σ x , H (0,0,0,1) =σ z ⊗I. (2.115) up to arbitrary baths. Again by replacing Eq. (2.115) to Table 2.6, we have 56 * I⊗I σ x ⊗σ x σ z ⊗I σ y ⊗σ x I⊗I I⊗I σ x ⊗σ x σ z ⊗I σ y ⊗σ x I⊗σ x I⊗σ x σ x ⊗I σ z ⊗σ x σ y ⊗I I⊗σ z I⊗σ z σ x ⊗σ y σ z ⊗σ z σ y ⊗σ y I⊗σ y I⊗σ y σ x ⊗σ z σ z ⊗σ y σ y ⊗σ z Table 2.8: 16 error types~ r 4 for the 4-layer NUDD scheme with the MOOS{σ z ⊗σ z ,I⊗ σ x ,σ z ⊗I,σ x ⊗I} 2.5.3 Decoupling order of each error type As Ref. [43], each error type is quantified by E ~ r 4 =kTr S (U(T)H ~ r 4 )k F , (2.116) wherekAk F is the Frobenius norm ofA, i.e., kAk F = Tr √ A † A, (2.117) the sum of singular values ofA. (The choice of norm is somewhat arbitrary; we could have used any other unitarily invariant norm ). Theoretically, E ~ r 4 ∼O(T ˇ N ~ r 4 +1 )∼O(τ ˇ N ~ r 4 +1 ) (2.118) where ˇ N ~ r 4 is given by the formula Eq. (2.40) andτ ≡ Ts j ` =1 s j `−1 =1 ...s j 2 =1 s j 1 =1 is the minimum delay time between pulses. Therefore, the numerical decoupling order of ~ r 4 -type error,n ~ r 4 , is obtained by n ~ r 4 = (log 10 E ~ r 4 )−1 (2.119) 57 2.5.4 Overall decoupling order The overall performance of NUDD is quantified with respect to the state-independent distance measure D = 1 √ d S d B min Φ kU(T)−I⊗Φk F , (2.120) where d S is the system Hilbert space dimension, d B is the environment dimension, and Φ is the bath operator. The minimum order scaling of NUDD is expected to be D ∼ O[T N+1 ] for a total sequence durationT . In other words, the numerical overall decoupling order of NUDD scheme,n min , is obtained by n min = (log 10 D)−1 (2.121) In the subsequent simulations, the scaling of D is extracted by varying the minimum pulse delay instead of total time T. 2.5.5 Comparison of theoretical predictions with numerical results We examine the performance of 4-layer NUDD scheme for two contrasting MOOS as mentioned above. We shall compare the numerical results with our theoretical predic- tions Eq. (2.40) and naive expectations, ˇ N ~ r ` = max[{r i N i } ` i=1 ], which assumes the suppression ability of each UDD layer is it’s corresponding sequence order. For each error, its actual decoupling order which is obtained by numerical simulations is marked as black; the order obtained by our decoupling order formula Eq. (2.40) is marked as blue; the naive decoupling order is marked as red. If a blue (red) number is absent, it means our theoretical decoupling order (naive decoupling order ) is exactly the same as the actual decoupling order. 58 N (0,0,0,0) ˇ N (0,0,1,0) ˇ N (0,0,0,1) ˇ N (0,0,1,1) 0 6 8 8 ˇ N (1,0,0,0) ˇ N (1,0,1,0) ˇ N (1,0,0,1) ˇ N (1,0,1,1) 2 6 8 8 ˇ N (0,1,0,0) ˇ N (0,1,1,0) ˇ N (0,1,0,1) ˇ N (0,1,1,1) 4 6 8 8 ˇ N (1,1,0,0) ˇ N (1,1,1,0) ˇ N (1,1,0,1) ˇ N (1,1,1,1) 4 6 8 8 Table 2.9: For 4-layers NUDD with N 1 = 2,N 2 = 4,N 3 = 6,N 4 = 8, theoretical and numerical decoupling order for all error types are in complete agreement ˇ N (0,0,0,0) ˇ N (0,0,1,0) ˇ N (0,0,0,1) ˇ N (0,0,1,1) 0 6 3 6,7 ˇ N (1,0,0,0) ˇ N (1,0,1,0) ˇ N (1,0,0,1) ˇ N (1,0,1,1) 2 6 3 6,7 ˇ N (0,1,0,0) ˇ N (0,1,1,0) ˇ N (0,1,0,1) ˇ N (0,1,1,1) 4 6 4,5 6,7 ˇ N (1,1,0,0) ˇ N (1,1,1,0) ˇ N (1,1,0,1) ˇ N (1,1,1,1) 4 6 4,5 6,7 Table 2.10: For 4-layers NUDD withN 1 = 2,N 2 = 4,N 3 = 6,N 4 = 3, theoretical and numerical decoupling order (black) for all error types are in complete agreement. The red ones are naive decoupling orders whose values are different than numerical ones. The Table 2.9, Table 2.10, and Table 2.11 are typical examples where our theoretical predictions and numerical results exactly match, i.e., our prediction for the decoupling order of each error type and the the overall decoupling order are the same as the actual orders obtained by numerical results. And it happens that e N i = N i for all these three types of examples. First example, Table 2.9, is one of the 4-layer NUDD with all even sequence orders. As discussed before, its decoupling order formula ˇ N ~ r ` = max[{r i N i } ` i=1 ] Eq. (2.50) coincides with the naive decoupling order formula; interference between layers is not observed. From Table 2.9, theoretical, numerical decoupling, and the naive decoupling orders for all error types are the same. The theoretical overall decoupling order ˇ N min = min[2,4,6,8] = 2 is found to agree with the actual overall decoupling order as well. The second example we consider, Table 2.10, is one of the 4-layer NUDD with all even sequence orders except the outer-most UDD layer, namely,[N i ] 2 = 0 for alli<` and [N ` ] 2 = 1. Theoretical and numerical decoupling orders (black) are in complete agreement for all error types. There is one order difference between the naive decoupling order (red) and the actual decoupling order (black) in the last column and the last two 59 ˇ N (0,0,0,0) ˇ N (0,0,1,0) ˇ N (0,0,0,1) ˇ N (0,0,1,1) 0 3 1 3,4 ˇ N (1,0,0,0) ˇ N (1,0,1,0) ˇ N (1,0,0,1) ˇ N (1,0,1,1) 7 7,8 7,8 7,9 ˇ N (0,1,0,0) ˇ N (0,1,1,0) ˇ N (0,1,0,1) ˇ N (0,1,1,1) 5 5,6 5,6 5,7 ˇ N (1,1,0,0) ˇ N (1,1,1,0) ˇ N (1,1,0,1) ˇ N (1,1,1,1) 7,8 7,9 7,9 7,10 Table 2.11: For 4-layers NUDD withN 1 = 7,N 2 = 5,N 3 = 3,N 4 = 1 theoretical and numerical decoupling order (black) for all error types are in complete agreement. The red ones are naive decoupling orders whose values are different than numerical ones. ˇ N (0,0,0,0) ˇ N (0,0,1,0) ˇ N (0,0,0,1) ˇ N (0,0,1,1) 0 1 6,3,2 6,1 ˇ N (1,0,0,0) ˇ N (1,0,1,0) ˇ N (1,0,0,1) ˇ N (1,0,1,1) 2 2,3 6,5,2 6,3 ˇ N (0,1,0,0) ˇ N (0,1,1,0) ˇ N (0,1,0,1) ˇ N (0,1,1,1) 4 4,5 6,4 6,5 ˇ N (1,1,0,0) ˇ N (1,1,1,0) ˇ N (1,1,0,1) ˇ N (1,1,1,1) 4 4,5 6,4 6,5 Table 2.12: For the case withN 1 = 2,N 2 = 4,N 3 = 1,N 4 = 6, all the numerical decou- pling orders (black) in the third column are larger than their corresponding theoret- ical predictions (blue). The reason that the actual decoupling orders in the fourth col- umn are all smaller than their naive decou- pling orders (red) come from that the sup- pression ability of the fourth UDD is totally ineffective. entries in the third column. From its decoupling order formula Eq. (2.51), the difference comes from that the outermost (4 th ) UDD layer with odd order 3 boost the decoupling order for error types, that are also addressed by one of the inner layers. Clearly, the deviation from the standard UDD scaling can be attributed to the asymmetry of the outer layer. The third example, shown in Table 2.11, is for all odd sequence orders such that N 1 > N 2 > N 3 > N 4 with [N i ] 2 = 1 for all i. The theoretical decoupling order formula is given in Eq. (2.53). Comparing the theoretical/numerical results with the naive decoupling orders, we find that the increase in decoupling order is attributed to the outer-odd-UDD suppression effect, the second term in Eq. (2.53). Discrepancies between numerical results and theoretical predictions occur when the inner UDD layers with odd parity sequence orders that are smaller than the orders of their outer layers. In Tables 2.12 and 2.13, we consider two cases where deviations from the theoretical predictions occur. It is important to note that although the numerical and 60 theoretical decoupling orders differ, we do not predict error suppression beyond what is achievable. The formula given by Eq. (2.40) can be thought of as a lower bound for the actual decoupling order. Essentially, we are predicting the minimum order of error suppression for each error type. Table 2.12 is an example where the third layer is the only odd parity layer. Our theoretical prediction Eq. (2.40) for this case is ˇ N ~ r 4 = max[{r i N i +r 3 } 2 i=1 ,r 3 N 3 ,r 4 (r 3 [N 3 ] 2 ⊕1) e N 4 ], (2.122) where the suppression ability of the fourth UDD layer, e N 4 = min[N 4 ,N 3 +1], is hin- dered by the third UDD layer if N 3 < N 4 − 1. For the particular sequence orders, N 1 = 2,N 2 = 4,N 3 = 1,N 4 = 6, the theoretical prediction is e N 4 = min[6,1+1] = 2. Comparing the theoretical results to numerical simulations, it appears that the theoreti- cal decoupling orders (marked in blue) are lower than the numerically calculated values for certain error types; see column 3 in Table 2.12. However, despite the discrepancy, Eq. (2.122) captures the inhibiting characteristics of the odd parity inner sequence order on the suppression ability of the outer-most layer. As theoretically predicted, the outer- odd-UDD suppression effects are observed on the error types, which anti-commute with the third and its inner UDD layers, in the second column by comparing their naive and the numerical decoupling orders. For the error types with r 3 = r 4 = 1 in the fourth column of Table 2.12, due tor 3 [N 3 ] 2 ⊕1 = 0, our theoretical formula Eq. (2.122) shows that the fourth layer of UDD sequence is totally ineffective owing to the odd sequence order of the third UDD layer. Indeed, the numerical decoupling order for the error types in the fourth column of Table 2.12 is smaller than 6, the sequence order of the fourth UDD layer. 