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Towards efficient fault-tolerant quantum computation

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Towards efficient fault-tolerant quantum computation

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Content TOWARDS EFFICIENT FAULT-TOLERANT QUANTUM COMPUTATION by Yi-Cong Zheng A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) August 2015 Copyright 2015 Yi-Cong Zheng To my parents Ming Zheng and Ying Xu and to my wife and son Yi-Zuo Chen and Andrew Zheng ii Acknowledgments The road to this Ph.D has been a long and tough journal, with numerous hard working and wonderful memory. I would not have completed it without many help and support from enormous people. First of all, I would like to thank my advisor, Todd Brun, for oering a great envi- ronment of studying and working. His deep knowledge in quantum information science, with broad scope, is indispensable and I am indebted that he took me as his student. He always gives valuable suggestion and encouragement on my research in our meeting, especially when I felt frustrated. After these years working with him, now I can safely say that Todd has set up an upper bound for how good an advisor can be. I am also greatly thankful to Daniel Lidar who has enormous impact on my research during the past few years. He inspired many of the studies in this thesis. I have learned a lot from his course \Quantum Error Correction" and \Open Quantum System". I can feel my research stepping up after these courses sharpening my knowledge and understanding of the eld. The discussion we have, either in person or through email, is a great help for my work. I am also grateful to Ben Reichardt, to whom I owe a signicant insight of quantum algorithm and topological quantum error correction. Ben is one of the smartest people I have ever met, and I learned so much from him. It was a great pleasure to have chance to discuss with him. I want to thank Anthony Levi and Armand R. Tanguay. Jr for introducing research opportunity studying quantum optics at the early days when I arrived at USC. iii Special thanks are due to Aiichrio Nakano and Michelle Povinelli, who not only served in my candidacy committee but also trained me well to obtain strong background of device physics and numerical analysis. I thank to David Hocker, Alireza Shafaei and Martin Suchara for our collaboration on project of error model analysis on dierent physical implementations in IARPA QCS program. I also thank Joshua Job helping me working on Titan supercomputer. I also thank everyone in the group, Ching-Yi Lai, Shengshi Pang, Kung-Chuan Hsu, Jos e Raul Gonzalez Alonso, Scout Kingery, Jan Florjanczyk, Shesha Raghunathan, Bilal Shaw, Martin Varbanov, and Christopher Cantwell. Ching-Yi is always being patient in discussing error correcting codes and fault-tolerant quantum computation with me. He set up a model to show me how research can be done. I have a wonderful time collaborating with Kung-Chuan on classical codes decoding. Shengshi taught me a lot of quantum metrology and information theory. Jos e helps me out so much from all aspects during my Ph.D program, especially with physics and L A T E X problems. I thanks all people whom I benetted from, both in research and daily life, including, but certainly not limited to: Hui-Khoon Ng, Jim Harrington, David Poulin, Austin Folwer, Stephen Jordan, Guanyu Zhu, Iman Marvian, Zhi-Hui Wang, Tameem Albash, Hao Feng, Jian-Ming Wen, Huo Chen, Chen-Xi Lin, Ningfeng Huang and Jing Ma. Thanks are due also to all members in Communication Science Institute (CSI), Elec- trical Engineering department, especially Anita Fung, Corine Wong, Gerrielyn Ramos, Susan Wiedem and Diane Demetras. Finally, I would thank my family: my parents, my wife and my son. Without my parents' long time support, this work would have been impossible. They ensured me the best education I can receive, gave me guide and condence, supported all my endeavors throughout my life. I thank my wife, Yizuo, for her unconditional love, her steady belief in me and her sel ess contribution to my family. I greatly thank my extremely cute son, Andrew, whose coming to this world brings endless excitation and forever happiness. iv Table of Contents Dedication ii Acknowledgments iii List of Tables viii List of Figures ix Abstract xvi I Preliminaries 1 Chapter 1: Introduction 2 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Quantum computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Quantum Stabilizer Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Fault-tolerant Quantum Computation (FTQC) . . . . . . . . . . . . . . . . 9 1.5 Holonomic Quantum Computation . . . . . . . . . . . . . . . . . . . . . . . 11 II Quantum Computation in Cold Atoms 15 Chapter 2: Quantum Computation in Clouds of Atoms 16 2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3 A Universal Set of Geometric Gates . . . . . . . . . . . . . . . . . . . . . . 21 2.3.1 One Qubit Phase Gate . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.2 Y-Rotation Gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3.3 Two Qubit Controlled Phase Gate . . . . . . . . . . . . . . . . . . . 33 2.4 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.5 Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.5.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.5.1.1 Spontaneous Emission . . . . . . . . . . . . . . . . . . . . . 45 2.5.1.2 Cavity Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 v 2.5.1.3 Trap Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.5.1.4 Stray Electric Fields . . . . . . . . . . . . . . . . . . . . . . 48 2.5.2 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.5.2.1 Phase Gate . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.5.2.2 Y-Rotation Gate . . . . . . . . . . . . . . . . . . . . . . . . 53 2.5.2.3 Controlled Phase Gate . . . . . . . . . . . . . . . . . . . . 55 2.6 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.7 Appendix: Spectrum of Rydberg States . . . . . . . . . . . . . . . . . . . . 58 III Ecient Fault-tolerant Quantum Computation: Part I 68 Chapter 3: Fault-tolerant Quantum Computation in Large Block Codes 69 3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.2 Universal FTQC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.3 Steane Syndrome Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.4 Logical Teleporataion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.5 Logical Cliord Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.6 Error Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.7 Estimate of the Logical Error Rate . . . . . . . . . . . . . . . . . . . . . . . 83 3.7.1 Decoding of Stabilizer Codes . . . . . . . . . . . . . . . . . . . . . . 83 3.7.2 Quantum BCH Codes and Golay Codes . . . . . . . . . . . . . . . . 89 3.7.3 [[15; 1; 3]] Reed-Muller Code . . . . . . . . . . . . . . . . . . . . . . . 91 3.8 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Chapter 4: Distillation of Ancilla States for Steane Syndrome Extraction 97 4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.2 Ancilla State Distillation using Classical Codes . . . . . . . . . . . . . . . . 100 IV Ecient Fault-tolerant Quantum Computation: Part II 107 Chapter 5: Fault-tolerant Holonomic Quantum Computation in Stabilizer Codes 108 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.2.1 HQC: A Geometrical Formalism . . . . . . . . . . . . . . . . . . . . 111 5.2.2 Stabilizer Codes and Fault-Tolerant Quantum Computation . . . 117 5.3 Fault-tolerant Holonomic Quantum Computation . . . . . . . . . . . . . . 121 5.3.1 Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.3.2 Fault-Tolerance of the Scheme . . . . . . . . . . . . . . . . . . . . . 128 5.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.4.1 3-qubit Repetition Code . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.4.2 The Steane Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.4.2.1 CNOT Gate . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.4.2.2 Lowering the Weight of the Hamiltonian . . . . . . . . . . 141 vi 5.5 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Chapter 6: Fault-tolerant Holonomic Quantum Computation in Surface Codes 145 6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6.2 Surface Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 6.3 Sketch of the Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 6.3.1 Adiabatic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 6.3.2 Error Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 6.3.3 Parallelism of Adiabatic Operation . . . . . . . . . . . . . . . . . . . 156 6.4 HQC in Surface Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.4.1 Creation of S+⟩ (S0⟩) State for X (Z)-cut Double Qubit . . . . . . 159 6.4.2 Enlarging the Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 6.4.2.1 Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 6.4.2.2 Error Propagation . . . . . . . . . . . . . . . . . . . . . . . 164 6.4.3 Moving Logical Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . 165 6.4.3.1 Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 6.4.3.2 Error Propagation and Fault Tolerance . . . . . . . . . . 168 6.4.4 Creation of S0⟩ (S+⟩) State for X (Z)-cut Double Qubit . . . . . . 172 6.4.5 Logical Z (X) Measurement for X (Z)-cut Double Qubit . . . . . 174 6.4.6 Holonomic Logical CNOT . . . . . . . . . . . . . . . . . . . . . . . . 174 6.4.7 Measurement of Z (X) Basis for Z (X)-cut Double Qubit . . . . 177 6.4.8 Ancilla Recycling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 6.4.9 State Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 6.4.10 State Distillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 6.4.11 Logical Phase and T Gates . . . . . . . . . . . . . . . . . . . . . . . 182 6.4.12 Hadamard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 6.5 Fault-tolerance of the Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 186 6.5.1 Error Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 6.5.2 Local Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 6.5.3 Adiabatic Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 6.6 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 6.7 Appendix: Proof of Lemma 1, 2, 3 . . . . . . . . . . . . . . . . . . . . . . 192 6.7.1 Lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 6.7.2 Lemma 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 6.7.3 Lemma 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 Chapter 7: Epilogue 198 Bibliography 200 vii List of Tables Tab. 1.1 Stabilizer generators of Steane's code. . . . . . . . . . . . . . . . . 11 Tab. 2.1 Decoherence mechanism for the atom chip system. Some of the typical values are based on [49] and [47]. . . . . . . . . . . . . . . 47 Tab. 2.2 The matrix element of operator V dip in atomic units. . . . . . . . 61 Tab. 6.1 The related transformation of stabilizer generators {S i } and log- ical operators {L i } of a Z-cut qubit in Fig. 6.4 is shown under gate operation {g i }. . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 Tab. 6.2 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 viii List of Figures Fig. 2.1 Schematic of trapped ensembles of atoms in magnetic atom traps inside plano-concave optical microcavities (horizontal). Each atomic cloud is a qubit, and forms a processing cell. Dierent qubits couple to each other mediated by the photonic modes of another Fabry-Perot cavity (vertical). . . . . . . . . . . . . . . . . . . . . . 18 Fig. 2.2 Storage of single photon qubit. Level scheme of a single 87 Rb atom is shown. The input qubit is encoded as S0⟩ if there is no photon in the input mode and S1⟩ if there is a photon in the input mode. Here, we use the hyperne levels of manifold 5S 1~2 : clock states Sg⟩ = SF = 1;m F = −1⟩ and Ss⟩ = SF = 2;m F = 1⟩; Sa − ⟩ = SF = 1;m F = 0⟩ and Sa + ⟩ = SF = 2;m F = 0⟩ are used as ancillary states. And a state of manifold 5P 1~2 serves as an intermediate state Se⟩ (SF = 1;m F = 0⟩). The storage process mapping single photon signals into collective excited atomic states of Ss⟩ can be accomplished by adiabatically turning on the classical laser eld (t) to a value much larger than the coupling constant g when the photon arrives. The reverse process can be used to read out the photon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Fig. 2.3 Left: Level structure and laser coupling diagram for a single atom. Right: The equivalent coupling diagram for an ensemble of atoms in second quantized representation. . . . . . . . . . . . . . . . . . . 22 Fig. 2.4 (a) The evolution of the relative phase between state S1⟩ L and state S0⟩ L for three dierent values of , to realize a ~16 gate. Note that after the cyclic evolution, we get an additional pure geometric phase, which ts the theoretical value quite well for larger than 200MHz. (b) The population change of state S1⟩ L during the process. We can see that only if is large enough to satisfy the adiabatic condition can we get high accuracy of the gate. Two short laser pulses are determined by (t)= 3 exp((t− 1) 2 ~0:15) and '(t)= 3 exp((t− 1:5917) 2 ~0:15) (the unit of t is s). 24 ix Fig. 2.5 The level diagram with the couplings to realize the R i y () gate. We apply an electric eld to the ensemble of atoms, and pick appropriate Stark eigenstates of rubidium to be the register states Sr⟩= Sr⟩ 70;0 , Sf⟩= Sr⟩ 72;0 and the intermediate Rydberg state Sm⟩= Sr⟩ 60;0 as shown the appendix. States Sg⟩, Sa + ⟩ and Ss⟩ are in the same manifold of the internal ground state. In the rst step, two conjugate laser beams p1 and p2 are used to adiabatically pump a single excitation from Sg⟩ to Sr⟩. In the second step, three laser beams 0 , 1 , a are used to adiabatically realize theR i y () gate. The excitation from Sg⟩ to Sr⟩ is then adiabatically reversed. 28 Fig. 2.6 The equivalent coupling diagram of the ensembles of atoms in cases of collective excitation for the case (a) system is initially in S0⟩ L and (b) system is initially in S1⟩ L . . . . . . . . . . . . . . . . . 29 Fig. 2.7 Adiabatic Pumping process. Note that at the end there is only a single excitation in the register state Sr⟩ when initially in S0⟩ L , and no excitation when initially in S1⟩ L . Here, we set rm = 400MHz and mm = 300MHz. Since we take the minimum energy shift as our simulation parameter, the actual performance should be better (i.e., the process could be nished in a shorter period of time). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Fig. 2.8 (a): The pulse shape of the control lasers of 0 , 1 and a respec- tively. (b)(c)(d): The evolution of the populations of Ss⟩, Se⟩, Sr⟩. We see that when is larger than 200MHz, the adiabatic con- dition is satised and the gate works with very high accuracy (close to the theoretical value) and the population of the short- lived state Se⟩ tends to zero. . . . . . . . . . . . . . . . . . . . . . . 33 Fig. 2.9 The schematic setup to realize the conditional phase gate, and the relevant atomic levels of the two clouds of atoms. . . . . . . . 34 Fig. 2.10 The evolution of the phase 3 of S10⟩ L . The pulse shapes of (t) and '(t) are also given. . . . . . . . . . . . . . . . . . . . . . . . . . 38 Fig. 2.11 The average number of photons in the cavity during gate opera- tion as a function of time and 1 for g 1 = 660MHz. (a) system initially prepared in S10⟩ L and (b) system initially prepared in S11⟩ L . 42 Fig. 2.12 The schematic setup to realize a projective measurement on a single qubit. Here, state Sd⟩ is a temporary state with a short lifetime. The transition between Sd⟩ and Ss⟩ is not allowed. The state Sd⟩ can be chosen for example, to be SF = 1;m F =−1⟩ of the manifold 5P 1~2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 x Fig. 2.13 Fidelity of a phase gate with 1 = ~8 as a function of for dierent values of not including stray elds. . . . . . . . . . . . 51 Fig. 2.14 Fidelity of a phase gate with 1 = ~8 for dierent values of with = 10Hz considering the stray eld eect. . . . . . . . . . . 52 Fig. 2.15 Fidelity of a phase gate with 2 =~4 for dierent values of , and for = 100Hz without considering the stray eld. . . . . . 52 Fig. 2.16 Fidelity of a phase gate with 2 = ~4 for dierent values of and considering the stray eld with = 1kHz and = 100Hz. . 53 Fig. 2.17 Fidelity of the controlled phase gate with 3 =~16 for dierent values of andg 1 without any other decoherence channels included. 55 Fig. 2.18 Fidelity of the controlled phase gate with 3 =~16 for dierent values of andg 1 including the stray eld eect, with = 100Hz and = 2:5kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Fig. 2.19 The energy level of rubidium around manifoldn= 15,m= 0. The circled states is our candidate states which have large energy shift. 66 Fig. 3.1 The architecture of our teleporation-based FTQC scheme. . . . 71 Fig. 3.2 The circuit for Steane syndrome extraction and its eective error model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Fig. 3.3 Diagram of the logical T gate using logical state teleportation. The red blocks represent joint measurements of logical qubits, and the blue one represents bitwise T or T applied to the processor block. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Fig. 3.4 (a) Hadamard. (b) CNOT from qubit i to j. The red blocks represent joint measurements of logical qubits. . . . . . . . . . . . 76 Fig. 3.5 (a) Phase gate. (b) SWAP gate. The red blocks represent joint measurements of logical qubits. . . . . . . . . . . . . . . . . . . . . 77 Fig. 3.6 The eect of measurement errors on the ancilla qubit can be eec- tively replaced by errors before and after the circuit. . . . . . . . 79 Fig. 3.7 The eect of gate errors (from (a) to (f)) on the rst CNOT, and ancilla preparation errors (from (g) to (l)). Both types of errors can be replaced by errors before and after the circuit. . . . . . . . 80 Fig. 3.8 Eective error model in a single time step of circuit . . . . . . . . 81 xi Fig. 3.9 The logical error rate of the memory blocks for the [[2047,23,77]] code (blue), [[2921,57,77]] code (red) and [[5865,143,105]] code (green) versus physical error rate p. The number of samples for each point is up to 4×10 8 . The dashed lines are from extrapolation of linear tting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Fig. 3.10 The logical error rate of the logical T gate performed on the concatenated [[15; 1; 3]] code of two levels (blue) and three levels (red) using up to 3× 10 7 samples for each point. The green point represents a numerical upper bound for three levels when the physical error rate is p = 7× 10 −4 . The dashed lines are from extrapolation of linear tting. . . . . . . . . . . . . . . . . . . . . 93 Fig. 4.1 The circuit for ancilla distillation by the classical [3; 1; 3] repe- tition code to distill ancilla free of X errors. The rst two S0⟩ L serve as check bits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Fig. 4.2 The circuit for ancilla distillation by the classical [3; 1; 3] repeti- tion code to distill ancilla free from X and Z errors. . . . . . . . 103 Fig. 4.3 The circuit for S + ⟩ ancilla distillation by the classical [3; 1; 3] repetition code to distill ancilla free of X errors. . . . . . . . . . . 104 Fig. 4.4 The circuit for S ij ⟩ L ancilla distillation by the classical [3; 1; 3] repetition code to distill X ancilla from X errors and Z ancilla from Z errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Fig. 5.1 Horizontal lift as a specied curve in S N;K (C) whose projection isP(t). The initial condition V (0) becomesV (T), which is gen- erally dierent from V (0). The dierence is the holonomy. . . . . 116 Fig. 5.2 A logical unitary quantum operation is realized as a series of quantum gates from a universal set of gates in the circuit model. 119 Fig. 5.3 The variation of the energy diagram at the beginning of the third and fourth step to break the degeneracy of space {s 1 = 1;s 2 =−1} and {s 1 =−1;s 2 = 1} . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 xii Fig. 6.1 A surface code based on an 8× 8 lattice with 113 physical qubits on the edges. This code contains 1 logical qubit and has distance d=L= 8, where d is the distance of the code. The four-body (or three-body) plaquette stabilizer generator (Z p ) and vertex stabi- lizer generator (X s ) are indicated as cyan and yellow plaquettes, respectively inside the lattice (or on the boundary). A particular choice of logical operators X L and Z L is shown. A number of qubits are aected by x (red dots) or z (purple dots) errors, leading to excited Z p operators (or m anyons) and X s operators (or e anyons). Measuring these operators yields the positions of the excited vertices and plaquettes but reveals no information about the actual physical errors which cause them. A minimum- weight matching error correction procedure applies x and z to the qubits marked by the larger red and purple circles. While the z errors are annihilated properly (up to a trivial loop of multipli- cation ofZ p operators), the red pair underneath is connected by a topologically non-trivial path across the surface. This introduces a logical error in the state to be protected. . . . . . . . . . . . . . 148 Fig. 6.2 Four-body plaquette operator Z p (a) and vertex operator X s (b) as stabilizer generators of surface code inside the lattice. The black dot in the center of the plaquettes are syndrome qubits used to do stabilizer measurement. . . . . . . . . . . . . . . . . . . 149 Fig. 6.3 Creation of S+⟩ for X-cut double qubit. System Hamiltonians before and after are shown in (a) and (b) respectively. Colored squares indicate that the correspondingX s (yellow) andZ p (cyan) stabilizer generators are turned on. . . . . . . . . . . . . . . . . . 160 Fig. 6.4 Enlarging a hole of an X-cut logical double qubit adiabatically. Colored squares indicate that the corresponding X s (yellow) and Z p (cyan) stabilizer generators are turned on. The yellow qubits in (b) and (c) indicate that x for that qubit is turned on in the Hamiltonian. Adiabatic evolution between (a) and (b) maps X L to X ′ L and Z L to Z ′ L . Similarly, adiabatic evolution between (b) and (c) maps X ′ L to X ′′ L and Z ′ L to Z ′′ L . . . . . . . . . . . . . . . . 161 Fig. 6.5 Error propagation during an adiabatic process to enlarge a hole of an X-cut logical qubit. Colored squares indicate that the corre- spondingX s , x (yellow) andZ p , z (cyan) operators are turned on. The purple circle around a qubit indicates a z error occurs on that qubit. (a) A z error occurs on qubit 1. (b) Eective errors after the adiabatic process. (c) An additional z error occurs on qubit 4. (d) Eective errors after the adiabatic procedure to enlarge the hole will cause a logical error after decoding. . . . . . 164 xiii Fig. 6.6 Adiabatic process for moving a Z-cut logical qubit hole horizon- tally right. Colored squares indicate that the corresponding X s , x (yellow) and Z p , z (cyan) operator are turned on. Logical operators of the qubit are X L and Z L in (a). An adiabatic pro- cess between (a) and (b) mapsX L toX ′ L andZ L toZ ′ L . Similarly, an adiabatic process between (c) and (d) mapsX ′ L toX ′′ L andZ ′ L to Z ′′ L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 Fig. 6.7 Error propagation during adiabatic process to move a hole of an X-cut logical double qubit horizontally right. Colored squares and qubits indicate that the corresponding X s , x (yellow) and Z p , z (cyan) operators are turned on. The purple circle around qubit indicates a z error occurs on that qubit and a red one indicates a x error occurs. (a) z errors occur on qubit 1 and 2. (b) Eective errors caused by z 1 and z 2 after adiabatic process. (c) x errors occur on qubit 1 and 2. (d) Eective errors caused by x 1 and x 2 after adiabatic process. . . . . . . . . . . . . . . . . 169 Fig. 6.8 Scheme to fault-tolerantly detect errors occurring on the bound- ary. (a) Before the movement, an error occurs on the boundary. (b) After expanding the holed~8 units rightward, the error propa- gates to a strip of errors. We measure all qubits in the dashed box and determine if the corresponding error happened on the bound- ary based on the majority vote of the measurement outcomes of each row of qubits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 Fig. 6.9 Creation of S0⟩ state for X-cut double qubit. (a) Create a S+ X DL ⟩ with two holes attached to each other. Measure z 1 , z 2 and z 3 and do majority vote to determine whether S0 X DL ⟩ or S1 X DL ⟩ is prepared. (b) Move two holes apart to increase error correction ability of z errors of the logical qubit. Note that both z and x errors on qubits between two holes during the adiabatic movement have no uncorrectable eect on logical state S0 X DL ⟩ or S1 X DL ⟩ and can be left for future error correction. . . . . . . . . . . . . . . . . 172 Fig. 6.10 Adiabatic braiding process of aZ-cut hole (dark blue) around an X-cut hole (orange) . The operator X L 1 has been stretched to multiply a loop of x operators which is equivalent to X L 2 up to multiplication by X s stabilizer generators (yellow) inside the loop, while X L 2 remains the same under the transformation. . . 175 xiv Fig. 6.11 Adiabatic braiding process of a Z-cut hole (dark blue) around an X-cut hole (orange). The operator Z L 2 has been stretched to form a strip of z operators, which is equivalent to Z L 1 up to multiplication by Z p stabilizer generators (cyan) inside the strip, while Z L 1 remains the same under the transformation. . . . . . . 176 Fig. 6.12 State injection for a X-cut qubit. Colored squares indicate that the corresponding X s (yellow) and Z p (cyan) operator is turned on. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Fig. 6.13 Circuits for logical SY ⟩ distillation from imperfect S ̃ Y ⟩ states. . . 181 Fig. 6.14 Circuits for logical SA⟩ distillation from imperfect S ̃ A⟩ states. . . . 182 Fig. 6.15 A dislocation in the geometry of the Hamiltonian produced by shifting the stabilizer generators along a line between two twists. The stabilizer generators corresponding to two dierent parallelo- grams (yellow/cyan and cyan/yellow) and a pentagon (dark gray) are shown on the right side. A pair of anticommuting strings of Pauli operators L 1 (solid red) and L 2 (dashed blue) that com- mute with all stabilizer generators forms the logical operators of the extra qubitF attached to the pair of twists. . . . . . . . . . . 184 Fig. 6.16 Adiabatically moving a pairs of holes of aZ-cut qubit (dark blue holes) across a twist on the surface to get a logical Hadamard gate. This process will transform a Z-cut qubit to an X-cut qubit (orange holes). . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Fig. 6.17 Logical error rate perm time steps for various values ofm andd. The dashed lines are forcJ = 8 and solid lines forcJ = 12. The blue (top), green (second top), red (third) and yellow (bottom) lines are for d= 7, d= 11, d= 15 and d= 19, respectively. . . . . 188 xv Abstract The threshold theorem indicates that if errors are all local and their rates are below a certain threshold, it is possible to implement large scale quantum computation with arbi- trarily small error based on active quantum error correction (QEC). However, most fault- tolerant quantum computation (FTQC) schemes require enormous overhead to achieve this rate by increasing either the concatenation levels for concatenated codes or the distance (hence the size) for topological codes. As a result, a logical qubit is usually encoded in more than thousands of physical qubits, even when the error rate is below the threshold. In this thesis, we try to reduce the resource overhead of FTQC. We rst study a quantum computation scheme using clouds of atoms trapped on atomic chips. It suggests that this scheme is robust to noise to some extent and has potential to be scalable. Secondly, we propose a FTQC scheme of encoding many logical qubits into large code blocks. The error performance of such scheme is numerically studied. Prepa- ration of various ancilla states necessary for computation is also explored. Thirdly, we explore the idea of introducing energy gap to suppress thermal noise on each qubit. We show that fault-tolerant holonomic quantum computation (HQC) can be implemented in stabilizer codes with the existence of such energy protection. Especially, we studied fault-tolerant HQC in surface codes in detail. This scheme opens the possibility for self-correcting quantum computation in 2D lattice. xvi Part I Preliminaries 1 Chapter 1 Introduction This Chapter aims to give a brief introduction to the background of quantum computing, quantum error correction, fault-tolerance, and holonomic quantum computation, which are particularly relevant to the rest of thesis. 1.1 Background The eld of quantum computation has grown rapidly during the last few decades. It was Feynman [46] who noted that simulating quantum system on a classical computer seems to require exponential growth of the calculation with the size of the system, while it may possibly be eciently simulated by another well controlled quantum system. This system designed to take full advantage of quantum mechanics, might oer much more computational power. The discovery of Shor's algorithm [124] for nding the factors of prime numbers eectively strengthened this intuition and greatly inspired research in this eld. Since then, more quantum algorithms have been found [63, 88, 6] and a great deal of theoretical and experimental research has been done to build real quantum computers. Roughly speaking, a quantum computer is composed of quantum bits (qubits), which encode information in a quantum state. Intuitively, it takes advantage of interference and entanglement: qubits are prepared in superposition of many dierent states during the computation. In carefully designed algorithms, the paths to the state encoding correct answers interfere constructively while paths to the wrong ones interfere destructively and somewhat cancel out. 2 In contrast to the great robustness of classical computers, the processing of quantum information seems to be much more challenging. A quantum state is far more fragile than its classical counterpart. In real world, quantum states suer from decoherence and quantum gates are faulty. Although constructing and controlling quantum hardware is currently a huge experimental challenge, we nevertheless hope that large-scale quantum computation is, at least in principle, possible achieve in spite of a variety noise processes. The theory of quantum error-correcting code (QECCs) [125, 132] and fault-tolerant quantum computation (FTQC) have shown that, quantum computation with arbitrary large-scale can be achieved if errors are not strongly correlated and their rate is small enough to fall below a threshold [123, 2, 60, 73, 40, 75, 84]. This is achieved by encoding the state of computation non-locally in quantum error-correcting codes. The decoherence can not access the state protected by error-correcting code without aecting a large number of qubits. However, it is not an easy task: existing schemes for FTQC require too much overhead|too much redundancy to protect qubits [145]. Currently, all FTQC schemes require relatively low rates of error to satisfy threshold theorems. Most of them also require a extremely large overhead. As a result, a logical qubit is encoded in thousands or tens thousands of physical qubits, if not more [75, 52, 145, 80]. Even in principle large-scale quantum computer is possible, in practice, realizing a large-scale FTQC is a formidable challenge. In this thesis, we examine the method to control error in real physical system and try to cut down the resource redundancy with new FTQC architectures. In Chapter 2, we explore a new physical architecture of quantum computation using clouds of atoms trapped on atomic chip as qubits. This scheme has potential to be scalable. Meanwhile, it is robust to certain types of noises to some extent. Next, we explore two paths in reducing the resource overhead of FTQC. In Part III, we follow the rst path. We try to use more ecient codes with high performance. In Chapter 3, we propose a FTQC scheme of encoding many logical qubits into large code blocks to reduce the number 3 of physical qubits protecting each logical qubit. The error performance of such scheme is numerically studied, which suggests that the overhead can be reduced by an order compared to the current best known FTQC schemes. Preparation of various ancilla states necessary for computation is also explored in Chapter 4. In Part IV, we follow the second path. We try to suppress the error rate on the individual qubits. This will reduce the frequency of error correction (which itself can introduce errors) and the size of the code used in computation. In Chapter 5, we explore the idea of introducing an energy gap to suppress thermal noise on each qubit. We show that fault-tolerant holonomic quantum computation can be implemented in stabilizer codes with the existence of such constant energy protection. In particular, surface codes turns out to be perfect candidate for such scheme. In Chapter 6, we study fault-tolerant HQC in surface codes in detail. 1.2 Quantum computation Quantum systems are described by quantum states. The state space of an isolated quantum system is a Hilbert space. In particular, we consider a 2-dimensional Hilbert space H with an (ordered) orthonormal basis {S0⟩;S1⟩}. A quantum state Sv⟩ in H is called a qubit and is denoted by a unit vector Sv⟩=S0⟩+S1⟩ or Sv⟩= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ; where and are complex numbers and SS 2 + SS 2 = 1. The concept of qubits S0⟩ and S1⟩ is similar to the classical bits 0 and 1, except that a qubit can be in a superposition state of S0⟩ and S1⟩. The Pauli matrices I = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 0 0 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ; X = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 1 1 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ; Z = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 0 0 −1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ; Y = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 −i i 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ =iXZ 4 form a basis of the space of the linear operators on the single-qubit state space H. The Pauli matrices X, Y , Z have eigenvalues ±1 and they anticommute with each other. They satisfy conditions: X 2 = Y 2 = Z 2 = I, X = X, Y = Y , and Z = Z, where A denotes the Hermitian conjugate of an operator A. The operations of X and Z on the basis vectors are XS0⟩= S1⟩; XS1⟩= S0⟩; ZS0⟩= S0⟩; ZS1⟩=−S1⟩: (1.1) Thus X is often referred to as a \bit- ip" operator, and Z is often referred to as a \phase- ip" operator. Y is the combined \bit-phase ip" operator. Beyond the single qubit, the state space of n qubits is the tensor product of n single qubit state space and is denoted by H⊗⋯⊗H =H ⊗n . If Sv i ⟩ is the state of the i-th qubit, the joint state is Sv 1 ⟩⊗⋯⊗Sv n ⟩ or just Sv 1 ⟩ ⋯ Sv n ⟩ for simplicity. The evolution of a closed quantum system is described by a unitary transformation. After a unitary quantum operatorU applies to a state S ⟩, the state becomesUS ⟩. When a state of quantum system is not known for sure, it can be denoted as an ensemble of pure states S i ⟩ with probability p i . More often, it is described by a density operator =∑ i p i S i ⟩⟨ i S, where {p i ;S i ⟩} the evolution of the density operator under a unitary operator U is =Q i p i S i ⟩⟨ i S U →UU =Q i p i US i ⟩⟨ i SU : (1.2) Any quantum computation can be implemented by a universal set of unitary oper- ation, called elementary gates. For example, the Cliord operators Hadamard H and phase gate S, H = 1 √ 2 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 1 1 −1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ; S = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 0 0 i ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (1.3) 5 controlled NOT(CNOT) gate, ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (1.4) together with a gate outside the Cliord group such as the T gate, T = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 0 0 e −i~4 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ; (1.5) form a universal set. If a system is not isolated, it will interaction with the environment and its evolution is no longer unitary in general. A powerful mathematical tool to describe such evolution is the quantum operation (evolution map). A quantum operationE can be described by the operation-sum representation: E()=Q i E i E i (1.6) with a set of operators {E i } such that Q i E i E i =I: (1.7) After the computation process, we need to read out the computation results. Quan- tum measurement can be described by a collection of measurement operators. In partic- ular we consider the projective measurement described by a (Hermitian) observable M with a spectral decomposition M =Q m mP m (1.8) 6 where P m is the projector onto the eigenspace of M with eigenvalue m. Note that P m are orthogonal to each other and∑ m P m =I. The state S ⟩ will collapse to S m ⟩= P m S ⟩ » ⟨ SP m S ⟩ (1.9) after we read out the result m. 1.3 Quantum Stabilizer Codes Error correcting codes, classical or quantum, are methods of storing information redun- dantly. Even though part of the information has been corrupted or lost, the stored information can still recovered with high probability. A quantum error-correcting code is formally dened as a subspace C of some larger Hilbert space. A necessary and sucient condition for a set of errors {E i } to be cor- rectable is [100, 84]: PE i E j P = ij P; ∀i;j; (1.10) for some Hermitian matrix . Here P is the projector onto C. Since any linear combi- nation of {E i } is also correctable, we dene E = Span{E i } (1.11) to be a correctable error set for codeC. The codes we are interested in are the stabilizer codes [60]. We brie y review the formalism of stabilizer codes. Let G n be the Pauli group acting on n qubits: G n = {eM 1 ⊗⋯⊗M n ∶M j ∈ {I;X;Y;Z};e∈ {±1;±i}} (1.12) 7 be the n-fold Pauli group. An Abelian subgroup S of G n is called a stabilizer group if −I ∉S. The stabilizer group denes a subspace of the n-qubit Hilbert space by C = {S ⟩∶SS ⟩= S ⟩ for all S ∈S}: (1.13) This C is called the code space. C is nonzero since −I ∉ S. A state in C is called a codeword. This subspace is the simultaneous +1 eigenspace of the stabilizer group. If the subspace has dimension 2 k (k logical qubits), the stabilizer code can be specied by n−k commuting stabilizer generators, which are elements of G n . The group S can be represented by these stabilizer generators: S = ⟨{S j }⟩. All stabilizer codes can be characterized by three parameters [[n;k;d]], where d is the minimum distance of the code, which is equal to the minimum weight of all nontrivial elements in the normalizer group ofS in G n . Lets=s 1 ⋯s n−k ∈Z n−k 2 be a binary (n−k)-tuple. Since the stabilizer generators have eigenvalues ±1, we deneH s S to be a subspace ofH ⊗n such that for Sv⟩∈H s S , S j Sv⟩= (−1) s j Sv⟩ for j = 1;⋯;n−k: (1.14) That is,H s S lies in the (−1) s j -eigenspace of g j for j = 1;⋯;n−k and the projector onto H s S is P s = n−k M j=1 I+(−1) s j S j 2 : (1.15) Note that the P s 's are orthogonal to each other and ∑ s P s =I. Thus the n-qubit state spaceH ⊗n can be decomposed as H ⊗n = ? s=s 1 ⋯s n−k ∈Z n−k 2 H s S : (1.16) 8 Obviously, C corresponds to H s S with all s j = 1. For an error operator E ∈ G n and S ⟩∈C(S), we have ES ⟩∈H s S for some binary (n−k)-tuples=s 1 ⋯s n−k , wheres j = 1 ifE anti-commutes withg j , and s j = 0, otherwise. The (n−k)-tuple binary number s is called the error syndrome of E and can be obtained by measuring the stabilizer generators S 1 ;⋯;S n−k . If the error syndrome is nonzero, we can detect the error and hopefully, correct it. Otherwise, either no error or an undetectable error (logical error) happened. 