61 ˇ N (0,0,0,0) ˇ N (0,0,1,0) ˇ N (0,0,0,1) ˇ N (0,0,1,1) 0 5,3,2 7,3,2 7,4,3 ˇ N (1,0,0,0) ˇ N (1,0,1,0) ˇ N (1,0,0,1) ˇ N (1,0,1,1) 1 5,2 7,2 7,5,3 ˇ N (0,1,0,0) ˇ N (0,1,1,0) ˇ N (0,1,0,1) ˇ N (0,1,1,1) 3,2 5,4,3 7,4,3 7,4 ˇ N (1,1,0,0) ˇ N (1,1,1,0) ˇ N (1,1,0,1) ˇ N (1,1,1,1) 3,2 5,3 7,5,3 7,6,4 Table 2.13: For the 4-layers NUDD with N 1 = 1, N 2 = 3, N 3 = 5, N 4 = 7, all naive decoupling orders are greater or equal than their corresponding numerical decou- pling orders, and all the numerical decou- pling orders are greater or equal than their corresponding theoretical predictions. ˇ N (0,0,0,0) ˇ N (0,0,1,0) ˇ N (0,0,0,1) ˇ N (0,0,1,1) 0 5,4 1 5,6,5 ˇ N (1,0,0,0) ˇ N (1,0,1,0) ˇ N (1,0,0,1) ˇ N (1,0,1,1) 7 7,8 7,8 7,9 ˇ N (0,1,0,0) ˇ N (0,1,1,0) ˇ N (0,1,0,1) ˇ N (0,1,1,1) 3,4,3 5,4 3,4 5 ˇ N (1,1,0,0) ˇ N (1,1,1,0) ˇ N (1,1,0,1) ˇ N (1,1,1,1) 7,8 7,9 7,9 7,10 Table 2.14: For the case with N 1 = 7, N 2 = 3, N 3 = 5, N 4 = 1, the numerical decoupling order (black) and our theoretical predictions (blue) have different values for some error types. Naive decoupling order is marked as red. For NUDD with a sequence order set{1,3,5,7}, their corresponding suppression orders are e N 1 = N 1 = 1 and e N 2 = e N 3 = e N 4 = 2. It follows that even though the sequence orders of the last three UDD layers are larger than three, all their suppression abilities were hindered, and can only achieve second order decoupling, due to the odd sequence order, 1, of the first UDD layer. From Table 2.13, numerical decoupling orders (black) for most errors are smaller than the naive one (red), which shows that the sup- pression abilities of the outer UDD layers were inhibited. However, many theoretical predictions (blue) in this case are smaller than their associated actual decoupling orders, which means that those theoretical predictions made by our decoupling order formula are not tight. Moreover, it appears that the actual decoupling order is not only deter- mined by the sequence orders, but also by the error types in a complicated fashion not captured by the decoupling formula presented. Note that we still obtain a lower bound on the decoupling order for each error type. For NUDD with a sequence order set{7,3,5,1}, only the suppression order of the third UDD layer is not equal to its UDD sequence order 5, e N 3 = min[3 + 1,5] = 4. 62 From Table 2.14, our theoretical estimations (blue) are not tight with their numerical decoupling orders (black) for only a few error types. The theoretical overall decoupling orders completely agree with the numerical ones for all the above examples. 2.5.6 Discussions Note that our prediction for the decoupling order of each error type is consistent with the numerical one as long as the numerical decoupling order is greater (not smaller) than ours. The discrepancy exhibits that our analysis gives the lower bound of the decoupling order for each error type. And we attribute this discrepancy to the fact that the method we used in Sec. 2.4.6 does not use the full information contained in the integrands, i.e., we discard all Fourier coefficients. To be more specific, recalling the method we used in Sec. 2.4.6 and Sec. 2.4.7, we argue that if F (n) (~ r ` can be written as R π 0 c N ` ,n odd dθ with no constant 1 in c N ` ,n odd , then it’s guaranteed that F (n) (~ r ` = 0. However, if we take all Fourier coefficients into account in c N ` ,n odd or other forms listed in Table 2.5, it is still possible that F (n) (~ r ` vanishes because a sum of non-zero terms could be zero when combined with the right Fourier coefficients. Therefore, the decoupling orders obtained by our method of analysis are not tight. From those earlier numerical examples, it seems that the numerical suppression ability of each UDD layer could also depend on the error types for some NUDD schemes in contrast with our theoretical decoupling order formula Eq. (2.40) where the suppression ability of each UDD layer depends only on the sequence order set. And it seems challenging to conclude the actual decoupling order formula from a few examples. 63 2.6 Conclusions NUDD scheme, which nests multiple layers of UDD sequences, is a much more efficient scheme, compared to other known DD schemes, for general multi-qubit decoherence suppression. In this work, we give a rigorous analysis with a compact formulation for the university and performance of general NUDD sequence with a MOOS as our control pulses set. We show that the overall suppression order of NUDD with more general con- trol pulses is also the minimum of all the sequence orders of NUDD scheme. Moreover, we also obtain the decoupling order for each type of error, provided that a sequence order set of NUDD is given. Our decoupling order formula Eq. (2.40) shows that only for the NUDD with all even sequence orders, all the layers of NUDD work indepen- dently, i.e. the suppression ability of each UDD layer remains the same as the one without nesting other UDD sequences. For the remaining NUDD schemes with at least one odd sequence order, the interference phenomenon between UDD layers appears and is summarized as follows, 1. For a given UDD layer, say the i th UDD layer with sequence order N i , if there are inner layers with odd sequence order and the lowest odd order is smaller than N i −1, then the suppression ability of thei th UDD layer is hindered by its inner UDD layer with lowest odd order, i.e., it cannot achieveN th i -order decoupling. ( e N i =N k 0 <i o min +1<N i in Eq. (2.40) ) 2. For the i th UDD layer to be effective on a given error type, this error needs to not only anti-commute with the control pulses of thei th UDD layer (r i = 1) but also anti-commute with the total even number of UDD layers with odd sequence orders before thei th UDD layer (⊕ i−1 k=1 r k [N k ] 2 = 0). 3. For a given error type, if there is total odd number of UDD layers with odd sequence orders before the i th UDD layer that the error anti-commutes with 64 (⊕ i−1 k=1 r k [N k ] 2 = 1), then thei th UDD layer is totally ineffective (⊕ i−1 k=1 r k [N k ] 2 ⊕ 1 = 0) no matter if this error anti-commutes with this layer or not. 4. For a given error type, each odd order UDD layer, the error anti-commutes with and is nested outside thei th UDD layer, can enhance the suppression ability of the i th UDD layer one more order (outer-odd-UDD effect) on this error type. In other words, the outer-odd-UDD suppression effect is cumulative. ( P ` k=i+1 r k [N k ] 2 ) Since our analysis identifies the conditions under which the suppression ability of a given UDD layer is being inhibited, or totally ineffective, or enhanced by other UDD layers with odd sequence orders, one can design a NUDD scheme accordingly such that the full power of each UDD layer is fully exploited. To be more specific, sup- pose we would like to design a NUDD scheme from some UDD sequences whose con- trol pulse types and sequence orders are given. From the analysis, in order to reach optimal efficiency of NUDD, first, nest all the UDD layers with even sequence orders together where the nesting orders can be arbitrary, and denote this resulting sequence as NUDD even ; second, nest all the UDD layers with odd sequence orders together such that the sequence orders from the inner-most to the outer-most layers are decreasing, and denote this resulting sequence as NUDD odd ; Third, the final NUDD scheme is constructed by nesting NUDD even as the inner sequence with NUDD odd as the outer sequence. 2.7 Appendices 2.7.1 The outermost UDD interval decomposition We shall derive Eq. (2.69) by splitting each integral of F ⊕ n p=1 ~ r (p) ` [Eq. (2.32)] into a sum of sub-integrals over the normalized outermost layer intervalss j ` . SinceF ⊕ n p=1 ~ r (p) ` 65 comprises a series of time-ordered, nested integrals, our procedure for decomposing F ⊕ n p=1 ~ r (p) ` is to split its nested integrals one by one, fromη (n) toη (1) . We call the sub-integral over thej th ` outermost interval “sub-integral-j ` ”. Suppose the integral of the integration variableη (p) follows the sub-integral-j (p+1) ` of the previous variableη (p+1) . By splitting the integral ofη (p) with respect to the normalized outermost intervalss j ` , we have Z η (p+1) 0 ` Y i=1 f i (η (p) ) r (p) i dη (p) = j (p+1) ` −1 X j (p) ` =1 f ` (j (p) ` ) r (p) ` Z η j (p) ` η j (p) ` −1 `−1 Y i=1 f i (η (p) ) r (p) i dη (p) + f ` (j (p+1) ` ) r (p) ` Z η (p+1) η j (p+1) ` −1 `−1 Y i=1 f i (η (p) ) r (p) i dη (p) whereη j (p) ` is UDD N ` pulse timing andf ` (j (p) ` ) = (−1) j (p) ` −1 . For each sub-integral with the integration regime from [η j (p) ` −1 ,η j (p) ` ) with pulse intervals j (p) ` , we make the following linear change of variable, ˜ η (p) = η (p) −η j (p) ` −1 s j (p) ` , (2.123) 66 to linearly normalize the integration regime to be 1. Accordingly, each f i (˜ η (p) ) with i≤`−1 becomes the normalizedi th -layer modulation function for(`−1)-layers NUDD scheme, and is the same function for all the outermost pulse interval. Consequently, Z 1 0 d˜ η (p) `−1 Y i=1 f i (˜ η (p) ) r (p) i j (p+1) ` −1 X j (p) ` =1 f ` (j (p) ` ) r (p) ` s j (p) ` (2.124) + Z ˜ η (p+1) 0 d˜ η (p) `−1 Y i=1 f i (˜ η (p) ) r (p) i f ` (j (p+1) ` ) r (p) ` s j (p+1) ` (2.125) where R 1 0 Q `−1 i=1 f i (˜ η (p) ) r (p) i d˜ η (p) is taken out from the summation. Rewrite Eq. (2.124) and Eq. (2.125) to the similar forms by introducing the config- uration numberξ p whereξ p = 0 denotes Eq. (2.124) whileξ p = 1 denotes Eq. (2.125). Then Eq. (2.124) rewritten withξ p = 0 becomes Z (η (p+1) ) ξp 0 `−1 Y i=1 f 0 i (η (p) ) r (p) i dη (p) j (p+1) ` −(ξp⊕1) X j (p) ` =1 f ` (j (p) ` ) r (p) ` s j (p) ` (2.126) wheref 0 i (η (p) )≡f i (˜ η (p) ) and Eq. (2.125) withξ p = 1 becomes Z (η (p+1) ) ξp 0 `−1 Y i=1 f 0 i (η (p) ) r (p) i dη (p) j (p+1) ` −(ξp⊕1) X j (p) ` =ξpj (p+1) ` f ` (j (p) ` ) r (p) ` s j (p) ` (2.127) where j (p+1) ` −(ξp⊕1) X j (p) ` =ξpj (p+1) ` = j (p+1) ` X j (p) ` =j (p+1) ` (2.128) meaning thatj (p) ` =j (p+1) ` . Note that both forms of Eqs. (2.126) and (2.127) are naturally decomposed into inner part, which is a integral of the first(`−1) modulation functions for(`−1)-layers 67 NUDD scheme, multiplying the outer part, which is sum over the outermost (` th ) layer modulation function. In most cases, each integral ofF ⊕ n p=1 ~ r (p) ` can be split into Eqs. (2.126) and (2.127), with one exception: ifj (p+1) ` = 1, the subsequent sub-integrals of variables fromη (p) to η (1) will only contain Eq. (2.127) whose configuration number is 1. By substituting Eqs. (2.126) and (2.127) into each integral ofF ⊕ n p=1 ~ r (p) ` , in sequence fromη (n) toη (1) , taking that exception into account, and collect all the inner(outer) parts of all sub-integrals together, we obtain F ⊕ n p=1 ~ r (p) ` = X {ξ k =0,1} n−1 k=1 Φ in ξn...ξ 1 ({~ r (p) `−1 } n p=1 )Φ out ξn...ξ 1 ({r (p) ` } n p=1 ), (2.129) where Φ in ξn...ξ 1 ({~ r (p) `−1 } n p=1 ) = n Y p=1 Z (η (p+1) ) ξp 0 `−1 Y i=1 f 0 i (η (p) ) r (p) i dη (p) , (2.130) , Φ out ξn...ξ 1 ({r (p) ` } n p=1 ) = N ` +1 X j (n) ` = P n k=1 (ξ k ⊕1) f ` (j (n) ` ) r (n) ` s j (n) ` n−1 Y p=1 j (p+1) ` −(ξp⊕1) X j (p) ` =ξpj (p+1) ` +(ξp⊕1) P p k=1 (ξ k ⊕1) f ` (j (p) ` ) r (p) ` s j (p) ` (2.131) and P {ξ k =0,1} n−1 k=1 sums over all possible nested integration (sum) configurations ξ n ...ξ 2 ξ 1 withξ n ≡ 0 for the inner partΦ in ξn...ξ 1 (the outer partΦ out ξn...ξ 1 ). From Eq. (2.131), note that forξ p = 1, j (p+1) ` −(1⊕1) X j (p) ` =j (p+1) ` +(1⊕1) P p k=1 (ξ k ⊕1) = j (p+1) ` X j (p) ` =j (p+1) ` . (2.132) 68 which means that the variables η (p) and η (p+1) in Eq. (2.32) are in the same interval j (p) ` =j (p+1) ` , while forξ p = 0, j (p+1) ` −(0⊕1) X j (p) ` =0+(0⊕1) P p k=1 (ξ k ⊕1) = j (p+1) ` −1 X j (p) ` = P p k=1 (ξ k ⊕1) (2.133) which means that the variableη (p) in Eq. (2.32) locates in an earlier interval than the variableη (p+1) , namely,j (p) ` <j (p+1) ` . P p k=1 (ξ k ⊕1) counts the number ofξ k = 0 from k = 1 tok =p. Note that eachΦ in ξn...ξ 1 ({~ r (p) `−1 } n p=1 ) is a product of multiple nested integrals with mod- ulation functions of (`− 1)-layers NUDD scheme as integrands. As a matter of fact, each nested integrals contained in Φ in ξn...ξ 1 ({~ r (p) `−1 } n p=1 ) is actually one of (`− 1)-layers NUDD coefficients. For example, an inner term appeared in the expansion of the 8 th order`-layers NUDD coefficientsF ⊕ 8 p=1 ~ r (p) ` reads Φ in 01101011 ({~ r (p) `−1 } 8 p=1 ) = F ⊕ 8 p=6 ~ r (p) `−1 F ⊕ 5 p=4 ~ r (p) `−1 F ⊕ 3 p=1 ~ r (p) `−1 = F (3) ~ r 000 `−1 F (2) ~ r 00 `−1 F (3) ~ r 0 `−1 (2.134) where~ r 000 `−1 =⊕ 8 p=6 ~ r (p) `−1 ,~ r 00 `−1 =⊕ 5 p=4 ~ r (p) `−1 , and~ r 0 `−1 =⊕ 3 p=1 ~ r (p) `−1 . And it’s correspond- ing outer part reads Φ out 01101011 ({r (p) ` } 8 p=1 ) = N ` +1 X j (8) ` =3 f ` (j (8) ` ) ⊕ 8 p=6 r (p) ` s 3 j (8) ` × j (8) ` −1 X j (5) ` =2 f ` (j (5) ` ) ⊕ 5 p=4 r (p) ` s 2 j (5) ` × j (5) ` −1 X j (3) ` =1 f ` (j (3) ` ) ⊕ 3 p=1 r (p) ` s 3 j (3) ` (2.135) where Eq. (2.25) is being used. 69 As suggested by Eqs. (2.134) and (2.135), one can see that each configuration ξ n ...ξ 1 defines a way to separate n vectors {~ r (p) ` } n p=1 into several clusters. Order configuration numbers and error vectors as ξ n ~ r (n) ` ...ξ 1 ~ r (1) ` . Then a cluster of vec- tors is defined as a contiguous set of vectors only connected by configuration num- ber 1. Different clusters are separated by configuration number whose value is 0. For ξ 8 ...ξ 1 = 01101011 in Eqs. (2.134) and (2.135) as an example, we have 0~ r (8) ` 1~ r (7) ` 1~ r (6) ` 0~ r (5) ` 1~ r (4) ` 0~ r (3) ` 1~ r (2) ` 1~ r (1) ` (2.136) which divides the 8 vectors into three clusters. Suppose for a given configuration ξ n ...ξ 2 ξ 1 which separatesn vectors{~ r (p) ` } n p=1 intom clusters, thea th cluster (count- ing from right to left) containsn a number of vectors{~ r (p) ` } μ p=ν withn a = μ−ν + 1. Then according to the result of ⊕ μ p=ν ~ r (p) ` , its corresponding inner part is F (na) ~ r <a> `−1 with ~ r <a> `−1 ≡⊕ μ p=ν ~ r (p) `−1 and the corresponding outer part is P j (a+1) ` −1 j (a) ` =a f ` (j (a) ` ) r <a> ` s na j (a) ` with r <a> ` ≡⊕ μ p=ν r (p) ` . Note that for the last cluster, i.e.a =m, the upper bound of the sum j (m+1) ` −1 is replaced byN ` +1. Therefore, Eq. (2.129) can be re-expressed as a more compact form, Eq. (2.69). 2.7.2 F (n) ~ r ` in terms ofF (n 0 ) ~ r 0 `−1 Here, we shall prove Lemma 1: ˇ N ~ r `−1 ≤ P m a=1 ˇ N ~ r <a> `−1 where~ r `−1 =⊕ m a=1 ~ r <a> `−1 . Proof of Lemma 1. For a given~ r `−1 -type error with~ r `−1 = ⊕ m a=1 ~ r <a> `−1 Eq. (2.72), its decoupling order ˇ N ~ r `−1 Eq. (2.40) is max[{r i ((|ν i | ⊕1) e N i + `−1 X k=i+1 r k [N k ] 2 } `−1 i=1 ], (2.137) 70 where |ν i |≡⊕ i−1 k=1 r k [N k ] 2 (2.138) for shorthand notation. Suppose the maximum occurs at theM th UDD layer, i.e., ˇ N ~ r `−1 = e N M + `−1 X k=M+1 r k [N k ] 2 (2.139) where the coefficient of e N M is equal to 1, implying that r M = 1 (2.140) |ν M |≡⊕ M−1 k=1 r k [N k ] 2 = 0. (2.141) With Eq. (2.72), Eq. (2.140), Eq. (2.141) , we have the following equalities and inequal- ity: firstly, ⊕ m a=1 r <a> M =r M = 1, (2.142) secondary, |ν M | =⊕ m a=1 |ν <a> M | = 0 (2.143) due to |ν M | =⊕ M−1 k=1 r k [N k ] 2 = ⊕ M−1 k=1 (⊕ m a=1 r <a> k )[N k ] 2 = ⊕ m a=1 (⊕ M−1 k=1 r <a> k [N k ] 2 ) = ⊕ m a=1 |ν <a> M |, (2.144) 71 and thirdly, `−1 X k=M+1 r k [N k ] 2 = `−1 X k=M+1 (⊕ m a=1 r <a> k )[N k ] 2 ≤ `−1 X k=M+1 m X a=1 r <a> k [N k ] 2 ≤ m X a=1 `−1 X k=M+1 r <a> k [N k ] 2 . (2.145) By using the following properties: max[A,B,C]≥ max[A,B], (2.146) max[A+c,B +c] = max[A,B]+c, (2.147) we have m X a=1 ˇ N ~ r <a> `−1 = m X a=1 max[{r <a> i (|ν <a> i |⊕1) e N i + `−1 X k=i+1 r <a> k [N k ] 2 } `−1 i=1 ] ≥ m X a=1 max[{r <a> i (|ν <a> i |⊕1) e N i + `−1 X k=i+1 r <a> k [N k ] 2 } M i=1 ] ≥ m X a=1 max[{r <a> i (|ν <a> i |⊕1) e N i + M X k=i+1 r <a> k [N k ] 2 } M i=1 ] + m X a=1 `−1 X k=M+1 r <a> k [N k ] 2 (2.148) 72 Equivalently, m X a=1 ˇ N ~ r <a> `−1 ≥ m X a=1 ˇ N ~ r <a> M + m X a=1 `−1 X k=M+1 r <a> k [N k ] 2 . (2.149) In light of the inequality presented in Eq. (2.145), the remaining task is to show m X a=1 ˇ N ~ r <a> M ≥ e N M (2.150) which will immediately lead to Lemma 1: P m a=1 ˇ N ~ r <a> `−1 ≥ ˇ N ~ r `−1 . The condition⊕ m a=1 r <a> M = r M = 1 Eq. (2.142) implies that there must exist at least onea 0 such thatr <a 0 > M = 1. For this~ r <a 0 > M vector withr <a 0 > M = 1, it’s firstM−1 components either satisfy|ν <a 0 > M | = 0 or|ν <a 0 > M | = 1. For the ~ r <a> M -type error with r <a 0 > M = 1 and |ν <a 0 > M | = 0, the coefficient r <a 0 > M (|ν <a 0 > M |⊕1) of e N M in ˇ N ~ r <a 0 > M Eq. (2.40) is not zero. Therefore, with the property Eq. (2.146) ˇ N ~ r <a 0 > M = max[{···} M−1 i=1 , e N M ]≥ e N M (2.151) which leads to Eq. (2.150). However, for the~ r <a> M -type error with r <a 0 > M = 1 and v <a 0 > M = 1, the coefficient r <a 0 > M (|ν <a 0 > M |⊕1) of e N M in Eq. (2.40) is zero, leading to ˇ N ~ r <a 0 > M = max[{···} M−1 i=1 ,0] (2.152) which can not prove ˇ N ~ r <a 0 > M ≥ e N M . This notwithstanding,v <a 0 > M ≡⊕ M−1 k=1 r <a 0 > k [N k ] 2 = 1 indicates that among theM −1 components of the vector~ r <a 0 > M , there is a total odd number of components each of which is equal to 1 and associates with odd sequence 73 order. Suppose the o 0 component is the outer-most non-zero componentr o 0 = 1 with [N o 0] 2 = 1 of~ r <a 0 > M . Accordingly, |ν <a 0 > o 0 | =⊕ o 0 −1 k=1 r <a 0 > k [N k ] 2 = 0 (2.153) M X k=o 0 +1 r <a 0 > k [N k ] 2 = 0 (2.154) With Eq. (2.153) and Eq. (2.154), it follows that ˇ N ~ r <a 0 > M = max[{···} o 0 −1 i=1 , e N o 0,{···} M−1 i=o 0 +1 ]≥ e N o 0 ≥ e N M −1, (2.155) where the second inequality is derived as follows e N M = min[N M ,N k<M o min +1] = min[N M ,N k<o 0 o min +1] ≤N k<o 0 o min +1 = min[N k<o 0 −1 o min +1,N o 0 +1] ≤ e N o 0 +1 (2.156) Moreover, ifv <a 0 > M = 1, the condition|ν M | =⊕ M−1 a=1 |ν <a> M | = 0 Eq. (2.143) implies that there is another vector~ r <a”> M such that|ν <a 00 > M |≡⊕ M−1 k=1 r <a 00 > k [N k ] 2 = 1. Using the same argument, there must exist ar o” = 1 withv <a”> o” = 0 with a odd sequence order N o” ,o”<M. It follows that ˇ N ~ r <a 00 > M = max[{···} o”−1 i=1 , e N o” ,{···} M i=o”+1 ]≥ e N o” ≥ 1. (2.157) Therefore, with Eqs. (2.155) and (2.157), m X a=1 ˇ N ~ r <a> M ≥ ˇ N ~ r <a 0 > M + ˇ N ~ r <a 00 > M ≥ e N M (2.158) 74 which proves Eq. (2.150). 2.7.3 Fourier expansions after linear change of variable The ` th -layer modulation function f ` (θ) which switches its sign alternatively with the new outermost-layer pulse timing θ j ` = j ` π N ` +1 has a period of 2π N ` +1 , because the outermost-layer pulses are now equally spaced. Therefore, f ` (θ) has a simple Fourier expansion, f ` (θ) = ∞ X k=0 d ` k sin[(2k+1)(N ` +1)θ], (2.159) withd ` k = 4 (2k+1)π . The remaining modulation functions such asf i (θ) withi < ` which switches sign with the inner-layer UDD pulse timingθ j ` ,...,j i all have the same period, π N ` +1 . This is because the inner (`− 1)-layers NUDD pulse timing structure inside each [θ j ` −1 ,θ j ` ) is still preserved and actually becomes identical to each other, in that Eq. (2.81) just rescales the outermost layer interval [θ j ` −1 ,θ j ` ) linearly. In addition, due to time- symmetric structure of NUDD pulse timings, it follows that inside each[θ j ` −1 ,θ j ` ),f i (θ) withi < ` is an even function when its orderN i is even, and is an odd function when itsN i is odd. Therefore, it follows that the Fourier expansion off i (θ) has the following form, f i (θ) =            ∞ X k=0 d i k cos[2k(N ` +1)θ] N i even ∞ X k=1 d i k sin[2k(N ` +1)θ] N i odd (2.160) 75 withi < `. Note that the Fourier expansion coefficientsd i k in the even and odd cases are in fact different, we use the same notation for both since the exact values of these coefficients are irrelevant for our proof. ForG 1 (θ) = N ` +1 π s j ` withθ∈ [θ j ` −1 ,θ j ` ) Eq. (2.85), the symmetry property (2.79) implies thatG 1 (θ) is also time symmetric so it can be written as G(θ) = ∞ X p=1 g p sin[pθ]. (2.161) Let us now compute the expansion coefficients: g p ≡ 1 π/2 Z π 0 G(θ)sin[pθ]dθ = 2(N ` +1) π 2 N ` +1 X j ` =1 s j ` Z θ j ` θ j ` −1 sin[pθ]dθ = − 2(N ` +1) π 2 p sin π 2(N ` +1) N ` +1 X j ` =1 sin (2j ` −1)π 2(N ` +1) × (cospθ j ` −cospθ j ` −1 ) = − 2(N ` +1) π 2 p sin π 2(N ` +1) N ` +1 X j ` =1 sin (2j ` −1)π 2(N ` +1) × (−2)sinp θ j ` +θ j ` −1 2 sinp θ j ` −θ j ` −1 2 (2.162) 76 where we used the sum-to product formula in the third equality. Due to θ j ` −θ j ` −1 2 = π 2(N ` +1) and the product-to sum formula, we have g p = 4(N ` +1) π 2 p sin π 2(N ` +1) sin pπ 2(N ` +1) × N ` +1 X j ` =1 sin (2j ` −1)π 2(N ` +1) sinp (2j ` −1)π 2(N ` +1) = 4(N ` +1) π 2 p sin π 2(N ` +1) sin pπ 2(N ` +1) × (2.163) 1 2 N ` +1 X j ` =1 cos (p−1)(2j ` −1)π 2(N ` +1) −cos (p+1)(2j ` −1)π 2(N ` +1) . Considering the sum overj ` we have N ` +1 X j ` =1 cos (p±1)(2j ` −1)π 2(N ` +1) = N ` +1 X j ` =1 cos[ (p±1)j ` π (N ` +1) − (p±1)π 2(N ` +1) ] = Re[e −i (p±1)π 2(N ` +1) N ` +1 X j ` =1 e i (p±1)j ` π (N ` +1) ] = Re[e −i (p±1)π 2(N ` +1) e i (p±1)π N ` +1 1−e i(p±1)π 1−e i (p±1)π N ` +1 ] = Re[ 1−cos(p±1)π−isin(p±1)π e −i (p±1)π 2(N ` +1) −e i (p±1)π 2(N ` +1) ] = sin(p±1)π 2sin (p±1)π 2(N ` +1) , (2.164) 77 where in the third equality we used the geometric series formula. The last expression vanishes ifp6= 2k(N ` +1)∓1. The only values ofp for whichg p does not vanish are 2k(N ` +1)∓1. Therefore, the Fourier expansion ofG 1 (θ) is G 1 (θ) = ∞ X k=0 X q=−1,1 g k,q sin[2k(N ` +1)θ+qθ]. (2.165) According to Definition 2, the function type of f ` (θ), f Ω i<` (θ), and G 1 (θ), Eq. (2.159), Eq. (2.160), and (2.165), are identified as Eqs (2.86)- (2.88). 