1.4 Fault-tolerant Quantum Computation (FTQC) Quantum fault-tolerance is concerned not only with protection for storing information but also processing information in an encoded form. Theory has shown that if errors of each type are suciently local, and their rates are small enough to fall below a thresh- old, it is possible to carry out quantum computations of arbitrary size with arbitrarily small error, by so-called fault-tolerant methods using quantum error-correcting codes (QECCs) [40, 55, 84]. FTQC methods are the backbone to build quantum computer against decoherence. These schemes protect quantum information from decoherence by encoding it in quantum error-correcting codes [60, 125, 132], and replacing the quantum gates of an ideal quantum circuit by fault-tolerant circuits that implement the desired quantum gates on logical states without ever decoding the quantum information. The basic idea of fault-tolerant quantum computation is to compute on the encoded quantum states and perform quantum error correction periodically (or right after each logical operation) to prevent accumulation of errors. Decoding is never required during the computation. The goal of fault-tolerant circuit design is to prevent errors at physical level to build up at the logical level over a course of computation, even while gate operation and measurement used to do diagnose and correct errors themselves are faulty. 9 Since we are dealing with the encoded state of a certain quantum error-correcting code, the encoded version of those gates in universal gate set are required. Each encoded gate is composed of a series of non-perfect physical gates. Moreover, errors may prop- agate between qubits through multiple-qubit gates. For example, we know that errors will propagate through the CNOT gate: X 1 CNOT 1→2 Ð→ X 1 X 2 : Z 2 CNOT 1→2 Ð→ Z 1 Z 2 : A single qubit error could evolve to multiple qubit errors through a series of CNOT gates. Apparently, a quantum circuit with CNOT gates involving dierent qubits would be bad, since it will propagate errors to uncorrectable ones, which we try to avoid. Instead, we hope the error rate of the output of any encoded gate is small enough so that the followed error-correction routines can recover the quantum states. To analyze the error performance, we have to dene the notion of fault-tolerance rst. Denition 1. A procedure is fault-tolerant if it has the property that if only one compo- nent (or more generally, a small number of components) in the procedure fails, the errors produced by this failure are not transformed into an uncorrectable error by the procedure, before error correction is applied. A text book example is Steane's [[7; 1; 3]] code. It is the rst quantum code that has been shown to have an error threshold, which is of the magnitude 10 −5 . This quantum code comes from the classical [7; 4; 3] Hamming code and has a set of stabilizer generators listed in Tab. 1.1. The logical operators are X =XXXXXXX; Z =ZZZZZZZ: (1.17) 10 1 2 3 4 5 6 7 S 1 X I I X X X I S 2 I X I X I X X S 3 I I X I X X X S 4 Z I I Z Z Z I S 5 I Z I Z I Z Z S 6 I I Z I Z Z Z Table 1.1: Stabilizer generators of Steane's code. The logical Hadamard gate is H =HHHHHHH (1.18) so that H X H = Z and vice versa. The logical Phase gate is S = (ZS) ⊗7 : (1.19) The bitwise implementation of these gates are in a fashion called transversality. It can be observed that a transversal gate is fault-tolerant: an error on a qubit does not propagate to other qubits. The encoded CNOT gate is also in a bitwise fashion. We have shown that the encoded Cliord operators of the Steane code can be fault- tolerantly implemented. However, the implementation of a fault-tolerant encoded T gate involves the preparation of certain special ancilla states, called magic state, which is usually very dicult to prepare and certain technique called magic state distillation is needed [24]. 1.5 Holonomic Quantum Computation Besides active \software" error correction methods mentioned in previous sections, there have been proposals to deal with noise by designing the \hardware" to provide inherent robustness. One such method is holonomic quantum computing (HQC) [153]|an all- geometric, adiabatic method of computation that uses a non-Abelian generalization of 11 the Berry phase [15]. This approach has been shown to be robust against various types of errors during the process [130, 120, 129] and could in principle be done in several dierent systems [41, 155, 118]. Both closed-loop and open-loop HQCs can be compatible with active QEC [102, 103, 8, 9, 156], and can achieve fault-tolerant QC. This technique will be used a lot in this thesis. In this section, we give a brief introduction of HQC. Consider a Hamiltonian family {H } on anN−dimensional Hilbert space. The point , parametrizing the Hamiltonian, is an element of a manifold M called the control manifold, and the local coordinates of are denoted by i (1 ≤ i ≤ dimM). Assume there are only a xed number of eigenvalues {" k ()} and suppose the nth eigenvalue " n () is K n -fold degenerate for any . The degenerate subspace at is denoted by H n (). The orthonormal basis vectors ofH n () are denoted by {S n ;⟩}, satisfying H S n ;⟩=" n ()S n ;⟩; (1.20) and ⟨ n ;S m ;⟩= nm : (1.21) Assume the parameter is changed adiabatically, which means that (" n ((t))−" n ′((t)))T ≫ 1 (1.22) is satised for n≠n ′ during 0≤t≤T ). Suppose the initial state at t= 0 is an eigenstate S n (0)⟩= S n ;(0)⟩. The Schr odinger equation is i d dt S n (t)⟩=H((t))S n (t)⟩; (1.23) whose solution will have the form S n (t)⟩= Kn Q =1 S n ;(t)⟩U (t): (1.24) 12 where we have used the adiabatic approximation from Eq. (1.22). Substituting Eq. (1.24) into Eq. (1.23), one nds that U satises _ U (t)=−i" n ((t))U (t)−Q ⟨ n ;(t)S d dt S n ;(t)⟩U (t): (1.25) The solution can be expressed as U(t)= exp−i S t 0 " n ((s))ds×T exp− S t 0 A n ()d; (1.26) whereT is the time-ordering operator and A n (t)= ⟨ n ;(t)S d dt S n ;(t)⟩ (1.27) is the Wilczek-Zee (WZ) connection [149]. Dene the connection A n i; (t)= ⟨ n ;(t)S @ @ i S n ;(t)⟩; (1.28) through which U(t) can be expressed as U(t)= exp−i S t 0 " n ((s))ds×P exp− S (t) (0) Q i A n i d i ; (1.29) whereP is the path-ordering operator. Eq. (1.29) is a general description of both open loop and closed loop adiabatic state evolution. In particular, suppose the path (t) is a loop inM such that (0)=(T)= 0 (closed loop). Then after transporting through , states are transformed to S n (T)⟩= Kn Q =1 S n (0)⟩U (T): (1.30) The unitary matrix =P exp− c Q i A n i d i (1.31) 13 is called the holonomy associated with the loop (t). is a purely geometric object, and is independent of the parametrization of the path. A geometric formulation of the holonomic problem, which gives an alternative description as shown in Refs. [137, 138], is also given in Sec. 5.2.1, which is more useful in certain circumstances. 14 Part II Quantum Computation in Cold Atoms 15 Chapter 2 Quantum Computation in Clouds of Atoms 2.1 Preliminaries If we cannot build quantum computer in real world, it is just a curiosity of math. In recent decades, numerous candidates for physical implementation of quantum informa- tion processors have been proposed [32]. Because of long coherence times and exceptional control ability, quantum optical and atomic systems such as trapped ions [34, 131], neu- tral atoms [25, 67], cavity QED [108, 154], and superconducting qubits [10, 71, 35] have taken a leading role in implementing quantum logic. The photon is a remarkably robust candidate for a qubit, and easy to transport long distances; however, as we know, they interact only weakly with each other, which makes the realization of quantum gates based on photons dicult. For instance, in linear optical quantum computation(LOQC) [76], extra measurements are required. On the other hand, ensembles of trapped atoms or molecules may serve as convenient and robust quantum memories for photons. They can act as an interface with ying qubits, store and retrieve single photons, for example by electromagnetically induced transparency (EIT) [89, 47], which allows to keep coherence of quantum states up to several hundreds of millisecond [122]. One might therefore hope to use controlled interactions in the ensemble of atoms, once the ying qubits are stored, to realize universal quantum logic gates in a determinis- tic and scalable way. Several promising schemes for such operations based on combining EIT with Rydberg atoms were proposed in Ref. [90, 58], exciting atoms with lasers to high-lying Rydberg states and exploiting the strong long-range dipole-dipole interaction 16 between Rydberg states. Recent theoretical proposals include a single-step, high-delity entanglement of a mesoscopic number of atoms [98], and quantum simulation of both coherent dynamics and dissipative evolution processes for many body systems [148]. Remarkably, the building blocks of all such proposals have been demonstrated exper- imentally by several groups [54, 144]. In Ref. [109], an eective strong dipole-dipole blockade eect is achieved by coupling to microwave coplanar waveguide resonators to realize quantum logic on ensembles of molecules. However, dynamical control of such ensemble is dicult due to the inhomogeneous coupling between laser and atoms and it suers from control noise as well. In addition, proposals for geometric manipulation of ions [41] and neutral atoms [116] have been made for their natural robustness against certain control errors. However, single ion and atom in these proposals are very dicult to transfer the photon signal to the matter states and it needs a extremely good cavity to realize a high delity two-qubit quantum gate due to small coupling eciency. To overcome problems mentioned above, here, we put forward an alternative, scalable approach to realize universal quantum gates based solely on laser-controlled geometric manipulations of neutral atomic clouds trapped on the surface of atom chips. The main result was presented in Ref. [155]. 2.2 Model Atom chips supply a good platform on which precise control and manipulation of neutral atoms can be performed [49, 51]. We will show that an atom chip with integrated Fabry- Perot (FP) microcavity can realize a universal set of all-optical universal quantum gates for atomic ensemble qubits. Consider several plano-concave Fabry-Perot (FP) microcavities resonators [143] inte- grated on the atom chip, which are parallel to each other. Another large FP cavity is integrated so that its mode is perpendicular to all the plano-concave cavities. The dia- gram is shown in Fig. 2.1. Plano-concave resonator consists of an isotropically etched dip 17 Figure 2.1: Schematic of trapped ensembles of atoms in magnetic atom traps inside plano-concave optical microcavities (horizontal). Each atomic cloud is a qubit, and forms a processing cell. Dierent qubits couple to each other mediated by the photonic modes of another Fabry-Perot cavity (vertical). in a silicon surface, and the cleaved tip of a single-mode bre [142] serves as the input- output channel. Atomic ensembles are placed in the region of highest eld strength of the cavity modes, leading to higher values of the couplingg, and aQ value over 10 6 . The perpendicular FP cavity modes serve as a data bus to couple between dierent qubits. The atomic clouds must be conned in traps inside the cavities. We can introduce magnetic elds produced by current-carrying wires on the surface and coupled to the magnetic dipole of the atoms. The small scale of the structure produces strong magnetic eld gradients, which make tight traps for magnetic atoms. Atoms in a weak-eld-seeking state will be attracted and held in this region. The hyperne states SF = 2;m F = 1⟩ and SF = 1;m F =−1⟩ of 5S 1~2 of 87 Rb are ideal weak-eld-seeking states that can be trapped by a static magnetic potential [49]. In addition, these two states have opposite Land e factors, so that they experience identical magnetic potentials. Thus, the decoherence 18 g s e g photon a+ 1/2 5S 1/2 5P 1 F = 1 F = 2 F = F m -1 -1 -1 1 1 1 0 0 0 -2 a- 2 Figure 2.2: Storage of single photon qubit. Level scheme of a single 87 Rb atom is shown. The input qubit is encoded as S0⟩ if there is no photon in the input mode and S1⟩ if there is a photon in the input mode. Here, we use the hyperne levels of manifold 5S 1~2 : clock states Sg⟩ = SF = 1;m F = −1⟩ and Ss⟩ = SF = 2;m F = 1⟩; Sa − ⟩ = SF = 1;m F = 0⟩ and Sa + ⟩= SF = 2;m F = 0⟩ are used as ancillary states. And a state of manifold 5P 1~2 serves as an intermediate state Se⟩ (SF = 1;m F = 0⟩). The storage process mapping single photon signals into collective excited atomic states of Ss⟩ can be accomplished by adiabatically turning on the classical laser eld (t) to a value much larger than the coupling constant g when the photon arrives. The reverse process can be used to read out the photon. of an arbitrary superposition of these two states due to current intensity uctuations is strongly inhibited [31]. Coherent oscillations between these states have been observed with decoherence times as long as c = 2:8± 1:6s [141]. So it is natural to consider these two states of 87 Rb for quantum information processing. The state of the 87 Rb atom cloud can be initialized by inputting a single photon qubit to the plano-concave cavity through the bre, as shown in Fig. 2.2. The state S0⟩ ph corresponds to no photons in the input channel, and the state S1⟩ ph corresponds to one photon. An arbitrary qubit state can be represented as S0⟩ ph +S1⟩ ph . In general, the state of n input channels is an entangled state in a Hilbert space of dimension 2 n . An ensemble of N identical multi-state atoms is trapped in each cell. Using well- developed techniques, all atoms can be initially prepared and trapped in a specic sub- level (hyperne level Sg i ⟩, i= 1; 2; 3;:::;N) of the internal atomic ground space manifold 19 [90]. Relevant states of each atom include the other clock state Ss i ⟩, two ancillary states Sa −i ⟩;Sa +i ⟩ (which are chosen not be aected by the magnetic potential), and an interme- diate state Se i ⟩. (As mentioned earlier, Ss i ⟩ has a long coherence time.) We also assume the atomic density is not too high, so that interaction between atoms can be safely neglected when atoms are in ground state manifold. Atoms are manipulated by illumi- nating the entire ensemble, to excite all atoms with equal probability, so only symmetric collective states are involved in this process [90]. For loading quantum information into the ensembles, we consider only three states: Se i ⟩, Ss i ⟩ and Sg i ⟩, where Se i ⟩ and Sg i ⟩ are coupled to the cavity mode of the plano-concave cavity, while Se i ⟩ and Ss i ⟩ are coupled by a classical eld. Initially, all the atoms are in their ground states: Sg⟩= Sg N ⟩= Sg 1 ⟩:::Sg N ⟩: We dene operators ^ S = 1 √ N Q i Sg i ⟩⟨s i S; ^ E = 1 √ N Q i Sg i ⟩⟨e i S; ^ A ± = 1 √ N Q i Sg i ⟩⟨a ±i S: The storage state with excitation n is Ss n ⟩≡ Sg N−n ;s n ⟩= ¾ (N −n)!N n N!n! ( ^ S ) n Sg⟩: For the case when n= 1, this is Ss⟩= 1 √ N Ss 1 ;g 2 ;g 3 :::g N ⟩+Sg 1 ;s 2 ;g 3 :::g N ⟩+⋯+Sg 1 ;g 2 ;g 3 :::s N ⟩: (2.1) 20 The commutation relation of ^ S and ^ S is [ ^ S; ^ S ]= 1− 2n N : For a suciently small number of excitations (n ≪ N), these two operators can be approximated by bosonic annihilation and creation operators of quasiparticles corre- sponding to Ss⟩. Operators ^ E and ^ E , and ^ A ± and ^ A ± have the same properties. As shown in Ref. [48] and Ref. [91], when the photon arrives in the cavity, one can adiabatically turn on the classical eld coupling the states Se⟩ and Ss⟩, until the Rabi frequency is much larger than the cavity-atom coupling constant g, so that the state of photon is stored in the ensemble of atoms in the cavity in the form Sn⟩ ph → Ss n ⟩, where Sn⟩ ph represents the n-photon Fock state. This process is reversible: one can retrieve the photon from the ensemble of atoms. So, an arbitrary input state S0⟩ ph +S1⟩ ph can be mapped into the state Sg⟩+Ss⟩ with high delity [48]. We denote S0⟩ L = Sg⟩ and S1⟩ L = Ss⟩ for the logical representation of a qubit in each cloud of atoms. 2.3 A Universal Set of Geometric Gates We now discuss how to geometrically implement quantum computation (holonomic quan- tum computation) in our system. As we learned in Sec. 1.5, the rst step is to construct a Hamiltonian that has an eigenspace with eigenvalue 0 to avoid dynamic phase during the cyclic evolution. This eigenspace can either be nondegenerate, which would introduce simple Abelian phase factors (Berry phases), or degenerate, which could cause general non-Abelian operations [149]. Although CNOT, Hadamard, Phase andT forms a univer- sal gate set, they are not easy to implement geometrically in this system. On the other hand, it is well known that single-qubit operations, together with a nontrivial two-bit gate, make another universal set of quantum logic gates for quantum computation [87]. By constructing an appropriate set of looped paths in the parameter space we can obtain an arbitrary unitary transformation in the computational space. 21 We choose for our universal set of gates R (i) z ( 1 )= expi 1 S1 i ⟩ LL ⟨1 i S; R (i) y ( 2 )= expi 2 y i ; U (jk) ( 3 )= expi 3 S1 j 0 k ⟩ LL ⟨1 j 0 k S: (2.2) The states S0 i ⟩ L and S1 i ⟩ L are the computational basis states for qubit i, and y i is the Pauli operator in they direction for theith qubit. It is well know that if we can implement these gates with arbitrary 1 ; 2 ; 3 , we can implement all unitary transformations [87]. The three operators correspond to a rotation in the z direction, a rotation in the y direction, and a conditional phase rotation when qubits j and k are in the state S10⟩ L . In this section, we show how to holonomically realize these three gates. 2.3.1 One Qubit Phase Gate e N,0,0,0 gesa = 0 L g s + a N-1,0,0,1 gesa N-1,1,0,0 gesa 1 L = N-1,0,1,0 gesa a Ω a Ω 1 Ω 1 Ω Figure 2.3: Left: Level structure and laser coupling diagram for a single atom. Right: The equivalent coupling diagram for an ensemble of atoms in second quantized repre- sentation. We rst consider how to perform a simple one qubit phase gate: S1⟩ L →e i 1 S1⟩ L . To simplify the model, we only consider the energy levels we are interested in. The total 22 pulse sequence and resonant coupling diagram is shown in Fig. 2.3. The Hamiltonian of of this system, in the interaction picture and rotating wave approximation, is H 1 (t)=Q i i 1 (t) (i) se + H.c+Q i i a (t) (i) a+e + H.c; (2.3) where the (i) jk = Sj i ⟩⟨k i S are transition operators from state Sk⟩ to Sj⟩ for atom i, and ̵ h= 1. Here, the two lasers are assumed to be phase matched, and their Rabi frequencies are real numbers. For simplicity, assuming the laser beams illuminate each atom with same intensity, we can rewrite the Hamiltonian in the second quantized representation: H 1 (t)= 1 (t) ^ E ^ S+ ∗ 1 (t) ^ S ^ E+ a (t) ^ E ^ A + + ∗ a (t) ^ A + ^ E; (2.4) and set 1 = sin and a =− cose i' . The parameters and' are functions of time, and the absolute magnitude is a constant that must be large enough to satisfy the adiabatic condition. ^ E; ^ S; ^ A + are bosonic operators dened in last section corresponding to the single atom states Se⟩;Ss⟩;Sa + ⟩, respectively. We can regard the system as three coupled harmonic oscillators. We are only inter- ested in singly excited states; the Hamiltonian discussed above is closed in that basis, which for simplicity we represent as {Se⟩= SN−1; 1; 0; 0⟩ gesa , S1⟩ L = Ss⟩= SN−1; 0; 1; 0⟩ gesa , Sa + ⟩ = SN − 1; 0; 0; 1⟩ gesa }. The coupling diagram of the ensemble of atoms is equivalent to a simple three-state coupling diagram for a single atom, as shown in Fig. 2.3. Representing the Hamiltonian in matrix form in this basis, we have H 1 (t)= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 sin − cose i' sin 0 0 − cose −i' 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ : (2.5) 23 0.0 0.5 1.0 1.5 2.0 2.5 3.0 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 Geometric Phase t( s) (a) =8MHz =10MHz =200MHz Theoretical 0.0 0.5 1.0 1.5 2.0 2.5 3.0 -0.25 0.00 0.25 0.50 0.75 1.00 =8MHz =10MHz =200MHz population of |1 Figure 2.4: (a) The evolution of the relative phase between state S1⟩ L and state S0⟩ L for three dierent values of , to realize a ~16 gate. Note that after the cyclic evolution, we get an additional pure geometric phase, which ts the theoretical value quite well for larger than 200MHz. (b) The population change of state S1⟩ L during the process. We can see that only if is large enough to satisfy the adiabatic condition can we get high accuracy of the gate. Two short laser pulses are determined by(t)= 3 exp((t−1) 2 ~0:15) and '(t)= 3 exp((t− 1:5917) 2 ~0:15) (the unit of t is s). The zero-energy eigenspace of the Hamiltonian is non-degenerate. The dark state is SD(t)⟩= cosS1⟩ L + sine −i' Sa + ⟩: (2.6) Note that the Hamiltonian is decoupled from the state S0⟩ L = SN; 0; 0; 0⟩ grsa . Assume the initial state is S (−∞)⟩ = SD(−∞)⟩ = S1⟩ L (which means (−∞) = 0 at the very beginning). Also, assume that initially ' = 0. Now, turn on two laser beams, 24 using the standard formula to calculate the geometric phase, where the parameters make a cyclic evolution with the starting and ending point = 0;'= 0: 1 =i c dR⟨D(t)S∇ R SD(t)⟩; (2.7) where R(t)= ((t);'(t)). We have ⟨D(t)S∇ R SD(t)⟩=−i sin 2 () ^ ' and 1 = c sin 2 ()d': According to Green's theorem, this geometric phase is exactly the enclosed solid angle: 1 = U @S sindd'= U @S d : A pulse sequence consisting of two stimulated Raman adiabatic passage (STIRAP) pulses with relatively large width (to guarantee that the process is adiabatic) can be used for this purpose. A numerical simulation of resonance coupling to realize a ~16 gate ( 1 = ~8) is shown in Fig. 2.4. Since in the laboratory the Rabi frequency can be as high as 200 MHz, the adiabatic process could be nished in several microseconds, which is much shorter than the decoherence time of the hyperne ground states. We can exchange the population of S1⟩ L and Sa + ⟩ back and forth without losing any information, and we can get a geometric phase with high accuracy if is large enough. In the previous method of using ensembles of atoms or molecules to realize quan- tum information processing, a major shortcoming is that it is very hard to uniformly illuminate all the atoms, thus limiting the accuracy of the gate operation. But in our proposal, each atoms interacts individually with 1 and a , and the geometric phase is independent of the absolute value of . If the spatial distribution of the electric elds 25 of the two laser beams match each other, say ̃ 1 (r) = ̃ E 1 (r)d es = ̃ E a (r)d ea+ = ̃ a (r) for every point in space r, the eect of non-uniform illumination can be naturally eliminated. This is an advantage of adiabatic control. 2.3.2 Y-Rotation Gate Now we show how to achieve the gate U =e i 2 y . This gate is much more complicated to realize than the one qubit phase gate. It is dicult to couple the state S0⟩ L to S1⟩ L , since directly coupling them by applying a laser beam to the ensemble of atoms would result in unwanted higher collective excitations. In Ref. [90], a scheme taking advantage of the dipole-dipole interaction of Rydberg states between atoms was used to realize this gate. Here we propose a geometric gate by following three steps: 1. Adiabatically pump the ground state S0⟩ L to a highly excited Rydberg state Sr⟩, using the dipole-dipole interaction to assure that only a single collective excitation is achieved. 2. Adiabatically control the coupling between the single excitation states Sr⟩, Ss⟩, and Sa + ⟩ to geometrically realize the gate. 3. Reverse step 1 to transfer Sr⟩ back to S0⟩ L . To carry out step 1, consider Rydberg states of a hydrogen atom within a manifold of xed principle quantum numbern with degeneracyn 2 . This degeneracy can be removed by applying a constant electric eld E along a certain axis (linear Stark eect), such as the z axis. For electric elds below the Ingris-Teller limit, the mixing of adjacent n manifolds can be ignored, and energy levels are split according to E nqm = 3nqea B E~2 with parabolic and magnetic quantum numbers q and m, respectively. Here, q can take values n− 1−SmS;n− 3−SmS;:::;−(n− 1−SmS), e is the electron charge, and a B the Bohr radius. These states have dipole moments of = 3nqea B e z ~2. 26 Consider two atomsk andl seperated by a distance R. The dipole-dipole interaction between them is V kl dip (R)= 1 4 0 ^ k ⋅ ^ l SRS 3 − 3 (^ k ⋅ R)(^ l ⋅ R) SRS 5 ; (2.8) where ^ k and ^ l are dipole moment operators for atoms k and l. Suppose the electric eld is suciently large that the energy splitting between two adjacent Stark states is much larger than the dipole-dipole interaction. For two atoms initialized in Stark eigenstates, the diagonal terms of V kl dip (R) provide an energy shift, while the nondiagonal terms couple adjacent m manifolds with each other, (m;m)→ (m± 1;m∓ 1). The o-diagonal transition can cause decoherence, and is suppressed by an appropriate choice of initial Stark eigenstate [67]. (For hydrogen, the state is Sr⟩ n = Sn;q =n− 1;m = 0⟩.) For a xed distance R = Re z , the dipole-dipole interaction of two atoms k and l in the states Sr k ⟩ n 1 and Sr l ⟩ n 2 is u kl n 1 n 2 (R)= n 1 ⟨r k S n 2 ⟨r l SV kl dip (R)Sr k ⟩ n 1 Sr l ⟩ n 2 =− 9a 2 B e 2 R 3 8 0 [n 1 n 2 (n 1 − 1)(n 2 − 1)]: (2.9) For n 1 and n 2 suciently large, u(R) ∝n 2 1 n 2 2 . For alkali atoms such as rubidium, the situation is more complicated, but this kind of analysis still works. In the appendix of this Chapter, we discuss how to pick a state which has characteristics similar to Sr⟩ n for rubidium. We will make use of this energy shift later. Consider the diagram shown in Fig. 2.5. For step 1, there should be a single excitation in state Sr⟩ after pumping if the initial state is S0⟩ L , and no excitation in state Sr⟩ if the initial state is S1⟩ L . To achieve this goal, we introduce two other Rydberg states Sm⟩ and Sf⟩, and adiabatically transfer the amplitude of S1⟩ L to Sf⟩ by coupling Ss⟩ and Sf⟩ to some intermediate state. Note that this pumping process can also be realized adiabatically. This process would also be used when we do measurement, more details will be given in the measurement section. 27 Figure 2.5: The level diagram with the couplings to realize the R i y () gate. We apply an electric eld to the ensemble of atoms, and pick appropriate Stark eigenstates of rubidium to be the register states Sr⟩= Sr⟩ 70;0 , Sf⟩= Sr⟩ 72;0 and the intermediate Rydberg state Sm⟩= Sr⟩ 60;0 as shown the appendix. States Sg⟩, Sa + ⟩ and Ss⟩ are in the same manifold of the internal ground state. In the rst step, two conjugate laser beams p1 and p2 are used to adiabatically pump a single excitation from Sg⟩ to Sr⟩. In the second step, three laser beams 0 , 1 , a are used to adiabatically realize the R i y () gate. The excitation from Sg⟩ to Sr⟩ is then adiabatically reversed. The Hamiltonian of the cloud of atoms as described in the left side of Fig. 2.5 in the rotating wave approximation is: H p (t)=Q i ( p1 (t) (i) gm + H.c)+Q i ( p2 (t) (i) mr + H.c) +Q k>l u kl mm (R kl )Sm k ⟩Sm l ⟩⟨m k S⟨m l S+u kl rr (R kl )Sr k ⟩Sr l ⟩⟨r k S⟨r l S) +u kl ff (R kl )Sf k ⟩Sf l ⟩⟨f k S⟨f l S+u kl rm (R kl )Sr k ⟩Sm l ⟩⟨r k S⟨m l S +u kl fm (R kl )Sf k ⟩Sm l ⟩⟨f k S⟨m l S+u kl fr (R kl )Sf k ⟩Sr l ⟩⟨f k S⟨r l S: (2.10) The two-body interaction term here is due to the energy shift of the states. Just as in the case of the phase gate, we represent the Hamiltonian in second- quantized form. Again, assuming the laser elds are uniformly coupled to each atom, and using the 28 standard second quantization procedure to deal with the two-body interaction, we apply the minimum energy shift for all pairs of atoms (as the worst case) and obtain H e p (t)≈ p1 (t) √ N( ^ M + ^ M)+ p2 (t)( ^ R ^ M + ^ R ^ M) +u mm ^ M ^ M ^ M ^ M+u rr ^ R ^ R ^ R ^ R +u ff ^ F ^ F ^ F ^ F+u rm ^ R ^ R ^ M ^ M +u fm ^ F ^ F ^ M ^ M+u fr ^ F ^ F ^ R ^ R; (2.11) where u rr , u mm , u ff , u rm , u fm and u fr are chosen to be the minimum energy shifts from the dipole-dipole interaction. Figure 2.6: The equivalent coupling diagram of the ensembles of atoms in cases of collective excitation for the case (a) system is initially in S0⟩ L and (b) system is initially in S1⟩ L . In the case of low excitation, in the Bogoliubov approximation we can regard the operators G;G on state Sg⟩ as the C-number √ N, where N is the total number of the atoms in the cloud. When the system is initially in S0⟩ L (and hence Sf⟩ is not excited), the equivalent state coupling diagram for this pumping process is shown in Fig. 2.6(a). Unlike the phase gate, we see that this Hamiltonian is no longer closed in a small subspace of states because p1 will continuously pump atoms of the ensemble from the ground state to the excited states. However, because of the large energy shift, the rate to 29 0.0 0.2 0.4 0.6 0.8 1.0 0 5 10 15 20 25 p1 N 1/2 p2 <n r >(|0> L ) <n r >(|1> L ) t( s) Rabi Frequency (MHz) 0.00 0.25 0.50 0.75 1.00 n r 0.0 0.5 1.0 1.0x10 -6 2.0x10 -6 Figure 2.7: Adiabatic Pumping process. Note that at the end there is only a single excitation in the register state Sr⟩ when initially in S0⟩ L , and no excitation when initially in S1⟩ L . Here, we set rm = 400MHz and mm = 300MHz. Since we take the minimum energy shift as our simulation parameter, the actual performance should be better (i.e., the process could be nished in a shorter period of time). excite SN − 2; 1; 0; 1⟩ grsm and SN − 2; 1; 1; 0⟩ grsm is strongly suppressed. Physically, this means there cannot be an atom excited to the Rydberg state Sr⟩ while another atom be excited to state Si⟩ or Sm⟩. Only a single excitation to the register state can be achieved (S0⟩ L → Sr⟩) during this adiabatic process. If the system is initially in the state S1⟩ L (and hence there is a single excitation in Sf⟩), both Sr⟩ and Sm⟩ cannot be excited due to the energy shift caused by the dipole-dipole interaction, as shown in Fig. 2.6(b). A numerical simulation of the pumping process for the minimum energy shifts is shown in Fig. 2.7. After the pumping process, and the adiabatic transfer of the population of Sf⟩ back to S1⟩ L , we can then realize the R i y () gate on the ensemble of atoms by applying three controlling laser beams. The coupling diagram is shown in the right side of Fig. 2.5. The Hamiltonian of step 2 can be obtained directly in second-quantized form: H 2 (t)= 0 (t)( ^ E ^ R + ^ E ^ R)+ 1 (t)( ^ E ^ S + ^ E ^ S)+ a (t)( ^ E ^ A + + ^ E ^ A + ): (2.12) 30 We can directly apply the technique of geometric transformations to realize this gate, as introduced in Ref. [41] for four-level systems. Here, we choose 0 = sin cos', 1 = sin sin', a = cos. This time, the Hamiltonian is closed in the basis {Se⟩;S0⟩ L = Sr⟩;S1⟩ L = Ss⟩;Sa + ⟩}: H 2 (t)= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 sin cos' sin sin' cos sin cos' 0 0 0 sin sin' 0 0 0 cos 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ : (2.13) The eigenspace corresponding to the zero-energy eigenvalue (dark space) is spanned by basis vectors {SD 1 ⟩, and SD 2 ⟩}, where SD 1 ⟩= sin'S0⟩ L − cos'S1⟩ L ; SD 2 ⟩= cos(cos'S0⟩ L + sin'S1⟩ L )− sinSa + ⟩: (2.14) We can use the formula for the degenerate subspace under the cyclic evolution of , '. Suppose S (t)⟩=C 1 (t)SD 1 ⟩+C 2 (t)SD 2 ⟩. We have the equation ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ _ C 1 _ C 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ D 11 (t) D 12 (t) D 21 (t) D 22 (t) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ C 1 C 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ; (2.15) where D ij (t)= ⟨D i S _ D j ⟩. Let's dene dA 2 (t)= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ D 11 (t) D 12 (t) D 21 (t) D 22 (t) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ dt= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 − cos _ ' cos _ ' 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ dt; (2.16) and also the unitary matrix U =P exp(i c C dA 2 (t)): (2.17) 31 After the cyclic evolution, SD 1 (T)⟩=U 11 SD 1 (0)⟩+U 21 SD 2 (0)⟩; SD 2 (T)⟩=U 12 SD 1 (0)⟩+U 22 SD 2 (0)⟩: (2.18) At the beginning, if we set (0) = 0 and '(0) = 2 , we will have SD 1 (0)⟩ = S0⟩ L and SD 2 (0)= S1⟩ L . We can use the Dyson expansion to get the unitary matrixU. This gives us: U 11 = 1− 2 2 2 + 4 2 24 − 6 2 720 :::= cos 2 ; U 12 = 2 − 3 2 6 + 5 2 120 :::= sin 2 ; U 21 =− 2 + 3 2 6 − 5 2 120 :::=− sin 2 ; U 22 = 1− 2 2 2 + 4 2 24 − 6 2 720 :::= cos 2 ; (2.19) where 2 = c C cosd' is a pure geometric phase. Thus we get the unitary transformation we want. Step 3 is simply the reverse of step 1, to return us to our original basis. A simulation to realize 2 = ~4 was done for a certain pulse shape of the cou- pling laser beams, where the qubit was initially prepared in the state S0⟩ L , as shown in Fig. 2.8(a). We show the evolution of the populations of states Ss⟩, Se⟩ and Sr⟩ in Fig. 2.8(b),(c), and (d), respectively. If is large enough (≥ 200 MHz, which is practical in current experiments), we could implement the gate operation in 1s with extremely high accuracy. Thus, the total procedure takes less than 5s to complete, while the lifetime of Sr⟩ is approximately equal to the lifetime of Sr⟩ [89], which is estimated to be 300-400s in a cryogenic environment [119]. 32 Figure 2.8: (a): The pulse shape of the control lasers of 0 , 1 and a respectively. (b)(c)(d): The evolution of the populations of Ss⟩, Se⟩, Sr⟩. We see that when is larger than 200MHz, the adiabatic condition is satised and the gate works with very high accuracy (close to the theoretical value) and the population of the short-lived state Se⟩ tends to zero. We simplied the calculation and simulation by assuming that all atoms are uniformly illuminated in the laser beam. However, just like the argument made in the previous section, the geometric phase is independent for large , if the spatial distribution of the electric elds of the three laser beams match each other at each point in space. Thus, we can in principal eliminate the error introduced by nonuniform illumination. 2.3.3 Two Qubit Controlled Phase Gate Now, we are ready to construct the controlled phase gate. Recently, an adiabatic SWAP operation between states of two clouds of atoms in a cavity QED system was experimen- tally achieved [126]. Here, we extend this method to geometrically realize the controlled 33 Figure 2.9: The schematic setup to realize the conditional phase gate, and the relevant atomic levels of the two clouds of atoms. phase gate, which together with the earlier one-qubit gates gives a universal set of quan- tum gates. The basic idea is to couple two qubits (that is, clouds of atoms) by virtually emitting and absorbing a cavity photon; and after an adiabatic evolution, the second qubit obtains an extra relative phase when the rst qubit is in S0⟩ L . The coupling dia- gram of the scheme is shown in Fig. 2.9. Suppose the length of the cavity is about 200m, which means, it could contain about 10 qubits. Consider two dierent clouds of atoms in the cavity. For a single atom in each cloud, pick a manifold of states of p. When an electric eld is applied to these two cloud of atoms, the manifold splits into Sr + ⟩ and Sr − ⟩. For atom cloud 1, we set a constant electrical eld E z1 and a constant magnetic eld B z1 in the z direction. The Stark Eect splits the manifold p, and the Zeeman eect changes the energy of Sg⟩ and Ss⟩. Assume the energy dierence between Sg⟩ and Ss⟩ to be . Set the appropriate non- zero value of E z1 and B z1 so that Sr + ⟩ is resonantly coupled to Sg⟩ and Ss⟩ by a cavity 34 mode a and a control laser 1 respectively. For atom cloud 2, E z2 is set so that Sr + ⟩ is resonantly coupled to Sg⟩ and Ss⟩ by the same cavity mode a and control laser 2 , and Sr − ⟩ is resonantly coupled to Ss⟩ and Sa − ⟩ by the cavity mode and a control laser 3 . Note that for Rb 87 , the hyperne structure energy split of the ground state manifold is ~2=6.835GHz, and the energy dierence of Sr + ⟩ and Sr − ⟩ should set to be equal to for atom cloud 2 for the purpose of resonant coupling, while the frequencies of 1 , 2 and 3 should be ! cav −, ! cav − and ! cav + , where ! cav is the frequency of cavity mode. Since the cavity mode is inhomogeneously distributed, the thermal motion (even at extremely low temperature) of atoms, causes the coupling rate between atoms and cavity photons to vary greatly from one to another (by roughly a factor of 2) and thus makes the system dicult to control accurately. To overcome these diculties, instead of directly applying the laser beam to the clouds of atoms, we use the idea of external laser driving control [42] as shown in schematic setup in Fig. 9. Three classical laser elds E 1 (t), E 2 (t) andE 3 (t) are incident on one mirror of the cavity to drive the transition Sr + ⟩→ Ss⟩ for cloud 1, and Sr + ⟩→ Sa + ⟩ and Sr − ⟩→ Sa − ⟩ for cloud 2 through another cavity mode a ′ . We assume for simplicity that a and a ′ have the same spatial mode structure (r) with the same frequency ! cav , so they can be dierent only in polarization. The driving elds E 1 (t), E 2 (t) and E 3 (t) are resonant with frequencies ! cav −, ! cav − and! cav +. So they are far-o-resonant to the cavity mode with large detuning. E 1 (t), E 2 (t) and E 3 (t) control the time evolution of the Rabi frequencies 1 , 2 and 3 . To see this, consider the input-output equation for the cavity mode a ′ [146]: da ′ (t) dt =−i[a(t);H sys ]− 2 a ′ (t)+ √ a ′ in (t); (2.20) where a ′ in (t) is the eld operator for the input driving pulse coupling to the mode a ′ , with ⟨a ′ in (t)⟩=E 1 (t)+E 2 (t)+E 3 (t); 35 and [a ′ in (t);a ′ in (t ′ )]=(t−t ′ ): Since a ′ is driven by strong classical pulses, we can treat the mode as classical and thus assume that the interaction between a ′ and the atoms will not change the state of the cavity mode a ′ . Eq. (2.20) therefore can be modied to be da ′ (t) dt =−i! cav a ′ (t)− 2 a ′ (t)+ √ (E 1 (t)+E 2 (t)+E 3 (t)): (2.21) First, we suppose there is only one classical pulse E 1 (t) = ⟨a ′ in (t)⟩ = " 1 (t)e −i(!cav−)t , where" 1 (t) is the slowly varying amplitude ofE 1 (t). Then, we can write the mean value of a ′ as ⟨a ′ (t)⟩=(t)e −i(!cav−)t ; which has the solution (t)= S t 0 " 1 () exp(−i−~2)(t−)d ≈ " 1 (t)−e (−i−~2)t " 1 (0) i+~2 (2.22) when the characteristic time T 1 of " 1 (t) satises T 1 ≫ 1. Assume that" 1 (t) gradually increases from zero with" 1 (0)= 0. Thus we can conclude (t)∝" 1 (t). By choosing an appropriate phase of " 1 (t), we set (t) to be real. If we inputE 2 (t) orE 3 (t), the solution will have a similar form. Now, consider the case when three classical pulses are incident on the cavity mirror. Eq. (2.20) is a linear equation, so, when the three input pulse have dierent resonant frequencies, the solution should be a superposition of the solutions for three single pulses. Thus, ⟨a ′ (t)⟩ could be represented as ⟨a ′ (t)⟩= 1 (t)e −i(!cav−)t + 2 (t)e −i(!cav−)t + 3 (t)e −i(!cav+)t : (2.23) This gives Rabi frequencies i (r;t) =d i i (t)(r) for i = 1; 2; 3, where d i is a coecient mainly determined by the dipole moment for the corresponding transition. 