2.7.4 min[{ ˇ N ~ r `−1 ∈~ v 1 `−1 }] =N k 0 <` o min We shall prove Lemma 4: min[{ ˇ N ~ r `−1 ∈~ v 1 `−1 }] =N k 0 <` o min . Proof of Lemma 4. For all~ r `−1 ∈~ v 1 `−1 , their components by definition satisfy the con- dition⊕ `−1 k=1 r k [N k ] 2 = 1, which implies that there is a total odd number of non-zero components with odd sequence orders in~ r `−1 . Suppose before the`-layer, theo th 1 ,o th 2 , ... , ando th j UDD layers where 0 < o 1 < o 2 < ··· < o j < ` are the layers with odd sequence orders, namely,[N o i ] 2 = 1 fori = 1,2,...,j. ~ e o i is the~ r `−1 -type vector withr o i = 1 and allr j6=o i = 0. For~ e o i -type errors which obviously belongs to~ v 1 `−1 -type error, we have ˇ N ~ eo i = e N o i according to the decoupling order formula Eq. (2.40). For the remaining~ v 1 `−1 vectors, they have 3 or more non-zero components associated with odd orders. For a given~ r `−1 6=~ e o i ∈~ v 1 `−1 , suppose the inner-most component with odd order isr o i = 1. Then its decoupling order follows ˇ N ~ r `−1 6=~ eo i ∈~ v 1 `−1 = max[..., e N o i + `−1 X k=o i +1 r k [N k ] 2 ,...] ≥ e N o i = ˇ N ~ eo i (2.166) 78 Therefore, min[{ ˇ N ~ r `−1 ∈~ v 1 }] will occur among the decoupling orders of the errors, which has only one non-zero component with an odd sequence order, i.e., min[{ ˇ N ~ r `−1 ∈~ v 1 `−1 }] = min[ ˇ N ~ eo 1 , ˇ N ~ eo 2 ,..., ˇ N ~ eo j ] = min[ e N o 1 , e N o 2 ,..., e N o j ] (2.167) = min[N o 1 ,N o 2 ,...,N o j ] (2.168) where the property such as min[A,min[B,C]] = min[A,B,C] is used in Eq. (2.167). Eq. (2.168) is the same expression of N k 0 <` o min ≡ min 6=0 [{[N k 0] 2 N k 0} `−1 k 0 =1 ] Eq. (2.42), which completes the proof of Lemma 4. 79 Chapter 3 Adiabatic Quantum Brachistochrone 3.1 Introduction of adiabatic quantum computing (AQC) It’s widely believed that quantum computation based on quantum mechanics can solve certain problems much faster than classical computers. There are many different models for quantum computation. The most standard model is the model of quantum circuits which performs quantum computation by applying a sequence of unitary logic quantum gates[38]. Another alternative computational model called Adiabatic Quantum Compu- tation (AQC) which based on quantum adiabatic theorem [34] was proposed by Farhi et al [19, 18]. It has been proved that AQC’s computation power is polynomially equivalent to the conventional gate model approach [2, 35]. The basic strategy of AQC is to solve computational problems based on adiabatic evolution. Prepare a system starting from a readily accessible ground state of some initial HamiltonianH i ; vary time dependent HamiltonianH(t) slowly enough fromH i into the final HamiltonianH p , which encodes a solution to the problem of interest; then the system will remain closely to the ground state of instantaneousH(t) in the course of the evolution, and finally reach the desired ground state of H p , as long as there is no energy level crossing during evolution. This final state corresponds to the solution 80 of a hard problem, while a short timeT corresponds to an efficient AQC strategy. The conventional path to connectH i withH p is linear interpolation H(t) = (1− t T )H i + t T H p . (3.1) However, differentH i ,H p andH(t) can be used and correspond to different adiabatic algorithms. This also implies that AQC exhibits inherent resistance to the noise. In this work, given a classical computation problem, we devise a variational time- optimal strategy to obtain an optimal path H(t) called quantum adiabatic brachis- tochrone (QAB) between H i and H p such that the total time taken to perform com- putation is shortest. Furthermore, we found that adiabatic algorithm can be recast in a natural differential-geometric framework. Specifically, we can construct a Riemannian geometry, along with its corresponding metric induced by the adiabatic evolution. In this way, the geometric method converts a quantum optimal control problem, i.e., find- ing QAB into finding geodesic (the shortest path) on the Riemannian manifold. We revisit Grover’s problem to illustrate the advantage of this optimal approach. 3.2 Time optimal AQC Consider a physically accessible family of HamiltoniansH(~ x) which include our initial H i and problem HamiltoniansH p . SupposeH(~ x) can be parametrized continuously by d-dimensional controllable parameters, ~ x = (x 1 ,x 2 ,...,x d ), such as electric or mag- netic fields, laser beams, or any other experimental knob. The set of thosed-dimensional parameters constitute a parameter manifold structureM. Suppose that our initialH i has coordinatesA = (a 1 ,a 2 ,...,a d ), while the coordinate of finalH p isB =(b 1 ,b 2 ,...,b d ) on this manifold. A path~ x(s) (s∈R can be viewed as the length parameter onM) from A toB specifies a sequence of HamiltoniansH(~ x(s)). 81 For the evolution being adiabatic, the rate of varying Hamiltonian should be suffi- ciently slow so that the state of the system stay close to the instantaneous ground state |E 0 (~ x(s(t)))i of Hamiltonian at instant time t during evolution. For global adiabatic evolution, the time variation of the Hamiltonian is chosen as constant and should satisfy max t∈[0,T] |hE 0 (t)| dH(t) dt |E 1 (t)i| min t∈[0,T] Δ(t) 2 6 (3.2) where Δ(t)≡E 1 (t)−E 0 (t) is the energy difference between the ground state and the first excited state. However, we adopt the local adiabatic approach [45] by adjusting the evolution rate of the driving Hamiltonian to fulfill the condition of adiabaticity locally, |hE 0 (t)| dH(t) dt |E 1 (t)i| Δ(t) 2 6. (3.3) In order to make the adiabatic dynamics have geometric structure ofM, the adia- batic condition is modified by replacing|hE 0 (t)|dH/dt|E 1 (t)i| withkdH/dtk F , where the Frobenius norm is defined as kAk F = p Tr[A † A] = s X ij |A ij | 2 . (3.4) As our approach is to use the adiabatic condition as a heuristic for finding optimal tra- jectories, the exact form of the adiabatic condition is in fact not essential. In addition, our choice of the Frobenius norm ensures its analyticity, and simplifies our calculations. In our multi-parameters setting, the modified adiabatic criterion, k dH(~ x(s)) ds k F Δ(~ x(s)) 2 ds dt 6, (3.5) 82 gives rise to the maximum rate of changing Hamiltonianv ad (s) at each point along~ x(s) v ad (s)≡ ds dt = Δ(~ x(s)) 2 k ˙ H(~ x(s)k F = Δ(~ x(s)) 2 k∂ i H(~ x(s)˙ x i k F (3.6) with ˙ H ≡ dH ds and ˙ x i (s)≡dx i (s)/ds. We namev ad (s) the adiabatic speed. Therefore, for a given curve~ x(s) which corresponds toH(~ x(s)) connectingH i and H p , we can speed up the computation by adopting the adiabatic speedv ad (s) locally to each point on~ x(s). Then the optimized total evolution time for a state evolving fromA toB along this given curve~ x(s) reads T[~ x(s)] = Z T 0 dt = Z ~ x(1)=B ~ x(0)=A ds v ad [~ x(s)] (3.7) = 1 Z 1 0 k ˙ H(~ x(s)k F Δ(~ x(s)) 2 ds| ~ x(s) (3.8) wheres is chosen as the normalized time, with end points being0 and1. Among all possible paths fromA toB on manifoldM, there exists a path, which takes the shortest time for a state reaching the desired solution state. From variational calculus, this optimal trajectory ~ x QAB (s) called quantum adiabatic brachistochrone (QAB) is obtained by δT[~ x(s)] δ~ x(s) = 0 (3.9) which leads to Euler-Lagrange equation. 83 3.3 Geometrization of AQC AQC can be recast into a differential-geometric language. We would like to endow a metric tensor on the manifoldM, such that the length of any path `[~ x(s)] = Z ~ x(1)=B ~ x(0)=A L(~ x(s); ˙ x(s))ds (3.10) = Z ~ x(1)=B ~ x(0)=A q g ij (~ x(s))˙ x i (s)˙ x j (s)ds (3.11) coincides with the optimal entire adiabatic evolution time required for the same path, i.e., `[~ x(s)] =T[~ x(s)]. (3.12) Identifying the infinitesimal length p g ij (~ x(s))˙ x i (s)˙ x j (s) with k∂ i H(~ x(s))˙ x i k F /Δ(~ x(s)) 2 in (3.8) gives rise to the adiabatic-compatible metric tensor. For each~ x on manifoldM, we have k ˙ Hk F = q Tr[ ˙ H 2 ] = r Tr[ ∂H(~ x(s)) ∂x i ∂H(~ x(s)) ∂x j ]˙ x i ˙ x j . (3.13) Therefore, the metric tensor induced by this adiabatic criterion is g ij (~ x) = 1 Δ(~ x) 4 Tr[ ∂H(~ x) ∂x i ∂H(~ x) ∂x j ]. (3.14) This metric tensor is meaningful in that it did satisfy the properties of a metric [37], 1. Positive definite:L(~ x; ˙ x) = k ˙ H(~ x)k F Δ(~ x) 2 > 0 2. Symmetric tensor:g ij (~ x) =g ji (~ x) due to Tr[ ∂H(~ x) ∂x i ∂H(~ x) ∂x j ] =Tr[ ∂H(~ x) ∂x j ∂H(~ x) ∂x i ] 3. Differentiability: This adiabatic condition requires thatH(x(s)) is a smooth func- tion (i.e., infinitely differentiable orC1) 84 4. Homogeneity:L(~ x;k˙ x) =|k|L(~ x; ˙ x) since L(~ x;k˙ x) = q Tr[ ∂H(~ x(s)) ∂x i ∂H(~ x(s)) ∂x j ]k˙ x i k˙ x j Δ(~ x) 2 =|k|L(~ x; ˙ x) (3.15) 5. detg ij 6= 0 : Our case generally obeys this condition. Accordingly, we equip AQC with a Riemannian manifold structure. In this frame- work, the problem of finding the QAB is equivalent to finding the geodesic over (M, g), which can be solved by those existing numerous numerical techniques that have been designed to solve