36 The frequencies of the three components of a ′ dier signicantly, so we can regard them as three separate pulses. In addition, the energy structure of atoms in cloud 1 is dierent from those in cloud 2, so 1 will not interact with the atoms in cloud 2. Similarly, 2 and 3 will not interact with atoms in cloud 1. All other clouds in the cavity are all far o resonant from both the cavity mode and the control lasers, and thus can be safely excluded by our gate operation. Now, we can represent the Hamiltonian for the situation of of Fig. 2.9 in the interaction picture: H 3 (t)=Q j 1 1 (r j 1 ;t) (j 1 ) sr+ +Q j 1 g + (r j 1 )a (j 1 ) gr+ +Q j 2 ( 2 (r j 2 ;t) (j 2 ) a+r+ + 3 (r j 2 ;t) (j 2 ) a−r− ) +Q j 2 (g + (r j 2 )a (j 2 ) gr+ +g − (r j 2 )a (j 2 ) sr− )+ H.c:; (2.24) wherea anda are annihilation and creation operators of cavity mode a, andg + (r) and g − (r) are coupling rates of cavity mode a to the transition Sg⟩→ Sr + ⟩ in cloud 1 and Ss⟩→ Sr − ⟩ in cloud 2. We can represent g + (r)(g − (r))= ̃ g + (r)(̃ g − (r)), where ̃ g + and ̃ g − are constants. So we can rewrite the Hamiltonian as: H 3 (t)=Q j 1 (r j 1 )(d 1 1 (t) (j 1 ) sr+ +̃ g + a (j 1 ) gr+ ) +Q j 2 (r j 2 )(d 2 2 (t) (j 2 ) a+r+ +d 3 3 (t) (j 2 ) a−r− +̃ g + a (j 2 ) gr+ +̃ g − a (j 2 ) sr− )+ H.c:; (2.25) We will see soon that the dark state of such a Hamiltonian should be independent of (r), and the adiabatic evolution of the system is determined only by the minimum value of (r j ) for all j, say m . So, the Hamiltonian can eectively be represented as: H e 3 (t)=Q j 1 ( 1 (t) (j 1 ) sr+ +g + a (j 1 ) gr+ )+Q j 2 ( 2 (t) (j 2 ) a+r+ + 3 (t) (j 2 ) a−r− ) +Q j 2 (g + a (j 2 ) gr+ +g − a (j 2 ) sr− )+ H.c:; (2.26) 37 0 1 2 3 4 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 phase /16 (t) 2 (t) time( s) Geometric phase 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Radian of and Figure 2.10: The evolution of the phase 3 of S10⟩ L . The pulse shapes of (t) and'(t) are also given. where i (t) = d i i (t) m for i = 1; 2; 3 and g + (g − ) = ̃ g + m (̃ g − m ). As in the previous cases, we transform the Hamiltonian to second quantized representation. For two clouds of atoms in the Bogoliubov approximation, we get the new Hamiltonian H e 3 (t)≈ 1 (t)( ^ S ) 1 ( ^ R + ) 1 + 2 (t)( ^ A + ) 2 ( ^ R + ) 2 + 3 (t)( ^ A − ) 2 ( ^ R − ) 2 +g + √ Na ( ^ R + ) 1 +g + √ Na ( ^ R + ) 2 +g − a ( ^ S ) 2 ( ^ R − ) 2 + H.c:; (2.27) where ^ S, ^ R − , ^ R + , ^ A − , and ^ A + are the bosonic annihilation operators of the single atom states Ss⟩, Sr − ⟩, Sr + ⟩, Sa − ⟩, and Sa + ⟩, respectively. We set 1 (t)= 1 sine −i' 1 ; 2 (t)= 2 cose −i' 2 ; 3 (t)= 3 cose −i' 3 : We also dene g 1 =g + √ N, g 2 =g + √ N and g 3 =g − for convenience. 38 As in the previous sections, for each cloud of atoms we have a set of basis states: S0⟩ L = SN; 0; 0; 0; 0; 0⟩ gr+r−a+a−s ; S1⟩ L = SN − 1; 0; 0; 0; 0; 1⟩ gr+r−a+a−s ; Sr + ⟩= SN − 1; 1; 0; 0; 0; 0⟩ gr+r−a+a−s ; Sr − ⟩= SN − 1; 0; 1; 0; 0; 0⟩ gr+r−a+a−s ; Sa + ⟩= SN − 1; 0; 0; 1; 0; 0⟩ gr+r−a+a−s ; Sa − ⟩= SN − 1; 0; 0; 0; 1; 0⟩ gr+r−a+a−s : The Hamiltonian is closed in a subspace of Hilbert space that can be divided into the direct sum of two closed subspaces: H= (H 1 ⊗H 2 + )⊕(H 1 ⊗H 2 − ): (2.28) Here, H 1 ⊗H 2 + is spanned by basis vectors {S100⟩, Sr + 00⟩, S001⟩, S0r + 0⟩, S0a + 0⟩}, and H 1 ⊗H 2 − is spanned by basis vectors {S110⟩, Sr + 10⟩, S011⟩, S0r − 0⟩, S0a − 0⟩}. The rst two degrees of freedom represent cloud 1 and cloud 2, and the last degree of freedom of the state is the Fock state of the cavity photons. Since the Hamiltonian doesn't couple these two subspaces, and has the same form in each subspace, we can just consider a single subspace, e.g.,H 1 ⊗H 2 + . The Hamiltonian can be represented in this subspace as H e 3 (t)= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 g 1 0 1 sin 0 0 g 1 2 cose i' 2 0 g 1 g 1 0 0 0 0 2 cose −i' 2 0 0 0 1 sin 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ; (2.29) 39 where we have set ' 1 = 0. A dark state exists for this system, since one eigenstate has eigenvalue 0: SD(t)⟩= g 1 1 cos ½ g 2 1 2 1 cos 2 + g 2 2 2 2 sin 2 + cos 2 sin 2 S100⟩ + e −i' 2 g 2 2 sin ½ g 2 1 2 1 cos 2 + g 2 2 2 2 sin 2 + cos 2 sin 2 S0a + 0⟩ − cos sin ½ g 2 1 2 1 cos 2 + g 2 2 2 2 sin 2 + cos 2 sin 2 S001⟩: (2.30) Just like the case of the phase gate, if the system is initially prepared in the dark state, after a cycle of adiabatic evolution a pure geometric phase can be obtained. In this case, if the initial state is S10⟩ L , we get a phase shift 3 =i c dR⟨D(t)S∇ R SD(t)⟩ = c d' 2 g 2 2 2 2 sin 2 g 2 2 2 2 sin 2 + g 2 1 2 1 cos 2 + cos 2 sin 2 : (2.31) Note that this phase is not aected by the spatially inhomogeneous distribution of the cavity mode. We also get a geometric phase for the subspaceH 1 ⊗H 2 − . However, we can set choose our path to set this phase to 0 independently of 3 . So, if we initially prepared the state S10⟩ L we get a geometric phase of 3 after the gate manipulations, and if the initial state is S11⟩ L the geometric phase is 0. The states S00⟩ L and S01⟩ L are not aected by this Hamiltonian, and hence also acquire no phase. The net eect is a two-qubit quantum gate, S00⟩ L → S00⟩ L ; S01⟩ L → S01⟩ L ; S10⟩ L →e i 3 S10⟩ L ; S11⟩ L → S11⟩ L ; 40 which is exactly the controlled phase gate we would like to implement. As we can see in Eq. (2.30), the two clouds of atoms interact with each other by virtually absorbing and emitting a photon in cavity modea, so leakage of cavity photons is the most important source of decoherence in our scheme. Analytically, the average photon number in the cavity during the process is n ph+ = cos 2 sin 2 g 2 1 2 1 cos 2 + g 2 2 2 2 sin 2 + cos 2 sin 2 ; (2.32) when the system is initially prepared in S10⟩ L , and n ph− = cos 2 sin 2 g 2 1 2 1 cos 2 + g 2 3 2 3 sin 2 + cos 2 sin 2 ; (2.33) when the system is initially prepared in S11⟩ L , respectively. If the initial state is S00⟩ L or S01⟩ L there are no photons in the cavity. Let's consider the concrete example of a conditional phase gate with 3 =~16. We set the parameters to be g + = 20MHz, g 3 = g − = 10MHz, and the number of atoms in each cloud to be N = 10 3 . Then we have g 1 = 660MHz, 1 = 40MHz, 2 = 50MHz and 3 = 300MHz. We choose (t)= 4 exp[−(t− 1:7) 2 ~0:5]; ' 2 (t)= 4 exp[−(t− 2:115) 2 ~0:5]; (2.34) (where time is expressed in s). A numerical simulation (not including cavity loss) is shown in Fig. 2.10 where the initial state is 1 4 S00⟩ L + 1 4 S01⟩ L + 1 4 S10⟩ L + 1 4 S11⟩ L . The whole process is very fast and can be nished in 4s, with extremely high accuracy approaching the theoretical value calculated in Eq. (2.31). Fig. 2.11 shows the average number of cavity photons n ph+ and n ph− during the process as a function of time and 1 (not including cavity loss). The photon number 41 Figure 2.11: The average number of photons in the cavity during gate operation as a function of time and 1 for g 1 = 660MHz. (a) system initially prepared in S10⟩ L and (b) system initially prepared in S11⟩ L . is less than 0.001 in general which means that, the probability of a photon loss in the cavity is bounded above by 0.001 for the parameters given previously. We see that when the state of system is initially in S11⟩ L , the average number of photons in the cavity is larger than in S10⟩ L , S00⟩ L and S01⟩ L . (Indeed, for the last two states the photon number is strictly zero.) 2.4 Measurement In fault-tolerant quantum computation, one must do syndrome measurement periodically [2, 40, 60], and the measurement results should be correct with high probability. It is also necessary to read out results at the end of a computation. Accurate measurement is therefore necessary. In this section, we propose a protocol to measure an atomic ensemble qubit in the computational basis. Collecting the uorescence from an atom is the most natural way to realize a mea- surement. For a single atom (or ion), many uorescence photons are needed to make the measurement reliable, so a \cycling" transition from the computational state (S0⟩ L or S1⟩ L ) to an unstable excited state is generally used. In our case, however, there is an 42 Figure 2.12: The schematic setup to realize a projective measurement on a single qubit. Here, state Sd⟩ is a temporary state with a short lifetime. The transition between Sd⟩ and Ss⟩ is not allowed. The state Sd⟩ can be chosen for example, to be SF = 1;m F =−1⟩ of the manifold 5P 1~2 . added complication. The computational states include superpositions of a single exci- tation over all the atoms of the cloud. The distance between atoms in the cloud can be several times larger than the wavelength of a photon emitted from the cloud, and this might make it possible (in principle) to distinguish which atom emitted the photon. This in turn could cause decoherence by collapsing the symmetric superposition, and take the system state outside the computational space. So we need a method to perform a reliable measurement while avoiding this problem. As shown in Fig. 2.12, the procedure can be realized in three steps: 1. Adiabatically transport the population of S1⟩ L to Sf⟩. (The procedure is similar to that used in realizing the gate R y ().) 2. Dynamically pump the atom cloud from the ground state to state Sm⟩ with a pulse 0 , and subsequently apply another pulse 1 between the state Sm⟩ and Sr⟩, nally, apply a third pulse 2 between state Sr⟩ and Sd⟩. Cycle this transition hundreds of times and collect the spontaneously emitted photons from the atom cloud. 43 3. Reverse step 1 to transfer the population of Sf⟩ back to S1⟩ L to prepare for next operation. If the qubit is initially in S0⟩ L , step 1 has no eect. The state of the cloud will transfer to Sm⟩ and then to Sr⟩ and Sd⟩ after three pulses (the energy between Sr⟩ and Sd⟩ is the same as that of Sr⟩ and Se⟩, therefore, no additional laser frequency is required for the measurement process.); the atom cloud will quickly decay to the ground state and emit a photon. By repeating this cycle many times, the probability of detecting the photons can be made quite high. If the qubit is initially in S1⟩ L , step 1 will produce a single excitation in Sf⟩, and by the dipole-dipole blockade eect the transition from the ground state to Sm⟩ will be blocked. Therefore we will not observe any emitted photons. Thus we observe uorescence only if the qubit is initially in the state S0⟩ L , and this procedure is a projective measurement. Note that non-uniform coupling is not a problem here since at the end of the procedure the state will be either S0⟩ L or S1⟩ L . The time for a single laser pulse (and the accompanying state transition) should be roughly 10-20ns when Rabi frequency of the pumping laser is hundreds of MHz. Assuming a life time of about 10ns for Sd⟩, each cycle of measurement can be nished in 30-40ns. After the cycling transitions, we must transfer the population of Sf⟩ back to S1⟩ L , so we need the total measurement time to be less than the lifetime of Sf⟩ (estimated to be about 400s). This gives an upper limit on the total number of cycles we can make during the process. If we want to limit the error to a reasonable level, we can perform at most a few hundred transition cycles. This should be adequate if the photodetector has suciently high eciency and large enough solid angle of detection. 44 2.5 Decoherence We will now consider the limitations imposed by decoherence on gate operation. 2.5.1 Mechanism For our system on atom chips to work, several main sources of noise have to be considered and kept under control, we treat them one at a time below. 2.5.1.1 Spontaneous Emission Like all ion trap and neutral atom schemes in quantum computation, our proposal suers from spontaneous emission when atoms are in excited states. In our scheme, the excited states (other than logical states) are never populated in the ideal case. However, in any real process (nite time, nite energy gap), the state of the system will precess around the dark state, instead of following it exactly in a perfect adiabatic way. This means that some population will reach the leaky excited states (including Rydberg states). We can calculate the population of such unwanted states during the adiabatic process and impose the condition: Q k S T 0 k P k (t)dt≪ 1; (2.35) where T is the duration of the gate, P k (t) is the population of the kth excited state of the atom, and k is the corresponding decay rate of that state. On the other hand, the condition for adiabatic process can be stated as: g T ≫ 1: (2.36) Here, E g = ̵ h g estimates the dierence between the dark state energy and the energy of the closest eigenstate. Suppose the kth excited state is an intermediate state used in our gate operation (Se⟩ for the phase gate and the y-rotation gate, Sr + ⟩ or Sr − ⟩ for the 45 controlled phase gate), We have max t {P k (t)} ≈ 1~ g T . Then the condition on k must be: k T ( g T) 2 ≪ 1: (2.37) 2.5.1.2 Cavity Loss Cavity loss may aect the controlled phase gate, which is realized based on the dark state in Eq. (2.30) through virtual emission and absorption of cavity photons. We note that even in the ideal adiabatic case, the state has nonzero projection onto the 1-photon cavity state, and so a lossy cavity will tend to destroy such a state. For the in uence of the cavity loss rate to be small, the condition S Tcp 0 ⟨D(t)Sa aSD(t)⟩dt≪ 1; (2.38) must be satised, where ⟨D(t)Sa aSD(t)⟩ is the population of the cavity mode during the adiabatic evolution, and T cp is the duration of the controlled phase gate. Since the integral above is always smaller than T cp max t {⟨D(t)Sa aSD(t)⟩}, the inequality in our case can be replaced by T cp max t {n ph- (t)}=T cp 2 1 2 3 4g 2 1 2 3 +g 2 3 2 1 ≪ 1: (2.39) Combined with Eq. (2.37), this inequality bounds the process time. Since our scheme has large g 1 , it has a good ability to tolerate cavity loss, while still allowing a long process time to overcome the spontaneous emission for the controlled phase gate. 2.5.1.3 Trap Loss It is crucial in our scheme to be able to trap the atoms inside the trap as long as possible. However, there are several possible sources of noise to drive the atoms out of the trap, as shown in Ref. [49]. These noise sources can be divided into two groups: 46 1. Noise-induced spin- ip. Our logical states are encoded in two weak-eld-seeking hyperne states. If the state of an atom is ipped to the strong-eld-seeking state, the magnetic potential is no longer a trap for that atom. Such spin- ips may be caused by uctuations produced by thermally excited currents in the metallic substrate, or simply by technical noise in the wire currents. 2. Heating. The energy exchange increases both the system's mean energy and its energy spread, and can excite atoms to a high energy vibrational mode. Since the trap cannot be made innitely deep, atoms may escape from the trap. We assume that the loss rate is not too high when the temperature is low, and that the noise from substrate and wire currents is well controlled. If the system is in the ground state Sg⟩, the loss of atoms will not cause decoherence at all. Only if the system is in a collective excited state does the trap loss mean information is lost to the environment. For simplicity in the following discussion, we assume the loss rate of internal state Sk⟩ to be k . We can see that loss of atoms is similar to spontaneous emission when the loss rate is low. Both eects can be treated as an amplitude damping channel. This observation will help us to analyze its eect by numerical simulation. Decoherence Channel Mechanism Eect Typical Rate Spontaneous emission Excited states decay 1kHz| to ground state. 100MHz Cavity loss Destroy the dark state 10kHz| through photon leakage. 100MHz Noise in the wire currents Flip the spin state through Trap loss. 1Hz| uctuation of magnetic potential. 1kHz Thermally excited currents Flip the spin state through Trap loss. 1Hz| uctuation of magnetic potential. 1kHz Heating Excite atoms to high Trap loss. 1Hz| energy vibrational mode. 1kHz Stray electrical eld Generate random phase. Dephasing. 1MHz|1GHz Table 2.1: Decoherence mechanism for the atom chip system. Some of the typical values are based on [49] and [47]. 47 2.5.1.4 Stray Electric Fields Stray electric elds are produced by the adsorbing of rubidium atom onto the substrate of the chip and mirrors of the cavity, and other uncontrollable mechanisms. In our case, we can treat such an electric eld as a spatially random distributed eld that causes extra detuning while there is optical coupling between dierent energy levels. To analyze this noise, we suppose the system we are interested in has Hamiltonian H sys , and that the total Hamiltonian including the stray electric elds can be written: H tot =H sys +Q i Q k k st (x i )Sk⟩⟨kS (2.40) in the interaction picture. Here, k st (x i ) is the extra detuning of the aected excited state Sk⟩ of atomi at positionx i relative to the center of the trap. Suppose the trap can be modeled as a harmonic oscillator. We have x i = ¾ 1 2m (b i +b i ); where is the frequency of the trap, and b i (b i ) is the lowering (raising) operator for atom i in the trap. Considering the limit where k st (x i ) changes slowly and smoothly spatially near by the trap, we can expand it around the center of the trap: k st (x i )≈ k st (0)+ k′ st (0)x i : If the temperature is low enough, i.e, kT ≪ ̵ h, most atom stay in the ground state, so b i (b i ) can be approximated as b i ≈ S0⟩ i ⟨1S (b i ≈ S1 i ⟩⟨0 i S), with S0 i ⟩ and S1 i ⟩ the ground state and rst excited state of the atom i, respectively. Techniques similar to those in Sec 2.2 can be applied here to put the Hamiltonian in the second quantizated representation: H tot =H sys +Q k k st (0) ^ K ^ K+ ~ k′ st (0) ^ K ^ K( ^ B + ^ B); (2.41) 48 for some random numbers k st (0) and ~ k′ st (0) ( k′ st (0)~ √ 2m). Here, we dene ^ K= 1 √ N Q i Sg i ⟩⟨k i S for dierent excited states Sk⟩, and ^ B = 1 √ N Q i S0 i ⟩⟨1 i S: Eq. (2.41) will be used in a numerical analysis of the eect of stray electric elds later. As a summary, Table 2.1 lists dierent decoherence channels, their mechanisms, their eects, and their typical rates on the atom chip system we are interested in. 2.5.2 Numerical Analysis We performed numerical simulations of the universal set of quantum gates described in previous sections. These simulation conrm what we stated in the previous section. The robustness of gates against noise is measured by the delity of the gate, i [100]: F i = min S ⟩ ¼ ⟨ SU i L(S ⟩⟨ S)U i S ⟩; (2.42) where U i is the desired gate operation and the superoperator L is determined by the master equation: d dt =−i[H sys ;]+Q j L j L j − 1 2 L j L j − 1 2 L j L j ≡L; (2.43) in the Markov approximation. Here, the fL j g are the Lindblad operators representing the interaction with the environment. In many cases, no analytical solution of such master equation is known and even a numerical solution can be hard. We solve this equation by unraveling the density operator evolution into quantum trajectories [121]. 49 The unraveling of the master equation implemented are given by the quantum state diusion (QSD) equation [57] : Sd ⟩=−iH tot dt+Q j ⟨L j ⟩ L j − 1 2 L j L j − 1 2 ⟨L j ⟩ ⟨L j ⟩ S ⟩dt+Q j L j −⟨L j ⟩ S ⟩d j : (2.44) Here, angular brackets denote quantum expectations ⟨L j ⟩ = ⟨ SL j S ⟩, andH tot is given by Eq. (2.41), which is a Hamiltonian with random parameters for each trajectory. The d j are independent complex Gaussian dierential random variables satisfying the conditions E[d j ]=E[d i d j ]= 0; and E[d ∗ i d j ]= ij dt; whereE denotes the ensemble mean. The density operator is given by the mean over the projectors onto thee quantum states of the ensemble: =E[S ⟩⟨ S]: In this Chapter, we present delity calculations for the given set of universal gates, considering several channels of decoherence which are important in experiments. The simulation was realized with the \Quantum State Diusion" C++ library [121] using a fourth-order Runge-Kutta integrator [112] and a pseudo random number generator to solve the master equation using quantum trajectories. For each trajectory, k st (0) and ~ k′ st (0) are Gaussian distributed random variables, i.e., k st (0) ∼ N( k ; 2 k ) and ~ k′ st (0)∼N( ′ k ; ′2 k ). 2.5.2.1 Phase Gate For the phase gate, spontaneous emission and trap loss, can aect the gate operation. The spontaneous emission rate e from the state excited state Se⟩ can destroy the super- position of hyperne ground states when e is large, though if we adiabatically transfer 50 the state, the population of Se⟩ is very small. As discussed in Ref. [89], the rate of sponta- neous emission of the ensemble of atoms should be the same as of single atoms. The loss rates of atoms from the trap e , s , a for dierent energy level must also be included. As mentioned earlier, loss of atoms and spontaneous emission have the same eect so that they should share the same form of Lindblad operator. For simplicity of analysis, we conservatively set = max( e ; a ), = e + e , and choose the Lindblad operators to be: L 1 = √ 2 ^ E, L 2 = √ 2 ^ S, and L 3 = √ 2 ^ A. Figure 2.13: Fidelity of a phase gate with 1 =~8 as a function of for dierent values of not including stray elds. We rst simulate the case where there is no stray electric eld. The simulation uses the initial state S ⟩ = S1⟩ L , which minimize the delity. The result of the calculation is shown in Fig. 2.13 for dierent values of and . (Each point is based on 100 trajectories, so this is not a highly precise calculation.) When is in the range between 10Hz and 100Hz, the delity doesn't change dramatically and can be as high as 0.9999 when is less than 1MHz. Next, we consider the eect of noisy stray electric elds by solving Eq. (2.41) for state Se⟩ with e st and ~ e′ st . Again, for simplicity we conservatively assume 51 0.5 1 2 4 8 16 32 64 128 0.98 0.985 0.99 0.995 1 =1MHz, =0.1MHz =10MHz, =1MHz =1GHz, =100MHz Fidelity Figure 2.14: Fidelity of a phase gate with 1 =~8 for dierent values of with = 10Hz considering the stray eld eect. Figure 2.15: Fidelity of a phase gate with 2 =~4 for dierent values of , and for = 100Hz without considering the stray eld. = max( e ; ′ e ) and = max( e ; ′ e ), with e st (0) ∼N(; 2 ) and ~ e′ st (0) ∼N(; 2 ). The simulation result shown in Fig. 2.14 is based on 100 trajectories for each point. The 52 noisy electric eld has almost no eect on the gate operation until it is about an order of magnitude larger than the Rabi-frequency we applied on the atoms. 2.5.2.2 Y-Rotation Gate For the y-rotation gate, we need to consider additional pumping processes as well as the adiabatic process to get the geometric phase. To simplify the calculation, we consider the delities dened in Eq. (2.42) for each process and multiply them together: F =F p 1 ⋅F p 2 ⋅F p 1 ⋅F pg ⋅F p 2 ; (2.45) whereF p1 is delity of adiabatic transfer from state S1⟩ L to state Sf⟩,F p 2 is the delity of the adiabatic pumping process from S0⟩ L to Sr⟩ andF pg is the delity of the non-Abelian geometric process. 0.5 1 2 4 8 16 32 64 128 0.76 0.78 0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 Fidelity z =200MHz, =2MHz, =0.2MHz =200MHz, =20MHz, =2MHz =1000MHz, =2MHz, =0.2MHz =1000MHz, =20MHz, =2MHz Figure 2.16: Fidelity of a phase gate with 2 = ~4 for dierent values of and considering the stray eld with = 1kHz and = 100Hz. According to Fig. 2.5, we need to consider both the loss from Rydberg states and from intermediate states. Just as for the phase gate, the parameters are chosen to be: 53 = max( e ; a+ ; s ; m ; r ; f ), = max( e ; d ) , = max( m + ; r + ; f + ) and the Lindblad operators to be: L 1 = √ 2 ^ M;L 2 = √ 2 ^ R;L 3 = √ 2 ^ F;L 4 = √ 2 ^ E;L 5 = √ 2 ^ D;L 6 = √ 2 ^ S, and L 7 = √ 2 ^ A + according to the corresponding loss channels. We rst simulate the case where there is no additional noisy stray electric eld. The simulation result is shown in Fig. 2.15 for = 100Hz. The delity is beyond 0.99 when is less than 1MHz, is 1kHz, and = 1000MHz. When is small, the error rate is dominated by . When is high, the error rate is dominated by . The larger the is, the smaller is the population of the intermediate state. Thus satisfying the condition (2.37) greatly reduces the error. The main reason why this gate can not achieve delity as high as 0.999 is due to the imperfect adiabatic pumping process p 2 . However, our calculation is based on taking the minimum value of the dipole-dipole interaction over all pairs of atoms in the cloud, more accurate simulation model of the dipole-dipole interaction would give better result. In this case, the noisy stray eld has dierent ways to aect our systems for dif- ferent processes. According to Eq. (2.41), we have Sk 1 ⟩ = Sd⟩ for process p 1 , Sk 2 ⟩ = Sm⟩ for process p 2 , and Sk 3 ⟩ = Se⟩ for process p g . The parameters are chosen to be = max( e ; ′ e ; m ; ′ m ; d ; ′ d ) and = max( e ; ′ e ; m ; ′ m ; d ; ′ d ), with k st (0) ∼N(; 2 ) and ~ k′ st (0)∼N(; 2 ) for k =d;m;e. The simulation results of the eect of noisy stray elds is shown in Fig. 2.16, with = 100Hz, = 1kHz and 100 trajectories for each point. The robust against the noisy electric eld is not as good as the other two gates (we will see the case for the controlled-phase gate next) because of p 2 . When the average amplitude of the noisy eld is comparable to the amplitude of the Rabi frequency of the laser used in adiabatic pumping (which in this case, is 20MHz), some extra Raman transition-like processes will occur to excite the ensemble to higher excited states even though the strong dipole-dipole interaction between atoms forbids these excitations. If can be reduced as low as a few MHz, we can safely neglect the eect of noisy electric elds. 54 2.5.2.3 Controlled Phase Gate For the controlled phase gate, besides the spontaneous emission and trap loss we ana- lyzed for the single qubit gates, cavity loss also plays an important role. In our analysis, as shown in Fig. 2.9, the parameters are chosen conservatively to be: = max( e ; a− ; a+ ; s ), = max( r− + r− ; r+ + r+ ) and we set the Lindblad operators to be L 1 = √ 2 ( ^ R + ) 1 ,L 2 = √ 2 ( ^ R + ) 2 ,L 3 = √ 2( ^ S) 1 ,L 4 = √ 2( ^ A + ) 2 ,L 5 = √ 2 ( ^ S ) 1 ( ^ R + ) 1 , L 6 = √ 2( ^ S ) 2 ( ^ A + ) 2 , L 7 = √ 2 ( ^ R − ) 2 , L 8 = √ 2 ( ^ S ) 2 ( ^ R − ) 2 , L 9 = √ 2 ( ^ A − ) 2 ( ^ R − ) 2 , L 10 = √ 2( ^ S) 2 , L 11 = √ 2( ^ A − ) 2 and L 12 = √ 2a cav according to the corresponding channels. We rst simulate the case where there is only cavity loss. As stated previously, the 0.25 0.5 1 2 4 8 16 0.90 0.92 0.94 0.96 0.98 1.00 g 1 =330MHz g 1 =660MHz g 1 =1000MHz g 1 =2000MHz Fidelity (MHz) Figure 2.17: Fidelity of the controlled phase gate with 3 =~16 for dierent values of and g 1 without any other decoherence channels included. initial state S11⟩ L emits more photons than the other three basis states during the gate operation. Hence, we take S ⟩= S11⟩ L as the worst case. The result of the numerical calculation in shown in Fig. 2.17, based on 100 trajectories for each point. From Eq. (2.31), we see that 3 is approximately independent of g 1 and 55 Figure 2.18: Fidelity of the controlled phase gate with 3 =~16 for dierent values of and g 1 including the stray eld eect, with = 100Hz and = 2:5kHz. g 2 when they are large enough, so we can compare the delity for dierent value of g 1 for the same operation. Fig. 2.17 shows the gate delity for dierent values of and g 1 . The larger the value of g 1 for this case, the easier condition (2.38) is to satisfy and the higher delity we get. The delity can exceed 0.9999 when is less than 500kHz for g 1 = 2000MHz, and exceed 0.999 when is less than 500kHz for g 1 = 660MHz, and when is less than 1MHz forg 1 = 1000MHz. For a FP cavity larger than 100m, values of under 1MHz are quite achievable, and even for large values of , the delity is still acceptable for some schemes of fault tolerant quantum computation. Wheng 1 = 330MHz, the performance is bad for large value of . Finally, we analyze the stray electric eld eect on Sr − ⟩ (Sr + ⟩) with r− st ( r+ st ) and ~ r−′ st ( ~ r+′ st ). The simulation result in shown in Fig. 2.18 with = 100Hz and = 2:5kHz. It's worth noting that the random electric eld does not aect the gate operation until it is an order of magnitude larger than the eective cavity coupling strength g 1 . For a smaller value of g 1 , such as 100MHz here, when is large, the photon no longer be able 56 to stay in the cavity for long enough to do the gate operation, In that regime the gate construction breaks down. However, when large stray elds exist in that regime, the detuning they induce further reduces the number of photons in the cavity, and give an even better delity. 2.6 Summary and Conclusion In this Chapter we have proposed a scheme to achieve quantum computation using geo- metric manipulation of ensembles of atoms in a cavity QED system. Adiabatic optical control can be used to obtain a geometric phase gate and controlled phase gate. Com- bining optical excitation with a dipole-dipole blockade between Rydberg states allows us to geometrically realize the R y ( 2 ) gate. Thus a universal set of quantum gates can be realized geometrically. We analyzed this scheme for ensembles of neutral rubidium atoms, magnetically trapped in planoconcave microcavities on an atom chip. Numerical simulations show that a single qubit gate can be performed in several microseconds with very low probability of gate error if the Rabi-frequency of coupling laser is larger than the average amplitude of stray electric elds and the rate of spontaneous emission. For the controlled phase gate, the operation is done by virtually emitting and absorbing a photon from the cavity mode, and can be completed in one single step operation, in a short time (about 4s) for a controlled~32, gate with very high delity even in a noisy environment thanks to the strong eective coupling eciency between cavity photon and atoms. An advantage of geometric manipulation is that by adiabatic parameter control, we can avoid certain kinds of errors, especially those caused by inhomogeneous distribution of the laser beam and cavity modes. The values of the elds can depend on atom position, but their ratio can be xed and controllable. The basic idea is to nd an appropriate adiabatic process, so that the relevant dynamics are either independent of, or depend only on the ratio of, the two coupling rates. 57 The ensemble of atoms eectively enlarges the coupling rateg by √ N, which greatly suppresses the likelihood of cavity photons and increases the delity of the operation. We analyzed the scheme for N = 10 3 atoms in each cloud. The parameters that we have assumed in our numerical simulations are all achievable by current experiments. This, together with the possibility of coupling stationary qubits (for computation) with ying qubits (for communication) makes this scheme look particularly promising for near-term quantum protocols. 2.7 Appendix: Spectrum of Rydberg States In this appendix, we discuss how to choose appropriate Stark eigenstates of alkali atoms for our purpose. Since alkali atoms have spectra similar to hydrogen, we rst analyze the eect of the dipole-dipole interaction on Rydberg Stark eigenstates for hydrogen. After that, we allow for the dierence between alkali atoms and hydrogen, and discuss a method of calculating their energy structure. Using this method, we will choose appro- priate Rydberg Stark eigenstates of rubidium as an example. (Please note that the ne structure of the p and d states of rubidium have observable eects on the spectrum. However, for simplicity, we do not take this into account.) The method of calculation is described in detail in Ref. [56]. Note that all quantities below are in atomic units for simplicity. We rst consider Stark states of a single hydrogen atom. The magnetic quantum numberm is a good quantum number. From perturbation theory, the rst order approx- imation of Stark eignstates are parabolic states Sn;n 1 ;n 2 ;m⟩ [56] with energies E n;n 1 ;n 2 ;m =− 1 2n 2 + 3 2 E(n 1 −n 2 )n: Here,n is the principal quantum number,E is the electric eld in thez direction, andn 1 , n 2 are non-negative integers satisfying the equalityn=n 1 +n 2 +SmS+1. Form= 0, allowed values of n 1 −n 2 are n− 1;n− 3;:::;−n+ 1 and for m= 1, they are n− 2;n− 4;:::;−n+ 2. 58 In the following discussion, we use the quantum number q =n 1 −n 2 instead ofn 1 andn 2 for simplicity. We expand the Stark parabolic states Sn;m;q⟩ in the spherical basis Sn;l;m⟩ as [56]: Sn;m;q⟩= n−1 Q l=0 Sn;l;m⟩⟨n;l;mSn;m;q⟩: (2.46) The coecients can be written in terms of Wigner 3J symbols [56]: c(n;l;m;q)≡⟨n;l;mSn;m;q⟩= (−1) (1−n+m+q)~2+l × √ 2l+ 1 ⎛ ⎜ ⎜ ⎝ n−1 2 n−1 2 l m+q 2 m−q 2 −m ⎞ ⎟ ⎟ ⎠ : (2.47) For example, if n= 2, m= 0, q = 1, we have c(2; 0; 0; 1)= √ 2~2 and c(2; 1; 0; 1)=− √ 2~2. So, S2; 0; 1⟩= ( √ 2~2)S2; 0; 0⟩−( √ 2~2)S2; 1; 0⟩. We will frequently use these parabolic states instead of spherical states in the following analysis. The dipole-dipole interaction V dip is proportional to ̂ r 1 ⋅̂ r 2 , where ̂ r 1 =r 1 ( x 1 r 1 e x + y 1 r 1 e y + y 1 r 1 e z )⊗I; ̂ r 2 =I⊗r 2 ( x 2 r 2 e x + y 2 r 2 e y + y 2 r 2 e z ); are coordinate operators for atoms 1 and atom 2. Replacing x, y, z with spherical coordinates r, , , x=r sin cos; y =r sin sin; z =r cos: (2.48) 59 gives us V dip ∝̂ r 1 ⋅̂ r 2 =r 1 ( x 1 r 1 e x + y 1 r 1 e y + y 1 r 1 e z )⋅r 2 ( x 2 r 2 e x + y 2 r 2 e y + y 2 r 2 e z ) =r 1 ¾ 4 3 ( e x +ie y √ 2 Y 1 1;−1 + ie y − e x √ 2 Y 1 1;1 +Y 1 1;0 e z )⋅ r 2 ¾ 4 3 ( e x +ie y √ 2 Y 2 1;−1 + ie y − e x √ 2 Y 2 1;1 +Y 2 1;0 e z ) = 4 3 r 1 r 2 (−Y 1 1;−1 Y 2 1;1 −Y 1 1;1 Y 2 1;−1 +Y 1 1;0 Y 2 1;0 ); (2.49) whereY i l;m is a spherical harmonic function for atomi. Generally speaking, for two atoms in the given initial Stark eigenstate, the diagonal terms of the dipole-dipole interaction give an energy shift, while its non-diagonal terms couple adjacentm manifolds with each other: (m;m) to (m±1;m∓1). The Stark states that are most useful for our scheme are those that maximize the energy shift while suppressing the transition between dierent m manifolds (which might introduce decoherence channels). So it is sucient for us to know the value of the matrix elements ⟨n;mS⟨n;m;qSV dip Sn ′ ;m+ 1⟩Sn ′ ;m− 1⟩; ⟨n;mS⟨n;m;qSV dip Sn ′ ;m− 1⟩Sn ′ ;m+ 1⟩; (2.50) and ⟨n;m;qS⟨n;m;qSV dip Sn ′ ;m;q⟩Sn ′ ;m;q⟩: (2.51) For hydrogen, we will only consider matrix elements with the samen for simplicity. Even so, it is too complicated (and unnecessary) to nd the general analytical form of these elements. Instead, we numerically do the integration to determine which states fulll our requirements. By symmetry, the transition strengths of (m;m)→ (m+ 1;m− 1) and (m;m)→ (m− 1;m+ 1) are the same, so we just give the rst value. The calculation results for some matrix elements are shown in Table 2.2 as an example. 60 Matrix element: Sm;q⟩→ Sm ′ ;q ′ ⟩ n =5 n = 10 n = 15 n = 20 n = 25 S0;n− 1⟩S0;n− 1⟩→ S1;n− 2⟩S− 1;n− 2⟩ 112.5 1012.5 3543.75 8550 16875 S0;n− 1⟩S0;n− 1⟩→ S0;n− 1⟩S0;n− 1⟩ 900 18225 99225 324900 810000 S0;n− 1⟩S0;n− 1⟩→ S1;n− 4⟩S− 1;n− 4⟩ 0 0 0 0 0 S0;n− 3⟩S0;n− 3⟩→ S1;n− 2⟩S− 1;n− 4⟩ 137.78 1350 4829.32 11769 23362.4 S0;n− 3⟩S0;n− 3⟩→ S1;n− 4⟩S− 1;n− 2⟩ 137.78 1350 4829.32 11769 23362.4 S0;n− 3⟩S0;n− 3⟩→ S1;n− 4⟩S− 1;n− 4⟩ 168.75 1800 6581.25 16200 32343.7 S0;n− 3⟩S0;n− 3⟩→ S1;n− 2⟩S− 1;n− 2⟩ 112.5 1012.5 3543.75 8550 16875 S0;n− 3⟩S0;n− 3⟩→ S0;n− 3⟩S0;n− 3⟩ 225.01 11025 72900 260100 680625 S1;n− 2⟩S1;n− 2⟩→ S2;n− 3⟩S0;n− 3⟩ 137.78 1350 4829.32 11769 23362.4 S1;n− 2⟩S1;n− 2⟩→ S1;n− 2⟩S1;n− 2⟩ 506.25 14400 85556 291600 743906 S1;n− 4⟩S1;n− 4⟩→ S2;n− 3⟩S0;n− 5⟩ 137.78 1350 4829.32 11769 23362.4 S1;n− 4⟩S1;n− 4⟩→ S2;n− 5⟩S0;n− 3⟩ 168.75 2062.16 7744.14 19281.9 38742.1 S1;n− 4⟩S1;n− 4⟩→ S2;n− 5⟩S0;n− 5⟩ 168.75 2062.16 7744.14 19281.9 38742.1 S1;n− 4⟩S1;n− 4⟩→ S2;n− 3⟩S0;n− 3⟩ 137.78 1350 4829.32 11769 23362.4 S1;n− 4⟩S1;n− 4⟩→ S1;n− 4⟩S1;n− 4⟩ 56.25 8100 61256.3 230400 620156 Table 2.2: The matrix element of operator V dip in atomic units. We have made several simplications in this table. First, we didn't calculate the transition strengths between dierentn manifolds, because they are several times smaller than their counterparts in the same n manifold. Second, we didn't calculate the case when two atoms are initially prepared in dierent Stark eigenstates, especially in two dif- ferent manifolds, which actually we have proposed to realize both the gatee 2 y and qubit measurement. Nevertheless, this table gives enough information about the characteris- tics of the state we are looking for. First, the transition strength must be much smaller than the energy shift inside the same n manifold. Second, those transitions between initial and nal states with a dierence in parabolic number q larger than one must be greatly suppressed. If we prepare an atom in the outermost state Sn;m= 0;q =n− 1⟩, we obtain the largest energy shift with the smallest transition strength to the (m+ 1;m− 1) state compared with other Stark eigenstates in the same manifold. In our proposed gate, since the Rydberg states of two atoms may not be in the same manifold, a natural solution to fulll our requirements is to choose the state S ⟩= Sn;m= 0;q =n− 1⟩Sn ′ ;m= 0;q =n ′ − 1⟩ for manifolds n and n ′ . Note that the energy shift term 61 in the Hamiltonian is a product of operators on two dierent atoms, so the analysis of two atoms in same manifold can be directly applied to show that the maximum energy shift is obtained if the two atoms are prepared in the state S ⟩. This gives an energy shift of ⟨ SV dip S ⟩∝n(n− 1)n ′ (n ′ − 1): Next, we consider the case of non-hydrogen alkali atoms. Physically, the main dif- ference between alkali atoms and hydrogen atoms is that the former have a nite sized ionic core that results in avoided crossings in the Stark spectrum. So, the Hamiltonian can be written as H =− ∇ 2 2 − 1 r +V d (r)+Ez: (2.52) Here, V d (r) is the dierence of the potential function between a hydrogen and an alkali atom, due to the nite-sized ionic core. We treat V d (r) as spherically symmetric and only nonzero near the nucleus. In the case where quantum defects are relatively small (e.g., n is large), we can use the hydrogenic parabolic states as our working basis. For diagonal terms of the Hamiltonian, we have: ⟨n;m;qSHSn;m;q⟩=− 1 2n 2 + 3 2 qnE +⟨n;m;qSV d (r)Sn;m;q⟩+O(E 2 ); (2.53) and we represent the non-diagonal terms using hydrogen spherical states Sn;l;m⟩: ⟨n;m;qSHSn ′ ;m;q ′ ⟩= n−1 Q l;l ′ ⟨n;m;qSn;l;m⟩ ×⟨n;l;mSV d (r)Sn ′ ;l ′ ;m⟩⟨n ′ ;l ′ ;mSn ′ ;m;q ′ ⟩; (2.54) 62 Denote the spherically symmetric eigenstates of the alkali atom as Sn;l;m⟩ al . For large n, we have − 1 2(n− l ) 2 = al ⟨n;l;mSHSn;l;m⟩ al ≈ ⟨n;l;mS− ∇ 2 2 − 1 r +V d (r)Sn;l;m⟩ =− 1 2n 2 +⟨n;l;mSV d (r)Sn;l;m⟩: (2.55) where l is the quantum defect for angular momentum l of the alkali atom [50], which is dierent for each element. For rubidium, if we neglect the ne structure eect, 0 ≈ 3:1, 1 ≈ 2:6, 2 ≈ 1:3, and 3 ≈ 0:02. For l> 3, l ≈ 0. On the other hand, − 1 2(n− l ) 2 ≈− 1 2n 2 − l n 3 : So we have: ⟨n;l;mSV d (r)Sn;l;m⟩≈− l n 3 : (2.56) Per Ref. [56], this expression may be generalized to ⟨n;l;mSV d (r)Sn ′ ;l;m⟩=− l √ n 3 n ′3 : (2.57) Here, we take advantage of the spherical symmetry of V d (r), so the matrix element vanishes when the l value on the two sides of the above equation are not the same. Observe that Eq. (2.51) can be represented as matrix multiplication. We dene matrices [C nm ] ql =⟨n;m;qSn;l;m⟩=c(n;l;m;q); [V nm ] lq =⟨n;l;mSn;m;q⟩=c(n;l;m;q); [D nn ′ ] ll ′ =⟨n;l;mSV d (r)Sn ′ ;l ′ ;m⟩; [S nn ′ m ] qq ′ =⟨n;m;qSHSn ′ ;m;q ′ ⟩: (2.58) 63 Then, we have S nn ′ m =C nm D nn ′ V n ′ m : (2.59) Note that these superscripts label a family of matrics, not matrix element indices. For the purpose of illustration, we show a simple example. We write the matrices C 20 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 √ 2 1 √ 2 1 √ 2 − 1 √ 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ; V 30 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 √ 3 1 √ 3 1 √ 3 1 √ 2 0 − 1 √ 2 1 √ 6 − ¼ 2 3 1 √ 6 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ; D 23 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ − 0 6 √ 6 0 0 0 − 1 6 √ 6 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ; and multiply them to get S 230 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ − 0 36 − 1 12 √ 6 − 0 36 − 0 36 + 1 12 √ 6 − 0 36 + 1 12 √ 6 − 0 36 − 0 36 − 1 12 √ 6 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ : MatrixS nn ′ m can be treated as a submatrix (or block) of the Hamiltonian matrix. Since the Hilbert space is innite-dimensional, we need to truncate the Hamiltonian matrix. For example, if we need to know eigenvalues and eigenstates of manifold n of rubidium, we need to consider only those manifolds that couple to it. We now calculate the energy structure of the n = 15 manifold for the purpose of illustration. We also need to consider the adjacent manifolds such as n = 14, n = 16, n= 17 and n= 18 since states like 16d, 17p, 17d, 18s and 18p are coupled to the n= 15 manifold. Dene the submatrix of the HamiltonianH nn ′ m as [H nn ′ m ] qq ′ = [S nn ′ m ] qq ′− nn ′ qq ′( 1 2n 2 − 3 2 qnE); 64 where, nm is the Kronecker delta. The Hamiltonian matrix is approximately represented as: ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ H 14;14;m H 14;15;m H 14;16;m H 14;17;m H 14;18;m H 15;14;m H 15;15;m H 15;16;m H 15;17;m H 15;18;m H 16;14;m