minimal distances on manifolds. The expression of the geodesic equation is ¨ x i +Γ i jk ˙ x j ˙ x k = 0 (3.16) where Γ i jk ≡ 1 2 g il ( ∂g lj ∂x k + ∂g lk ∂x j − ∂g jk ∂x l ) (3.17) is the Christoffel connection. Sinceg ij ∝ Δ −4 (if the numerator does not contribute a power ofΔ) we findΓ∼g −1 ∂g∼ Δ −1 ∂Δ. Given a family of controllable Hamiltonians, we can study its corresponding Rie- mannian geometric structure by the covariant Riemann curvature tensor R ijkl = 1 2 [ ∂g ik ∂x j ∂x l − ∂g jk ∂x i ∂x l − ∂g il ∂x j ∂x k + ∂g jl ∂x i ∂x k ]+g λη [Γ λ ik Γ η jl −Γ λ il Γ η jk ], (3.18) yieldingR ijkl ∼ Δ −6 . This geometrization not only gives the new insight of AQC, but also allows us to apply the powerful techniques and tools of differential geometry cultivated over many decades to the newer field, AQC. 85 3.3.1 Riemannian geometry of the whole Hamiltonian space From the form of (3.14), we can see that the metric tensorg ij (~ x) depends on the choice of Hamiltonian sets. Therefore, strictly speaking, our so-called QAB (geodesic) is time optimal on our chosen Hamiltonian manifold. The Hamiltonian manifold we choose is usually a sub-manifold of the whole Hamiltonian manifold, which includes all possible Hamiltonians. In other words, it’s possible that there exists a trajectory which is not on our selected Hamiltonians, but performs better then the geodesic on that certain family of Hamiltonians. Theoretically, we can get a ”real” QAB by applying our approach to the whole Hamiltonian space. However, in practice, we are not able to access the entire Hamiltonians. In spite of the implementation concern, it is still interesting to study the associated adiabatic geometric structure of this entire Hamiltonian manifold. Modeling AQC usually necessitates a parametrization of the Hamiltonian. This parametrization is in general not unique. The way we chose to parametrize all Hamil- tonian is to expand HamiltonianH in terms of tensor products of the normalized Pauli and identity matrices (normalized in the sense thatTr[σ i σ j ] =δ ij ), H(~ x) = X i x i σ i . (3.19) Note that another disadvantage of utilizing the whole Hamiltonian space is that the dimension of this manifold would be 4 n which increases exponentially withn, the size of the system . Substituting (3.19) to the metric form (3.14), we have g ij (~ x) = δ ij Δ(~ x) 4 (3.20) which is diagonal everywhere. Suppose the coordinates of the initial easily prepared HamiltonianH i and the problem HamiltonianH p areA = (a 1 ,a 2 ,...,a 4 n ), andB = 86 (b 1 ,b 2 ,...,b 4 n ), respectively. Then the geodesic which is probably not realizable is obtained by the following equation, ¨ x i − 4˙ x i Δ(~ x) ( 4 n X j=1 ∂Δ(~ x) ∂x j dx j ds )+2 P 4 n j=1 ˙ x 2 j Δ(~ x) ∂Δ(~ x) ∂x i = 0 (3.21) with boundariesA andB. 3.4 Adiabatic quantum search As an illustration, we shall apply our approach to the problem of searching in an unsorted database. The problem is this: there is an unsorted database containing N items, one of which is marked. The goal is to find this unknown marked item by access- ing the database a minimum number of times. Classically, an average of N/2 items must be tested before finding the right one. Grover’s quantum circuit model solution usesO( √ N) queries, which is provably opti- mal, and a quadratic improvement over the best possible classical strategy.[22]. An ana- log version of Grover’s algorithm in AQC model proposed by Farhi and co-workers [19] results in a complexity of orderN, which is the same order with classical algorithms. However, Roland and Cerf (RC) [45] recovered the quadratic speed-up of Grover’s algo- rithm by applying adiabatic evolution locally. In our approach, we also apply adiabatic condition locally but with a modified adi- abatic condition. Although we cannot analyze the complexity of our multi-parameters adiabatic model for a large system in general, the numerical simulation shows that the trajectory we obtained for a system with finite qubits do have better performance. In the following sections, we shall solve the geodesics on 1-dim and 2-dim manifolds for the quantum search problem and compare their performance. 87 3.4.1 Adiabatic search on 1-dimensional manifold We use the orthogonal computation basis states ofn qubits,|ii = 0,...N = 2 n to label N unsorted items, while the marked one is denoted by|mi. The initial Hamiltonian of the system is chosen as H i =I−|ψ 0 ihψ 0 | (3.22) where|ψ> is the ground state ofH i and is an equal superposition of all basis states, |ψ 0 >= 1 √ N N−1 X i=1 |i>. (3.23) The problem HamiltonianH p is chosen as H p =I−|m><m| (3.24) so that the final ground state is the marked state|m>. We selected1−parameter Hamil- tonian family ofH(x) –the conventional linear interpolation form, H(x) = (1−x)(I−|ψ 0 ihψ 0 |)+x(I−|m><m|) (3.25) with the energy gap between ground state and first exited state, Δ(x) 2 = 1−4x(1−x)(1− 1 N ). (3.26) The set of parameters constitutes a 1-dimensional parameter manifoldM 1 . The adia- batic compatible metric tensor in this case only has one component, g 11 (x) = 1 Δ 4 Tr[( dH dx ) 2 ]. (3.27) 88 Substituting energy spectrum and Tr[( dH dx ) 2 ] =Tr[(|ψ 0 ihψ 0 |−|mihm|)(|ψ 0 ihψ 0 |−|mihm|)] (3.28) =Tr[|ψ 0 ihψ 0 |−|ψ 0 ihψ 0 |mihm|−|mihm|ψ 0 ihψ 0 |+|mihm|] (3.29) = 2(1−|hψ 0 |mi| 2 ) (3.30) = 2(1− 1 N ) (3.31) to the metric form (3.27), we get g 11 (x) = 2(N−1) N[1−4x(1−x)(1− 1 N )] 2 . (3.32) From above, since the numeratork ˙ Hk 2 F =Tr[( dH dx ) 2 ] =2(1− 1 N ) is constant, the behavior of the metric is controlled only by the energy gap. The Christoffel connection in 1-dim case is Γ i jk = Γ 1 11 = 1 2 g 11 ∂g 11 ∂x (3.33) = 4(1−2x)(N−1) (1−2x) 2 (N−1)+1 (3.34) BecauseΓ i jk has only component, we can see that those terms in (3.18) cancel each other leading to R 1 111 =R 11 = 0 (3.35) which means that this space is a flat space. As a matter of fact, for 1-dimensional space, R ijkl always vanishes, i.e., there always exists a coordinate transformation which makes 89 the metricg being1 everywhere. Hence, in this trivial case, the geodesic is the only path on this 1-dim manifold. Our geodesic equation (QAB) is ¨ x+ 4(1−2x)(N−1) (1−2x) 2 (N−1)+1 ˙ x 2 = 0 (3.36) withx(0) = 0 andx(1) = 1 . It turns out that the optimal 1-d solution, t = r 2 N−1 N 1 2ε N √ N−1 [tan −1 { √ N−1(2x−1)}+tan −1 √ N−1], (3.37) coincides with the one derived by Roland and Cerf’s paper [45] up to some constant factor. This also indicates that our modified adiabatic condition is reasonable. The total required running time is given by takingx = 1 in (3.37), T = 1 ε N √ N−1 tan −1 √ N−1. (3.38) For a large system,N 1, we obtain T ' π 2ε √ N (3.39) which results in a complexity of the same order √ N as Grover’s algorithm. 3.4.2 Adiabatic search on 2-dimensional manifold We now extend the analysis by considering 2-d parametrizations. The problem is to find the optimal curves on a 2-d manifold. For simplicity, the 2-dimensional manifoldM 2 with coordinates(x,y) we pick is H(x,y) =x(1−|ψ 0 ihψ 0 |)+y(1−|mihm|). (3.40) 90 Note that the selection of Hamiltonian set is not unique and should be subject to various physical constraints. For the manifold Eq. (3.40), (1,0) is the starting point, which corresponds to initial Hamiltonian H i with ground state |ψ 0 i, while the end point is (0,1), which gives the problem Hamiltonian whose ground state is our desired marked state |mi. Then we shall find the geodesic which connects these two points on this manifold. It turns out that theM 1 , the 1-dim Hamiltonian set specified by(3.25), just happens to be the subspace of M 2 , the 2-dim Hamiltonian set we choose in (2.16). Therefore, the geodesic denoted as RC (or 1-d geodesic) on that 1-dim manifoldM 1 is also a path on the 2-dim manifoldM 2 , and as a matter of fact, RC is a straight line on M 2 with coordinate(x = 1−s,y =s). The energy gap in this 2-d version can be derived easily, Δ(x,y) 2 =x 2 +y 2 −2xy(1− 2 N ). (3.41) Hence, the components of adiabatic-compatible metric tensor read, g 11 (x,y) = 1 Δ 4 Tr[ ∂H ∂x ∂H ∂x ] (3.42) = 1 Δ 4 Tr[(1−|ψ 0 ihψ 0 |)(1−|ψ 0 ihψ 0 |)] (3.43) = N−1 Δ 4 , (3.44) g 22 (x,y) = 1 Δ 4 Tr[ ∂H ∂y ∂H ∂y ] (3.45) = 1 Δ 4 Tr[1−|mihm|] (3.46) = N−1 Δ 4 =g 11 (x,y), (3.47) 91 and g 12 (x,y) = 1 Δ 4 Tr[ ∂H ∂x ∂H ∂y ] (3.48) = 1 Δ 4 Tr[(1−|ψ 0 ihψ 0 |)(1−|mihm|)] (3.49) = 1 N (N−1) 2 Δ 4 . (3.50) Accordingly, the metric onM 2 are g ij = 1 Δ 4   N−1 (N−1) 2 N (N−1) 2 N N−1   (3.51) which is inversely proportional toΔ 4 . In two dimensions, the covariant Riemann curvature tensor R ijkl has only one independent component, which can be taken as R 1212 and the other components are expressed in terms ofR 1212 by R ijkl = (g ik g jl −g il g jk ) R 1212 g (3.52) whereg is the determinant ofg ij . Plugging the metric (3.51) into (3.18) yields R 1212 =− 4N 2 (N(2N−5)+3)((N−2)x 2 −2Nyx+(N−2)y 2 (N(x−y) 2 +4xy) 4 . The form of geodesic equations in this case are omitted due to its complicated form. Denote this 2-d optimal curve on this manifoldM 2 as 2-d geodesic (QAB). Fig. (3.1) depicts the RC and 2-d geodesic over the curvature R 1212 surface for 3 qubits case, namely, for the problem of searching the unknown object in N = 2 3 unsorted items. 