H 16;15;m H 16;16;m H 16;17;m H 16;18;m H 17;14;m H 17;15;m H 17;16;m H 17;17;m H 17;18;m H 18;14;m H 18;15;m H 18;16;m H 18;17;m H 18;18;m ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ; (2.60) which is an 80× 80 Hermitian matrix. Similar methods could be applied to much higher excitation states, like n= 80, that might be more suitable in practice. By diagonalizing the Hamiltonian matrix, we can obtain the spectrum and the eigenstates. As mentioned earlier, the state we use should have a large component of Sn;m= 0;q =n− 1⟩, which means we want the single-atom eigenstate Sr⟩ n;m =Q q C q Sn;m;q⟩ to have m= 0 with a large coecient SC q S for q =n− 1. Fig. 2.19 shows the spectrum around the manifold n = 15. The outermost Stark eigenstates Sr⟩ 15;0 , circled in the gure in the regime where the Stark eigenvalues are roughly linear in the electric eld (E ≤ 2:5× 10 −7 ), are good candidate states, since they should have behavior similar to the hydrogen state Sn;m= 0;q =n− 1⟩. Numerical calculation shows that the diagonal term of the dipole-dipole interaction is about 8×10 4 in Sr⟩ 15;0 Sr⟩ 15;0 (compared to 99225 for hydrogen in the previous table), which is larger than for the other states in the manifold. To put this in the context of our scheme for quantum computation, consider the case where there are 10 3 atoms in a volume of (6m) 3 . The smallest energy shift of a pair of atoms inside the cell is for those that are most distant. Suppose the two most distant atoms are in the state Sr⟩ 15;0 Sr⟩ 15;0 . The energy shift between them should be around 1MHz, depending on the spatial distribution of atoms. In our scheme, where two atoms 65 0.0000000 0.0000005 0.0000010 -0.0025 -0.0024 -0.0023 -0.0022 -0.0021 -0.0020 Energy Electrical Field 15f 18s 16d 17p 18p 17d Figure 2.19: The energy level of rubidium around manifold n= 15, m= 0. The circled states is our candidate states which have large energy shift. may also be in dierent single-atom states (actually in dierent manifolds), we could naturally extend our original analysis, but we expect a similar result. In practice, we would want to use higher energy Rydberg states; for instance an n = 60 state Sm⟩ = Sr⟩ 60;0 for the intermediate state and an n = 70 state Sr⟩ = Sr⟩ 70;0 for the register state, to get both longer lifetimes (more than 300-400s for n = 70 can be achieved in a cryogenic environment [119]) and a stronger dipole-dipole interaction. Since higher energy Rydberg states should behave more like the Rydberg states of hydrogen, we can pick the outermost Stark eigenstates in the linear Stark area for both atoms, whether or not they have the same principal quantum number. If the initial state of one atom is in the outermost state of manifold n= 60, and the other in either manifold n = 60 or n = 70, then by V dip ∝n(n− 1)n ′ (n ′ − 1) the energy shift should be roughly 66 200-300MHz or 300-400MHz, respectively. The distance of the most closely-spaced pair of atoms should be about 10 4 a 0 . This is larger than 2R, whereR= 4900a 0 is the radius of atoms in manifoldn= 70. This spacing should satisfy the assumption of our dipole-dipole interaction model. 67 Part III Ecient Fault-tolerant Quantum Computation: Part I 68 Chapter 3 Fault-tolerant Quantum Computation in Large Block Codes 3.1 Preliminaries In the previous Chapter, we have explored quantum computation at the physical level. In this Chapter and the next few Chapters, we will focus on the micro-architecture of quan- tum computation. More specically, we will focus on FTQC methods [123, 40, 2, 73], which largely determine the structure of quantum computer. Since the various threshold theorems have been established, many fault-tolerant schemes have been proposed, includ- ing those introduced in Refs. [60, 111, 75, 134, 52]. These schemes protect quantum infor- mation from decoherence by encoding it in quantum error-correcting codes [60, 125, 132], and replacing the quantum gates of an ideal quantum circuit by fault-tolerant circuits that implement the desired quantum gates on logical states without ever decoding the quantum information. Most of these schemes fall into two broad categories. In rst category, logical qubits are encoded in separate code blocks, generally constructed by concatenating fairly small one logical qubit codes like Steane code and Bacon-Shor code [7]. The size of these code blocks grows exponentially with the levels of concatenation, but so does their minimum distance; provided that the underlying physical error rate is suciently low (below the error threshold), the probability of an uncorrectable error falls o doubly exponentially. 69 This means that asymptotically the overhead for such schemes scales only modestly with the size of the computation, but for nite sized computation, the overhead for such codes can be extremely large. In the second group of fault-tolerant schemes, logical qubits are encoded into topological codes such as surface codes [73, 115, 52] and color codes [20, 82], which encode multiple logical qubits into one or more code blocks laid out as a lattice (usually two-dimensional) of qubits. The minimum distance of these codes typically scales with the linear size of the lattice. These codes have topological properties that make them robust against local errors. We will go back to topological codes in Part. IV. Currently, all schemes mentioned require relatively low rates of error, though the ability to tolerate errors has slowly been improved by a long string of theoretical works. Most of them also require an extremely large overhead. As a result, a logical qubit is encoded in thousands or tens of thousands of physical qubits [75, 52, 145, 80]. The main object of this thesis is trying to reduce the resource overhead, and thus making FTQC more ecient. From a practical viewpoint, the exact value of threshold may not be too important. We can say that an FTQC scheme is fault-tolerant if the logical error rates for each ele- mentary logical operation in the scheme are suciently low so that a quantum algorithm can be executed with a high probability of success. That is exactly the case for classical computers, where the error rate for each CMOS transistor at each time step is below 10 −18 , and hence makes the computation fault-tolerant without additional error correc- tion process, in general. Typically, a practical quantum algorithm using K qubits and Q elementary steps has KQ ≫ 10 10 , so the logical error rate for each logical operation should be much less than 10 −10 [145]. It was observed by Steane more than a decade ago that multi-qubit block codes can achieve signicantly higher code rates for comparable error protection ability, and hence greatly reduce the overhead. But logical gates in these codes are quite dicult to imple- ment [134, 136]. In this chaper, we propose a scheme that exploits the advantages of 70 Ancilla Factory Memory Array Processor Array M M M M M M P P P P A m A m A m A p A p ...... ...... Figure 3.1: The architecture of our teleporation-based FTQC scheme. multi-qubit block codes. Here, all logical operation are implemented by ancilla prepara- tion, transversal circuit, and bit-wise measurements, which can be inherent fault-tolerant. The main result of this Chapter is rst proposed in Ref. [27]. 3.2 Universal FTQC Similar to von Neumann architecture, our scheme has three components as shown in Fig. 3.1: a memory array of [[n;k;d]] CSS code blocks (C m ) [28, 132] with k ≫ 1; a processor array of an [[n ′ ; 1;d ′ ]] quantum code blocks (C p ) that support a transver- sal T (~8) gate (or other non-Cliord gate); and an ancilla factory that continuously produces a variety of fresh logical ancillas for error correction, teleportation, and logi- cal operator measurement. Another feature of our scheme is that magic state distilla- tion [52], which usually dominates the overhead of an FTQC scheme, is not required as in Refs. [105, 68, 5]. Quantum information is stored in the memory array, and error corrections using Steane syndrome extraction [133] are constantly performed. Logical Cliord operations can be implemented by measuring sequences of logical operators on the C m code blocks in the memory array. We will show that measuring logical Pauli operators of C m can be combined with error correction if some particular ancilla states are available. To implement a logical T gate on a particular logical qubit, that logical 71 qubit will be teleported to a C p code block, where a transversal T gate is performed, and then teleported back to its original memory block. Error corrections between these operations can be performed if necessary. Again, provided with some particular ancilla states, it is possible to simultaneously measure the logical operators and implement logi- cal teleportation betweenC m andC p code blocks. Thus, universal quantum computation is achieved. Also, this scheme needs only ancilla preparation, transversal circuits, and single-qubit measurements, and thus it is intrinsically fault-tolerant. Details of these logical operations will be given below. It is evident that a large number of clean ancillas of various types are required in this scheme. Fortunately, these logical ancillas are stabilizer states. They can be prepared by using quantum circuits of Cliord gates only and then distilled [81]. The distillation procedure is more like entanglement distillation [12] than magic state distillation, and has some advantages. For magic state distillation, there is a probability of failure, where everything has to be discarded, and it may need several iterations; while stabilizer states can be measured, and logical errors, if detected, can be corrected. We will not go through the details of this logical ancilla distillation in this Chapter, but simply assume that we have an ancilla factory capable of preparing all the ancillas with high delity. 3.3 Steane Syndrome Extraction First let us brie y review the Steane syndrome extraction, which leads to the other operations. It is used to measure the stabilizer generators for the [[n;k;d]] CSS code C m (similarly forC p if necessary) in our scheme. The procedure is as follows: 1. Prepare two ancilla states in the same codeC m (orC p ) where all logical qubits are set to the states S0⟩ ⊗k L and S+⟩ ⊗k L (called the X and Z ancillas respectively), for Z and X error syndrome measurements, respectively. 2. Perform a transversal CNOT from the information block to the Z ancilla. 3. Perform a transversal CNOT from the X ancilla to the information block. 72 4. Do single-qubit measurements on theX andZ ancilla qubits in theX andZ basis, respectively. Collecting the measurement outcomes and multiplying the correct subset of±1 results together reveals the eigenvalue of each stabilizer generator, and hence the error syndrome. This procedure is shown as the left circuit in Fig. 3.2. E 1 1 E 1 2 E 1 3 E 2 1 E 2 2 E 2 3 E 3 1 E E 3 2 E 3 3 data qubit Z ancilla X ancilla E i E f Mz Mz Mx Mx Figure 3.2: The circuit for Steane syndrome extraction and its eective error model. Measuring logical operators.{ Steane has shown that measuring a logical X u or Z u can be combined with the recovery operation in Steane syndrome extraction [134], where u is a binary k-tuple indicating which logical qubits are operated. If logical X u is to be measured, the prepared X ancilla S0⟩ L at step 1 is replaced with S0⟩ L + Su⟩ L , which is also a stabilizer state, and the rest of the steps are the same as before. As an example, suppose we wish to measure X i X j . Then the logical qubits i and j of the X ancilla are prepared in the state S + ⟩ L = 1 √ 2 (S0 i 0 j ⟩ L +S1 i 1 j ⟩ L ); (3.1) which is a joint+1 eigenstate of X i X j and Z i Z j , and the other logical qubits are prepared in the state S0⟩ L . This logical operator measurement is protected by the classical error- correcting code from whichC m (orC p ) is built up. Next we generalize this method to products of logical X and Z operators. To illus- trate, we show how to measure logical operators of the form X i Z j on logical qubitsi and j. This will allow us to do any Cliord gate as we shall see. In this case, if i ≠j, the X and Z ancillas at step 1 are prepared in a particular entangled logical state: logical 73 qubiti of theX ancilla and logical qubitj of theZ ancilla are prepared in the entangled state S ij ⟩ L = 1~2(S0 i 0 j ⟩ L +S0 i 1 j ⟩ L +S1 i 0 j ⟩ L −S1 i 1 j ⟩ L ); (3.2) which is the joint +1 eigenstate of X i Z j and Z i X j , while the other logical qubits of the X or Z ancillas are prepared in the state S0⟩ L or S+⟩ L , respectively. If i = j, we need to prepare the ancilla as a joint +1 eigenstate of Y i Z i and Z i X i . Again, these joint 2n-qubit states are a stabilizer states. It is similarly possible to measure any logical operator X u Z v , by preparing more complicated ancilla states. It is important to emphasize that these measurements are combined with error correction, as in the original Steane syndrome extraction. 3.4 Logical Teleporataion We show that logical qubits can be teleported between arbitrary code blocks (ofC m orC p ). To perform a non-Cliord quantum gate on a logical qubit (or qubits) of a C m memory block, the target logical qubit is teleported to aC p processor block that allows the non- Cliord gate to be implemented transversally. One can think of the two code blocks as a part of a larger code. Suppose the logical qubits ofC m have associated with them pairs of logical operators ( X 1 , Z 1 ), ( X 2 , Z 2 ),..., ( X k , Z k ), which are all Pauli operators. We reserve logical qubit 1 as a buer qubit used in teleporting qubits. Suppose the logical operators of the [[n ′ ; 1;d ′ ]] code C p are labeled as ( X 0 , Z 0 ). Here is the procedure to teleport logical qubit j from the storage block to the processing block: 1. Measure the operators X 0 X 1 and Z 0 Z 1 . This prepares a logical Bell state between the processor block and the buer qubit. 2. Measure the operators X 1 X j and Z 1 Z j . This does a logical Bell measurement on the buer qubit and qubit j, and teleports qubit j to the processor block. 74 3. If necessary, apply a logical Pauli operator to the processor block to correct the state. One would generally need to do correction before applying the non-Cliord gate. 4. Apply the non-Cliord gate by a transversal circuit on the processor block. 5. Measure the operators X 0 X 1 and Z 0 Z 1 . This prepares a logical Bell measurement on the processor block and the buer qubit, and teleports the transformed qubit back to logical qubit j of the memory block. 6. If desired, apply a logical Pauli operator to the memory block to correct. Unified Block T M X X 0 1 P M M Z Z 0 1 M X X 1 i M Z Z 1 i M X X 0 1 M Z Z 0 1 Figure 3.3: Diagram of the logical T gate using logical state teleportation. The red blocks represent joint measurements of logical qubits, and the blue one represents bitwise T or T applied to the processor block. The procedure is illustrated in Fig. 3.3 for a logical T gate. The steps of measur- ing logical operators are similar to what was described previously, except that the X and Z ancillas are prepared in a logical entangled state between C m and C p . Logical teleportation can also be applied, of course, to move logical qubits between memory blocks. 3.5 Logical Cliord Gates Cliord gates can be performed within a memory block solely by measuring logical operators, which is like a kind of simplied logical teleportation [59]. We demonstrate how to perform logical Hadamard gate, logical CNOT gate, Phase gate and the SWAP gate. 75 y y H (a) 0 qubit 0 qubit i M X Z 0 i M x i ± (b) y qubit 0 M X X 0 j M Z i Z j 0 qubit i M x i qubit j y j i C N O T ® ± Figure 3.4: (a) Hadamard. (b) CNOT from qubiti toj. The red blocks represent joint measurements of logical qubits. Suppose we wish to perform a logical Hadamard gate on logical qubiti of aC m block in the state S ⟩ L . Logical qubit 0 is reserved as a buer qubit in the state S0⟩ L . We do the following two measurements: 1. Preparing a buer S0⟩ as qubit 0. Measure X 0 Z i ; 2. Measure X i . Logical qubit 0 will be left in the state HS ⟩ L , up to a Pauli correction X 0 , Y 0 or Z 0 on logical qubit 0, according to the measurement outcomes. Logical qubit i will be left in the state S+⟩ L or S−⟩ L and can be reset to S0⟩ as a new buer qubit, if desired, by measuring Z i and applying an X i correction if necessary. A CNOT can be performed similarly. Suppose we wish to perform a CNOT from logical qubits i toj of aC m block, and again, suppose logical qubit 0 is a buer qubit in state S0⟩ L . Here is the procedure: 1. Preparing a buer S0⟩ as qubit 0. Measure X 0 X j ; 2. Measure Z i Z j ; 3. Measure X i . This does a CNOT from logical qubits i to j (up to a Pauli correction), shifts them to logical qubits j and 0, respectively, and moves the buer qubit to qubit i in the state S±⟩ L . 76 Similar schemes can be done for the Phase and SWAP gates. As shown in Fig. 3.5(a), assuming qubit 1 is in state S ⟩, the phase gate can be realized by: 1. Preparing a buer state S0⟩ as qubit 0; 2. measuring X 0 Y 1 followed by measuring Z i . It will leave the state as SS ⟩S0⟩ up to a Pauli operator correction. For the SWAP gate, we prepare a buer state initially in S0⟩, as shown in Fig. 3.5(b). To SWAP S 1 ⟩S 2 ⟩, we can measure X 0 X 1 X 2 followed by Z 0 Z 1 Z 2 , followed by X 0 . This will leave the state as S+⟩S 2 ⟩S 1 ⟩ up to a Pauli operator correction. The buer qubit can be reset to S0⟩ by measuring Z 0 . y y y S y 0 (a) M X X X 0 1 2 M z 0 (b) 0 qubit 1 qubit 2 qubit 0 1 y 2 y 0 2 y 2 y 1 y 0 qubit 0 qubit 1 M X Y 0 1 M z 1 M Z Z Z 0 1 2 M x 0 0 Figure 3.5: (a) Phase gate. (b) SWAP gate. The red blocks represent joint measure- ments of logical qubits. We can build any Cliord unitaries from Hadamard, Phase, CNOT, and SWAPs. But a complicated Cliord unitary can also be done directly by measuring more complicated combinations of logical operators. Each of these measurements requires the preparation of a particular ancilla state. Therefore a tradeo exists between the eciency of enabling a larger set of possible Cliord operations and the complexity of having to prepare and distribute more kinds of ancillas. Note also that it is not necessary to apply the Pauli corrections; we can just keep track of them and how they are transformed by Cliord unitaries (the `Pauli frame'). But if we do wish to correct, that is a transversal operation as well. 77 3.6 Error Model Steane syndrome extraction and its variations are used throughout the scheme. There are at least four kinds of errors in this scheme: memory errors in the code blocks, phys- ical gate errors, faulty ancilla preparation, and measurement errors. We model errors in physical gates, ancilla preparations, and measurements by treating them as perfect oper- ations followed or preceded by Pauli errors. In this Chapter, we represent the physical noise model as depolarizing errors. At each time step, every physical qubit indepen- dently undergoes a Pauli error X, Y or Z with probability ~3 or remains unchanged with probability 1−, where is called the memory error rate. We treat ancilla prepa- ration as perfect followed by each qubit of an ancilla block independently suering the same depolarizing errors with rate r afterwards. Similarly, we treat each single-qubit gate as perfect, followed by a depolarizing error with rate p g 1 for single-qubit gates. For a two-qubit gate, it is modeled as a perfect gate followed by one of the 15 possible single- or two-qubit error from IX, IY , IZ, XI, XX, XY , XZ, YI, YX, YY , YZ, ZI, ZX, ZY , andZZ with equal probability p g 2 ~15, or no error with probability 1−p g 2 . Finally, the measurement of a single physical qubit has a classical bit- ip error with probability p m (or equivalently, an X or Z error preceding a measurement in the Z or X basis, respectively). Note that we do not expect the form of the errors to greatly aect the performance; but the assumption of independence across qubits is very important. The measurement outcomes in Steane syndrome extraction can be erroneous due to either imperfect measurements or errors during the circuit. This makes error analysis dicult. Traditionally this is handled by repeated syndrome measurements. Herein we show that syndrome measurement can be done in a single shot. Actually, all errors during the syndrome extraction process can be mapped to errors occurring on the data qubits before and after the process, so that the ancillas, gates and qubit measurements in the circuit can be regarded as error-free, as illustrated in Fig. 3.2. This is formally stated as the following theorem. 78 Theorem 1. (Eective error) During the process of imperfect Steane syndrome extrac- tion and its variations, if errors in the same block (memory, processor, or ancilla) are uncorrelated, then the errors are equivalent to eective errors acting only on the data qubits before and after the process. Especially, if we assume =p m =p g 1 =p g 2 ≜p, the eective model on the data block at each time step can be approximated by E tot []≈ (1− 11p)+ 71 15 pXX+ 71 15 pZZ+ 23 15 pYY: (3.3) Proof. There are three types of errors introduced by imperfect circuits in syndrome extraction and logical state measurement in our scheme. These are: measurement errors, gate errors, and preparation errors. Theorem 1 states that it is possible to replace these errors with equivalent errors before and after the circuit if they are all independent in the same block. We can then use these equivalent error processes as our error model and treat the circuit as being ideal. We treat these errors one at a time. Every error in a Z measurement is equivalent to a single X error followed by a perfect measurement, while every X measurement error can be modeled by a single Z error followed by a perfect measurement. In Fig. 3.6, we can see that an error in the Z measurement is equivalent to twoX errors, before and after the circuit, on the codeword qubit. An error in an X measurement has a similar eect. M z M x X M z M x X X M z M x Z M z M x ( a ) ( b ) Z Z Figure 3.6: The eect of measurement errors on the ancilla qubit can be eectively replaced by errors before and after the circuit. 79 We treat a noisy gate as a perfect gate followed by an error. We treat the rst CNOT explicitly here, and the situation for the second is just the same. An error in the CNOT gate is represented as a tensor product of two operators from {I;X;Y;Z}. Each such operator can be equivalently replaced by errors before and after the circuit. See panels (a) to (f) in Fig. 3.7. Figure 3.7: The eect of gate errors (from (a) to (f)) on the rst CNOT, and ancilla preparation errors (from (g) to (l)). Both types of errors can be replaced by errors before and after the circuit. Preparation errors include errors in initializing the ancilla blocks, and any memory or transport errors that occur in storing or distributing them. The replacement of preparation errors on each qubit of the two ancilla blocks is shown in panels (g) to (l) in Fig. 3.7. 80 From the argument above, we see that we can replace the noisy circuit with a perfect circuit preceded or followed by a noisy process on the codeword qubits at a single time step. Note that the errors before and after a circuit are correlated. It may be possible to use this correlation to improve the estimation of the error, based on the entire time record of syndrome measurements. For now, we ignore this in nding the error process for a single time step. If there are also memory errors E m on qubits in codeword block, then the error process before the circuit isE tot =E i ○E m ○E f as in Fig. 3.8. Note thatE f here is the from previous circuit, and E m is the memory error. For simplicity, suppose that all errors are Pauli errors, and deneI[]=; X[]=XX; Z[]=ZZ; Y[]=YY . Since we model memory errors as depolarizing errors,E m = (1−)I+ 1 3 (X +Y+Z). E f Z ancilla X ancilla E i E f Mz Mx E m Figure 3.8: Eective error model in a single time step of circuit can be derived as follows: E f = [(1−p m )I+p m X]○[(1−p m )I+p m Z] ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ measurement ○1− 8 15 p g 2 I+ 8 15 p g 2 X ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ rst CNOT ○1− 12 15 p g 2 I+ 4 15 p g 2 (X +Y+Z) ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ second CNOT ○1− 2 3 rI+ 2 3 rX○(1−r)I+ 1 3 r(X +Y+Z) ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ ancilla preparation : (3.4) 81 Similarly, we could haveE i as: E i = [(1−p m )I+p m X]○[(1−p m )I+p m Z] ○1− 12 15 p g 2 I+ 4 15 p g 2 (X +Y+Z)○1− 8 15 p g 2 I+ 8 15 p g 2 Z ○(1−r)I+ 1 3 r(X +Y+Z)○(1− 2 3 r)I+ 2 3 rZ: (3.5) Thus, the entire noise process can be represented as: E tot ≡E i ○E m ○E f = 1 3 + 4 3 r+ 2p m + 16 15 p g 2 X + 1 3 + 4 3 r+ 2p m + 16 15 p g 2 Z+ 1 3 + 2 3 r+ 8 15 p g 2 Y +1−+ 10 3 r+ 4p m + 8 3 p g 2 I+Omax{ 2 ;r 2 ;p 2 m ;p 2 g 2 }: (3.6) If we set =p m =r =p g 2 =p,E total can be approximated byE total ≈ (1− 11p)I+ 71 15 pX + 71 15 pZ+ 23 15 pY. For non-Pauli errors, we can apply a similar trick by expanding the noise process in the Pauli basis and translating the errors term by term. In that case, each term also has a phase, which may depend on the measurement outcomes. This makes analysis more complicated but not really dierent in principle, especially since the syndrome measurements will tend to project into the Pauli basis. This theorem is applicable to quite general independent noise models and not just the depolarization channel we focus on in this Chapter. Consequently we only have to decode the eective error on the data qubits at each syndrome measurement. Since syndromes are measured at every step of the process, this avoids repeated syndrome measurement and greatly reduces the time overhead for syndrome extraction and implementing logical gates. On the other hand, it can also eliminate potential errors caused by repeated mea- surements, which is usually used to establish reliable syndromes in FTQC schemes. The estimated eective error will be a Pauli operator and can either be corrected instantly, 82 or kept track of in the cumulative Pauli frame on the data qubits. After every round, there will generally be a residual error that has not yet been detected, and that diers from the current error estimate. However, this is not a problem so long as the weight of this residual error is always small compared to the distance of the code. At the same time, it greatly simplies the analysis of error propagation. For simplicity, in the rest of the Chapter, we choose the error model to be a depolar- izing error with ratep e = (71~5)p in the following simulation, wherep is the underlying physical error rate. 3.7 Estimate of the Logical Error Rate The error rate for each logical step is determined by the failure rate of decoding for the eective error process. The performance and the physical resources of our scheme will depend heavily on the choice of quantum codes for the memory and processor blocks. We discuss the decoding process for each block in this section. Numerical analysis of logical error rate for each logical operation is also given. 3.7.1 Decoding of Stabilizer Codes The n−qubit Pauli group G n is the set of all operators of the form i k O 1 ⊗O 2 ⊗⋯⊗O n ; (3.7) where O j ∈ {I;X;Y;Z} for all j. We ignore the overall phase i k in this section, which represents a global phase. In general, for each [[n;k;d]] stabilizer code with stabilizer group S, we can always do encoding operation from canonical code S ⟩ = S0⟩ ⊗n−k S ⟩; where S ⟩ is a k−qubit state is the quantum state to be protected and used in the computation. This is a stabilizer code, whose logical operators are X i and Z i for i =n−k+ 1;:::;n and whose stabilizer generators are Z i for i = 1;:::;n−k. We can take the symplectic partners to be X i for 83 i= 1;:::;n−k. This is the \code" that we start with before the encoding operation. The encoding ofk qubits inton qubits can be specied by a unitary operationU E ∈C n , where C n is the normalizer of the Pauli group G n in U(2 n ) (Cliord group). Note that given a code, the encoding circuit is not unique. Here we choose a particular U E . Under the encoding operation, the rstn−k Z operators are mapped ton−k stabilizer generators with relationS i =U E Z i U E , whereas the images of theX operators acting on those qubits, T i =U E X i U E , are called pure errors or symplectic partners ofS i . The image of the Pauli operators acting on the last k qubits are logical Pauli operators X i = U E X n−k+i U E , Z i =U E Z n−k+i U E . Given a Pauli error E ∈ G n , we can always nd the corresponding syndromes (assuming no measurement errors), so we dene the syndrome extraction function S ∶G n → {−1; 1} n−k , whereS(E) gives the syndromes of errorE . We can also dene the following function: T ∶G n →G n , to represent pure error string. T (E) is uniquely deter- mined byE 's syndrome s ≡ S(E) = {s 1 ;s 2 :::s n−k }, and can be explicitly represented as: T (E)=U E n−k M i=1 X − s i −1 2 i U E ; (3.8) which is a product of pure errors. Dene another function: L∶G n →G n ; E ↦ET (E); (3.9) whereL(E) is a product of logical Pauli operators in the normalizer of stabilizer group N(S). It is easy to observe that theE can be uniquely decomposed as: E =L(E)T (E): (3.10) 84 Last, we dene a function which truncates the last k single Pauli operators of a length n Pauli operators string: L ∶G n →G k ; E ↦E n−k+1 ⋯E n (3.11) The images of the last k X and Z Pauli operators of the canonical code correspond to the logical operators of the code. If we run back the encoding circuit,L U E L(E)U E should indicate of logical errors actually happen according toE . Note that if forE and E ′ , the corresponding L and L ′ are dierent up to multiplication by some element in the stabilizer group, they should be regarded as equivalent. So when decoding, the only thing that matters is to estimate an equivalence classL containing all equivalentL. Each equivalence classL corresponds to a uniqueL∈G k , which is what logical errorL stands for, and can be represented as L=L U E LU E for anyL∈L. So the optimal decoding problem can be transformed to a maximum a posterior (MAP) probability problem: ^ L= arg max L P(LS s): (3.12) For small quantum codes concatenated together (like the concatenated [[15; 1; 3]] code that is used in our scheme), an optimal and ecient decoder called soft decision decoder does exist [110]. Consider an l−level concatenation code, and let the jth level be an [[n j ;k j ;d j ]] code with k j = 1 for j ≠ 1. The parameters of the corresponding concatenated code are [[∏ l j=1 n j ;k 1 ;∏ l j=1 d j ]]. Denes (i) j ∈ {−1; 1} n j −k j be the syndromes ofith block of thejth concatenation layer after syndrome extraction . Denote s (i) j be the collection of syndromes whose stabilizer generators act nontrivially on all physical qubits associated to the ith block of the jth concatenation layer. In other words, these sets of syndromes can be dened as: s (i) j = s (i) j ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ in j p=in j −i+1 s (p) j+1 ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ . Note that s (i) l =s (i) l are the syndromes of codes in bottom level. At last, denote s j = ∏ j−1 q=1 nq i=1 s (i) j to be the collection of all the syndromes from the layers j tol. Then it is easy to see that s 1 is the set of all 85 syndromes according to the concatenated code's stabilizer generators. Estimating L for the concatenated code is equivalent to estimating L 1 for the code at the top level. The decoding is equivalent to nding arg max L 1 P(L 1 Ss 1 ). Dene L 2 = (L (1) 2 :::L (n 1 ) 2 ) to be an array of logical operators at the second level. Then this probability can be factorized by conditioning on L 2 , P(L 1 S s 1 )=Q L 2 P(L 1 S s 1 ;L 2 )P(L 2 S s 1 ) =Q L 2 [L 1 =L 1 U 1 E L 2 U 1 E ] [s 1 =S 1 (L 2 )] P(s 1 S s 2 ) n M i=1 P(L (i) 2 S s (i) 2 ); (3.13) where L j , L j , U j E and S j are L, L , U E and S corresponding to the code at the jth level, and (⋅) is the indicator function. The derivation repeatedly uses the Bayes rule and the following facts: a) The syndromes and logical errors of level j are determined by the logical errors of layerj+ 1. b) The channel is not correlated (or more specically, that the error model can only correlate qubits in the same block but not across dierent blocks). The optimal decoding is reduced to a sum-product problem on a tree which can be exactly and eciently solved using message passing algorithm. If the estimated logical error is ^ L 1 =L(U E L(E)U E ), the decoding is a success, otherwise it is a failure. The error at the physical level then can be estimated as: ^ E =U E I ⊗n−k ⊗ ^ L 1 U E T (E); (3.14) which is used for actual correction at the physical level. In general, the optimal decoding problem is NP hard [14]. In practice, one must use codes with some structure (like the ones chosen for the memory block) so that an ecient hard decision decoding algorithm provides an error estimate ^ E based on the assumption that errors are all independent and the same type of error is equally likely to happen to dierent qubits. Ecient hard decision decoders for quantum BCH codes and the quantum Golay exist. The Golay code can be decoded up to its correctability 86 t= ⌊ d−1 2 ⌋= 3 using the Kasami error-trapping decoder. The BCH codes can be decoded using the Berlekamp-Massey algorithm . We can decompose ^ E into ^ E =L( ^ E)T s , where T s =T (E) =T ( ^ E). If ^ L =L U E L( ^ E)U E is equal to L =L U E L(E)U E , we declare that the error correction is correct, otherwise, we declare that a failure. To determine whether a correction of a random Pauli error is correct, we need to run the encoding circuit backward. So rst of all, we need to nd the encoding circuit of a stabilizer code. Using the symplectic form, this can be done in the following way. Let the symplectic matrix representation of the n−k stabilizer generators and logical operators be M. A Cliord circuit is an automorphism of Pauli group G n , which can be represented as a linear map that acts on the matrix: M→ M ′ = MC; (3.15) where C is a nonsingular 2n× 2n binary matrix representing the action of the Cliord circuit. Since C represents a unitary transformation, it must preserve the commutation relations of the operators it transforms; this restriction corresponds to the constraint CJC T = J with J= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 I I 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ : (3.16) Note that the n+k rows of the matrix M do not form a full basis. This re ects the fact that n+k Pauli operators they represent do not form a full set of generators. We can supplement them by adding an additional n−k symplectic partners of the stabilizer generators logical operators to M to form a new full rank matrix M. The symplectic partner T i of S i should satisfy [T i ;S j ]= 0 for i≠j; {T i ;S i }= 0: (3.17) 87 Start from the canonical code we described before, and dene the matrix M 0 to represent this \code". We use the following order for the operators: X n ;:::;X 1 ; Z n ;:::Z 1 . This gives us the very simple matrix M 0 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ I n−k 0 0 0 0 I k 0 0 0 0 I n−k 0 0 0 0 I k ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ≡ I 2n (3.18) where the rstn−k rows are the symplectic partners, the next k rows are the logicalX operators, the next n−k rows are the stabilizer generators and the last k rows are the logical Z operators. From this code, we produce the [[n;k;d]] code we are interested in by applying an encoding circuit U E , which has representation C E : M= M 0 C E = C E : (3.19) In other words, if we know all logical operators and symplectic partners of the stabilizer code, we can build the encoding matrix of that code directly. Suppose we know all of the stabilizer generators and logicalZ operators and X l , forl= 1;⋯i, we can recursively nd X i+1 by solving the following linear equation : X t i ⋯ X t 1 S t n+k ⋯ S t 1 Z T k ⋯ Z T 1 t ⊙ X i+1 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 ⋯ 0 ´¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¶ i S 0 ⋯ 0 ´¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¶ n+k S 0 ⋯ 0 ´¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¶ k−i−1 1 0 ⋯ 0 ´¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¶ i ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ t ; (3.20) 88 where ⊙ is the symplectic inner product between binary strings. Similarly, if we nd all S i , Z i , X i and T l for l= 1⋯i, we could nd T i+1 by solving the following equation: T t i ⋯ T t 1 X t k ⋯ X t 1 S t n+k ⋯ S t 1 Z t k ⋯ Z t 1 t ⊙T i+1 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 ⋯ 0 ´¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¶ i S 0 ⋯ 0 ´¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¶ k S 0 ⋯ 0 ´¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¶ n+k−i−1 1 0 ⋯ 0 ´¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¶ i S 0 ⋯ 0 ´¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¶ k ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ t : (3.21) 3.7.2 Quantum BCH Codes and Golay Codes For memory blocks, desirable properties include: 1) high distance, 2) good code rate, and 3) an ecient decoding algorithm. As preliminary research, we study three large block codes obtained by concatenating a medium-sized block code with a high-distance single- qubit code. By concatenating the [[89; 23; 9]], [[127; 57; 11]], and [[255; 143; 15]] quan- tum BCH codes [62] with the [[23; 1; 7]] quantum Golay code, we obtain CSS codes with parameters [[2047; 23; 63]], [[2921; 57; 77]] and [[5865; 143; 105]], respectively. Since the size of a concatenated code is the product of the code sizes at each layer, choosing codes that are not too big is important if one wishes to concatenate them. The Golay code and the BCH codes [36] are well-known classical cyclic codes due to their remarkable alge- braic structures and the ability to decode multiple errors that is especially useful when the size of the code is small. The scheme we consider for protecting the memory blocks against error is to use two-layer concatenation of a quantum BCH code at the top layer and the [[23; 1; 7]] quantum Golay code at the bottom layer. The quantum codes that we consider here are derived from their classical counterparts by the CSS construction. We pick the following three quantum BCH codes constructed from self-dual classical BCH codes that have reasonable code lengths, good code rates and good code distances: 1) [[89; 23; 9]] quantum BCH code, which is derived from the [89; 56; 9] classical BCH code. The generator polynomial of the classical code is X 33 +X 30 +X 27 +X 26 +X 25 + X 24 +X 22 +X 21 +X 20 +X 16 +X 15 +X 14 +X 11 +X 10 +X 9 +X 6 +X 3 +X 2 + 1. 89 2) [[127; 57; 11]] quantum BCH code, which is derived from the [127; 92; 11] classical BCH code. The generator polynomial of the classical code is X 35 +X 34 +X 33 +X 28 + X 24 +X 23 +X 22 +X 19 +X 17 +X 15 +X 12 +X 11 +X 9 +X 8 +X 6 +X 4 +X 2 +X+ 1. 3) [[255; 143; 15]] quantum BCH code, which is derived from the [255; 199; 15] classical BCH code. The generator polynomial of the classical code is X 56 +X 51 +X 50 +X 49 + X 46 +X 43 +X 41 +X 40 +X 39 +X 34 +X 30 +X 26 +X 25 +X 24 +X 22 +X 20 +X 17 +X 16 + X 11 +X 10 +X 8 +X 7 +X 4 +X 3 +X 2 +X+ 1. Note that the [[23; 1; 7]] quantum Golay code is derived from the [23,12,7] classical Golay code. The generator polynomial of the classical code is X 11 +X 10 +X 6 +X 5 + X 4 +X 2 + 1. In all four cases, logical Z i operators can be obtained by shifting the corresponding generator polynomials for classical codes. All three block codes on average encode a single logical qubit in less than 100 physical qubits. The Golay code is decoded using the Kasami error-trapping decoder [70], and the BCH codes are decoded using the Berlekamp-Massey algorithm [13, 95]. Fig. 3.9 shows the simulation of logical error rate for a memory block using Monte-Carlo simulation. However, the simulation complexity is too high to go beyond p e ≲ 10 −2 , even with the Titan supercomputing resource. Thus, we use linear extrapolation to estimate that region, and nd that atp e = 0:007 