92 Figure 3.1: ComponentR 1212 (x 1 ,x 2 ) of the curvature tensor for n = 3. The curves on the curvature surface show the critical line (vanishing gap as n → ∞), the RC interpolation, and the 2-d geodesic (QAB). R 1212 is the only independent component of the curvature tensor in this case [R ijkl = 2R 1212 (g ik g jl −g il g jk )/detg]. é é é é é é é í í í í í í í 1 2 3 4 5 6 7 1 100 10 4 10 6 10 8 n Ave@R 1212 D 0 1 0 R 1212 s 10 5 Figure 3.2: Average curvature component R 1212 vs n. Red diamonds or solid line (blue circles or dashed line) represent the 2- d QAB (RC) path. Inset: instantaneous cur- vature forn = 4. Results for other values of n are qualitatively similar. Clearly, the 2-d geodesic follows a path with lower curvature than RC curve. This is confirmed in Fig. 3.2, for different values ofn. 3.4.3 Performance comparison between 1-d and 2-d geodesics An naive efficiency comparison The conventional adiabatic theorem which uses ”global” adiabatic condition states that given a fixed << 1, provided (3.2), the fidelity between the desired ground state |E p = E 0 (T)i which encodes the solution of a problem and the real evolution state |ψ(T)i is F(T) =|hE p |ψ(T)i| 2 ≥ 1− 2 . (3.53) 93 î î î î î î î 1 2 3 4 5 6 7 0 5 10 15 number of qubits speedup Figure 3.3: The speedup as a function of the number of qubits,n. Thereby, one can view as the error between |E p i and |ψ(T)i. Note that from the expression of adiabatic velocity (3.6), larger values of allows faster adiabatic speed v ad and smaller total evolution time T , but leads to smaller fidelity. For a given error , the success is judged by the shortest total evolution timeT required to reach a state which is close to the solution state. However, this comparison could not be accurate since for local adiabatic condition, the actual relation betweenF(T) and is not clear. But it’s still worthy to compare the time taken by RC (1-d geodesic) with 2-d geodesic for a given. Provided a fixed, the speed up is defined as 100(T[~ x RC (s)]−T[~ x 2d−g (s)]) T[~ x RC (s)] . (3.54) The speedup vs the finite numbern of qubits which associatedN = 2 n unsorted objects is shown in Fig. (3.3). It is clear that the geodesic on 2-d manifoldM 2 does outperform the RC curve (1-d geodesic). However, the outperformance shrinks as the size of the system gets bigger. We can conjecture that for a large system , the complexity of our approach based on 2-d manifold is also √ N, the same as results in [45]. 94 This result is not surprising because the Hamiltonian families we chose for 1-d and 2-d manifolds are both constructed by the same Hamiltonian basis, H i and H p . The only difference is their coefficient freedoms. Adding another Hamiltonian which is independent of H i and H p could be useful. From [17], Farhi et al. have shown that the addition of H 0 can change the performance of the algorithm from unsuccessful to successful. More rigorous performance comparison The more rigorous way to compare the performance of RC(1-d geodesic) with 2-d geodesic more rigorously is to measure how long they take to accomplish the same task within an actual error , δ(T)≡ q 1−|hE p |ψ(T)i| 2 . (3.55) The procedure we use to get δ(T), which shows how error δ varies with the entire evolution timeT , is as follows. For a given, first solve geodesic (QAB) equation to get time-optimal curve~ x QAB (s) in terms of the normalized length parameter s. Then the corresponding evolution timeT for this path can be immediately obtained by calculating T[~ x QAB (s)] in Eq. (3.8). Next, solve the real evolution state of the system|Ψ(s)i by the Schrodinger equation, i d|Ψ(s)i ds ds dt =i d|Ψ(t)i ds v ad =H[~ x QAB (s)]|Ψ(s)i (3.56) with v ad = ds dt = Δ(~ x(s)) 2 k ˙ H(~ x(s)k F (3.57) 95 0 100 200 300 400 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 T Δ geodesic Figure 3.4: Final-time errorδ(T) for the RC and 2-d geodesic paths for the Grover search problem, forn = 3 qubits. 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Squares (cyan) indicate where the 2-d geodesic path outperforms the RC path (δ geodesic ≤ δ RC ); circles (red) correspond to the opposite case. Oscillations are due to δ(T) itself being highly oscillatory. and the initial condition |Ψ(0)i = 1 √ N P N i=1 |ii. Note that |Ψ(s = 1)i = |Ψ(t = T)i. Thereupon, by evaluating (3.55), we getδ for a given total timeT . By repeating the above steps to calculateδ for many different total timesT , we shall obtain a good approximation ofδ(T). Then we are able to compare the errorδ(T) for the RC interpolation with the error obtained from 2-d interpolation. For the3 qubits case, in Fig. (3.4),δ(T) for the 1-d RC curve is denoted by the yellow line,δ(T) of the 2-d geodesic is the red line, and the blue envelope is∝ 1 T . Oscillations occurs fromδ(T) itself being highly oscillatory. From the Fig. (3.4), roughly speaking, at a given evolution timeT , the error of RC path is larger than the error of the geodesic on our 2-d manifoldM 2 . For the case withn = 6 qubits,δ RC −δ 2dgeodesic ≥ 0 marked as cyan dots in Fig. (3.5) indicates that at a given timeT , the error of the RC path is greater than the error 96 of the 2-d geodesic. In other words, the 2-d geodesic outperforms the RC trajectory. Red dots correspond to the opposite case, δ RC −δ 2dgeodesic ≤ 0. For other values of n, results are qualitatively similar. However, the advantage of the optimal interpolation diminishes whenn qubits grows. In summary, the 2-d QAB results in a smaller error for most values of T. In other words, for most values of the error δ, the optimal 2-d interpolation requires a smaller evolution time than the one required for the RC curve. 3.4.4 Adiabatic search on 4-dimensional manifold In this section, we shall demonstrate how to find QAB by accessing the entire Hamil- tonian space for the same search problem. For simplicity, we consider the problem of searching one marked object out of2 unsorted objects. In other words, the Hamiltonian we consider for this problem is1 qubit system. For 1 qubit case, all the Hamiltonian are 2 by 2 matrices and can be expressed in terms of4 independent pauli matrices, H(~ x = (x 0 ,x 1 ,x 2 ,x 3 )) = 1 √ 2 (x 0 +x 1 σ 1 +x 2 σ 2 +x 3 σ 3 ) (3.58) = 1 √ 2   x 0 +x 3 x 1 −ix 2 x 1 +ix 2 x 0 −x 3 .   (3.59) Their energy eigenvalues are E 0,1 (x) = 1 2 [TrH∓ p (TrH) 2 −4detH] = 1 √ 2 [x 0 ∓ q x 2 1 +x 2 2 +x 2 3 )] (3.60) with the gap Δ(x) = q 2(x 2 1 +x 2 2 +x 2 3 ). (3.61) 97 The metric on this 4-d manifoldM 4 , which includes all possible Hamiltonians for 1 qubit, reads g ij (x) = 1 4(x 2 1 +x 2 2 +x 2 3 ) 2 δ ij (3.62) and the corresponding curvature scalar is R =−48(x 2 1 +x 2 2 +x 2 3 ). (3.63) The explicit form of geodesic equations are ¨ x 0 − 4 (x 2 1 +x 2 2 +x 2 3 ) ( 3 X j=1 x j ˙ x j )˙ x 0 = 0, (3.64) and ¨ x i − 4 (x 2 1 +x 2 2 +x 2 3 ) ( 3 X j=1 x j ˙ x j )˙ x i + 2x i (x 2 1 +x 2 2 +x 2 3 ) ( 4 X j=1 ˙ x j2 ) = 0 (3.65) forx i withi = 1,2,3. The initial Hamiltonian for 1 qubit Grover’s search problem, 1−|ψihψ|, in matrix form, reads H i = 1 2   1 −1 −1 1   . (3.66) Accordingly, by identifying (3.66) with (3.58), we obtain the corresponding coordinate (x 0 ,x 1 ,x 2 ,x 3 ) ofH i on 4-d manifoldM 4 , x 0 (0) = 1 √ 2 ,x 1 (0) =− 1 √ 2 ,x 2 (0) = 0 =x 3 (0). (3.67) 98 0 100 200 300 400 0.000 0.005 0.010 0.015 0.020 0.025 T Δ geodesic Figure 3.6: Final-time error δ(T) for the RC, 2-d geodesic, and 4-d geodesic paths, denoted as yellow, orange, and red lines, respectively, for then = 1 qubit Grover search problem. The envelopes of all the lines are proportional to 1 T . The final Hamiltonian ,1−|mihm|, in matrix form is H(1) =   0 0 0 1   with its corresponding coordinate on 4-d manifoldM 4 , x 0 (1) = 1 √ 2 ,x 1 (1) =x 2 (1) = 0,x 3 (1) =− 1 √ 2 . (3.68) Accordingly, the time-optimal curve ~ x QAB (s) for 1 qubit Grover’s search problem is obtained by solving geodesic equation (3.64) and (3.65) with the above boundary con- ditions (3.67) and (3.68). Fig. (3.6) shows δ(T) for 4-d, 2-d and RC geodesic paths indicated by red, orange, and yellow lines, respectively. For a given errorδ, 4-d geodesic requires the shortest time to perform the task as expected. These results provide a rather striking demonstration of the power of our formalism, as due to its highly optimized nature, the Grover example is one where hardly any improvement was to be expected. 99 3.5 Conclusion A time-optimal, differential-geometric framework for AQC is presented. The power of this new framework was illustrated via the searching in an unstructured database as an example. We have shown that the performance of an adiabatic algorithm can be improved by increasing the dimension of the control parameter space. Our approach is general and can in principle be used to optimize any adiabatic quantum algorithm for which the gap is known or can be estimated. 100 Chapter 4 Conclusions We started by laying down a mathematical formulation for analysis of dynamical decou- pling pulse sequences. Then the main part of this thesis was devoted to Nested Uhrig Dynamical Decoupling (NUDD), which is a highly efficient decoupling scheme that utilizes the decoupling characteristics of UDD (an optimal DD scheme for suppression of general dephasing on a single qubit) by multi-layer nesting. We provided a rigorous analysis of the decoupling order for each error type and overall performance for NUDD in the setting of generalized control pulses. An explicit formula for the decoupling order of each error type was given, which elucidates the relationship between the error suppression characteristics of each error type and sequence order for each nested UDD layer. The analysis conveyed that the error suppression capabilities of NUDD are clearly dependent on the parity and relative magnitudes of all sequence orders. Therefore, a NUDD scheme can be designed accordingly such that optimal efficiency of NUDD is achieved. We also discovered that the error cancellation mechanism relates to the under- lying group structure of the error types. We hope that a more efficient DD technique with realistic pulses for multi-qubit or multilevel general decoherence models can be stimu- lated by the better understanding of the efficient noise suppression mechanisms behind NUDD. Finally, we formulated a general time-optimal strategy for Adiabatic Quantum Com- putation model (AQC). Quantum Adiabatic Brachistochrone (QAB), the time-optimal algorithm obtained by this strategy, was shown to have better algorithmic performance than the conventional linear interpolation one. In addition, we found that AQC can be 101 recast naturally into a differential-geometric framework. Specifically, a given family of physically controllable Hamiltonians can be equipped with a Riemannian structure, along with a metric induced by the adiabatic evolution. Therefore, the problem of find- ing QAB (the time-optimal path) for a given computational problem is equivalent to finding the geodesic (the shortest path on the Riemannian manifold). This geometriza- tion gives new insights into AQC. Most importantly, numerous existing powerful tools of differential geometry can be applied to the field of AQC. In this thesis, we have, hopefully, provided helpful steps for realizing a scalable quantum computer. 102 References [1] Wenxian Zhang E. E. Haller J. W. Ager V . V . Dobrovitski S. A. Lyon A. M. Tyryshkin, Zhi-Hui Wang. Dynamical decoupling in the presence of realistic pulse errors. eprint arXiv:1011.1903, 2010. [2] Dorit Aharonov, Wim van Dam, Julia Kempe, Zeph Landau, Seth Lloyd, and Oded Regev. Adiabatic quantum computation is equivalent to standard quantum compu- tation. SIAM J. Comput., 37:166, 2007. [3] Panos Aliferis, Daniel Gottesman, and John Preskill. Quantum accuracy threshold for concatenated distance-3 codes. Quantum Info. Comput., 6(2):97–165, March 2006. [4] Ido Almog1, Yoav Sagi, Goren Gordon, Guy Bensky, Gershon Kurizki, and Nir Davidson. Direct measurement of the systemvenvironment coupling as a tool for understanding decoherence and dynamical decoupling. J. Phys. B: At. Mol. Opt. Phys, 44:154006, 2011. [5] Gonzalo A. ´ Alvarez, Ashok Ajoy, Xinhua Peng, and Dieter Suter. Performance comparison of dynamical decoupling sequences for a qubit in a rapidly fluctuating spin bath. Phys. Rev. A, 82:042306, Oct 2010. [6] Masashi Ban. Photon-echo technique for reducing the decoherence of a quantum bit. Journal of Modern Optics, 45:2315 – 2325, 1998. [7] C. Barthel, J. Medford, C. M. Marcus, M. P. Hanson, and A. C. Gossard. Interlaced dynamical decoupling and coherent operation of a singlet-triplet qubit. Phys. Rev. Lett., 105:266808, Dec 2010. [8] Michael J. Biercuk, Hermann Uys, Aaron P. VanDevender, Nobuyasu Shiga, Wayne M. Itano, and John J. Bollinger. Experimental uhrig dynamical decoupling using trapped ions. Phys. Rev. A, 79:062324, Jun 2009. [9] Michael J. Biercuk, Hermann Uys, Aaron P. VanDevender, Nobuyasu Shiga, Wayne M. Itano, and John J. Bollinger. Optimized dynamical decoupling in a model quantum memory. Nature, 458:996–1000, 2009. 103 [10] S. Blanes, F. Casas, J.A. Oteo, and J. Ros. The magnus expansion and some of its applications. Phys. Rep., 470:151, 2009. [11] H.-P. Breuer and F. Petruccione. The Theory of Open Quantum Systems. Oxford University Press, Cambridge, England, 2000. [12] Łukasz Cywi´ nski, Roman M. Lutchyn, Cody P. Nave, and S. Das Sarma. How to enhance dephasing time in superconducting qubits. Phys. Rev. B, 77:174509, May 2008. [13] Deepak Dhar, L. K. Grover, and S. M. Roy. Preserving quantum states using inverting pulses: A super-zeno effect. Phys. Rev. Lett., 96:100405, Mar 2006. [14] David P. DiVincenzo and Peter W. Shor. Fault-tolerant error correction with effi- cient quantum codes. Phys. Rev. Lett., 77:3260–3263, Oct 1996. [15] Jiangfeng Du, Xing Rong, Nan Zhao, Ya Wang, Jiahui Yang, and R. B. Liu. Preserving electron spin coherence in solids by optimal dynamical decoupling. Nature, 461:1265, 2009. [16] L.-M Duan and G. Guo. Suppressing environmental noise in quantum computation through pulse control. Phys. Lett. A, 261:139, 1999. [17] Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. Quantum adiabatic evolution algorithms with different paths. 2002. eprint arXiv:quant-ph/0208135v1. [18] Edward Farhi, Jeffrey Goldstone, Sam Gutmann, Joshua Lapan, Andrew Lund- gren, , and Daniel Preda. A quantum adiabatic evolution algorithm applied to random instances of an np-complete problem. Science, 292:472, 2001. [19] Edward Farhi, Jeffrey Goldstone, Sam Gutmann, and Michael Sipser. Quantum computation by adiabatic evolution. 2000. eprint arXiv: quant-ph/0001106v1. [20] R.P. Feynman. Quantum mechanical computers. Found. Phys., 16:507, 1986. [21] D. Gottesman. An introduction to quantum error correction and fault-tolerant quantum computation. In S.J. Lomonaco, editor, Quantum Information Science and Its Contributions to Mathematics, volume 68 of Proceedings of Symposia in Applied Mathematics, page 13. Amer. Math. Soc., Rhode Island, 2010. eprint arXiv:0904.2557. [22] L. K. Grover. Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett, 79:4709, 1997. [23] U. Haeberlen. High Resolution NMR in Solids. Advances in Magnetic Resonance Series, Supplement 1. Academic Press, New York, 1976. 104 [24] E. L. Hahn. Spin echoes. Phys. Rev., 80:580–594, Nov 1950. [25] Elizabeth R. Jenista, Ashley M. Stokes, Rosa Tamara Branca, and Warren S. War- ren. Optimized, unequal pulse spacing in multiple echo sequences improves refo- cusing in magnetic resonance. J. Chem. Phys, 31:204510, 2009. [26] Liang Jiang and Adilet Imambekov. Universal dynamical decoupling of multiqubit states from environment. Phys. Rev. A, 84:060302, Dec 2011. [27] K. Khodjasteh and D. A. Lidar. Fault-tolerant quantum dynamical decoupling. Phys. Rev. Lett., 95:180501, Oct 2005. [28] Kaveh Khodjasteh and Daniel A. Lidar. Performance of deterministic dynamical decoupling schemes: Concatenated and periodic pulse sequences. Phys. Rev. A, 75:062310, Jun 2007. [29] Emanuel Knill and Raymond Laflamme. Theory of quantum error-correcting codes. Phys. Rev. A, 55:900–911, Feb 1997. [30] Emanuel Knill, Raymond Laflamme, and Wojciech H. Zurek. Resilient quantum computation. Science, 279:342, 1998. [31] Wan-Jung Kuo and D.A. Lidar. Quadratic dynamical decoupling: Universality proof and error analysis. Phys. Rev. A, 84:042329, Oct 2011. [32] T. D. Ladd, F. Jelezko, R. Laflamme, Y . Nakamura, C. Monroe, and J. L. O/’Brien. Quantum computers. Nature, 464:45, 2010. [33] B. Lee, W. M. Witzel, and S. Das Sarma. Universal pulse sequence to mini- mize spin dephasing in the central spin decoherence problem. Phys. Rev. Lett., 100:160505, Apr 2008. [34] A. Messiah. Quantum Mechanics. Dover Publication, New York, 1999. [35] A. Mizel, D. A. Lidar, and M. Mitchell. Adiabatic quantum computation is equiv- alent to standard quantum computation. Phys. Rev. Lett, 99:070502, 2007. [36] Musawwadah Mukhtar, Wee Tee Soh, Thuan Beng Saw, and Jiangbin Gong. Pro- tecting unknown two-qubit entangled states by nesting uhrig’s dynamical decou- pling sequences. Phys. Rev. A, 82:052338, Nov 2010. [37] M. Nakahara. Geometry, Topology and Physics. Adam Hilger, New York, 1990. [38] M.A. Nielsen and I.L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, Oxford,, 2002. 105 [39] J.A. Oteo and J. Ros. From time-ordered products to magnus expansion. J. Math. Phys., 41:3268, 2000. [40] Stefano Pasini and G¨ otz S. Uhrig. Optimized dynamical decoupling for time- dependent hamiltonians. J. Phys. A: Math. Theor., 43:132001, 2010. [41] X. Peng, D. Suter, and D.A. Lidar. High fidelity quantum memory via dynamical decoupling: theory and experiment. J. Phys. B: At. Mol. Opt. Phys., 44:154003, 2011. [42] J. Preskill. Reliable quantum computers. Proc. R. Soc. London A, 454:385, 1998. [43] Gregory Quiroz and Daniel A. Lidar. Quadratic dynamical decoupling with nonuniform error suppression. Phys. Rev. A, 84:042328, Oct 2011. [44] A. T. Rezakhani, Wan-Jung Kuo, A. Hamma, D. A. Lidar, and P. Zanardi. Quantum adiabatic brachistochrone. Phys. Rev. Lett, 103:080502, 2009. [45] J. Roland and N. J. Cerf. Quantum search by local adiabatic evolution. Phys. Rev. A, 65:042308, 2002. [46] Peter W. Shor. Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A, 52:R2493–R2496, Oct 1995. [47] A. M. Steane. Error correcting codes in quantum theory. Phys. Rev. Lett., 77:793– 797, Jul 1996. [48] Barbara M. Terhal and Guido Burkard. Fault-tolerant quantum computation for local non-markovian noise. Phys. Rev. A, 71:012336, Jan 2005. [49] G¨ otz S. Uhrig. Exact results on dynamical decoupling by pulses in quantum infor- mation processes. New. J. Phys, 10:083024, 2008. [50] G¨ otz S. Uhrig. Concatenated control sequences based on optimized dynamic decoupling. Phys. Rev. Lett., 102:120502, Mar 2009. [51] G.S. Uhrig. Keeping a quantum bit alive by optimizedπ-pulse sequences. Phys. Rev. Lett., 98:100504, Mar 2007. [52] G.S. Uhrig and D.A. Lidar. Rigorous bounds for optimal dynamical decoupling. Phys. Rev. A, 82:012301, Jul 2010. [53] Lorenza Viola, Emanuel Knill, and Seth Lloyd. Dynamical decoupling of open quantum systems. Phys. Rev. Lett., 82:2417–2421, Mar 1999. [54] Lorenza Viola and Seth Lloyd. Dynamical suppression of decoherence in two-state quantum systems. Phys. Rev. A, 58:2733–2744, Oct 1998. 106 [55] Zhen-Yu Wang and Ren-Bao Liu. Protection of quantum systems by nested dynamical decoupling. Phys. Rev. A, 83:022306, Feb 2011. [56] Zhi-Hui Wang and V . V . Dobrovitski. Aperiodic dynamical decoupling sequences in presence of pulse errors. J. Phys. B: At. Mol. Opt. Phys, 44:154004, 2011. [57] Zhi-Hui Wang, Wenxian Zhang, A. M. Tyryshkin, S. A. Lyon, J. W. Ager, E. E. Haller, and V . V . Dobrovitski. Effect of pulse error accumulation on dynamical decoupling of the electron spins of phosphorus donors in silicon. Phys. Rev. B, 85:085206, Feb 2012. [58] Jacob R. West, Bryan H. Fong, and Daniel A. Lidar. Near-optimal dynamical decoupling of a qubit. Phys. Rev. Lett., 104:130501, Apr 2010. [59] Wen Yang and Ren-Bao Liu. Universality of uhrig dynamical decoupling for sup- pressing qubit pure dephasing and relaxation. Phys. Rev. Lett., 101:180403, Oct 2008. [60] Wen Yang, Zhen-Yu Wang, and Ren-Bao Liu. Preserving qubit coherence by dynamical decoupling. Front. Phys., 6:2, 2011. [61] Kevin C. Young and K. Birgitta Whaley. Qubits as spectrometers of dephasing noise. arXiv:1102.5115v1, 2011. [62] Paolo Zanardi. Symmetrizing evolutions. Phys. Lett. A, 258:77–82, 1999. [63] W.H. Zurek. Decoherence and the transition from quantum to classical. Physics Today, 44:36, 1991. 107

Abstract (if available)

Abstract Two topics in the field of quantum information processing, optimized dynamical error suppression and quantum algorithms, are considered here. ❧ The computational errors induced by the surrounding environment is one of the main obstacles in building a quantum computer. Engineering powerful techniques to combat errors in quantum devices is highly demanding. In the first part of this thesis, I focus on one quantum error correction technique, dynamical decoupling (DD), introduced in Chapter 1. Chapter 2 is dedicated to nested UDD (NUDD), a highly efficient decoupling scheme that utilizes the decoupling characteristics of UDD by multi-layer nesting. UDD (1-layer NUDD) is an optimal DD method for eliminating single-qubit general dephasing, and QDD (2-layer NUDD) is a near-optimal DD method for eliminating one qubit general decoherence. I present a rigorous analytical proof of the performance and universality of QDD/NUDD, and obtain an explicit formula for the decoupling order of each error type, which elucidates the relationship between the error type and characteristics of NUDD. From the explicit formula, a NUDD scheme can be designed accordingly such that optimal efficiency of NUDD is achieved. Moreover, the highly efficient error cancellation mechanism is revealed by the analysis. The proof of QDD has been published in [31], and the proof of NUDD will be submitted for publication shortly. ❧ Chapter 3 is devoted to the Adiabatic Quantum Computation (AQC). In this work (published in [44]), a general time-optimal strategy, which in principle can optimize any quantum adiabatic algorithm for which the gap is known or can be estimated, is formulated. In addition, I present a natural differential-geometric framework for AQC.

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Asset Metadata

Creator Kuo, Wan-Jung (author)

Core Title Towards optimized dynamical error control and algorithms for quantum information processing

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Physics

Publication Date 07/25/2012

Defense Date 06/18/2012

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

Tag adiabatic quantum computing,dynamical decoupling,OAI-PMH Harvest,quantum error correction,quantum information,quantum open systems

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Lidar, Daniel A. (committee chair), Daeppen, Werner (committee member), Dappen, Werner (committee member), Haas, Stephan W. (committee member), Jonckheere, Edmond A. (committee member), Takahashi, Susumu (committee member)

Creator Email astroskykuo@gmail.com,wanjungk@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-65763

Unique identifier UC11290381

Identifier usctheses-c3-65763 (legacy record id)

Legacy Identifier etd-KuoWanJung-998.pdf

Dmrecord 65763

Document Type Dissertation

Rights Kuo, Wan-Jung

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA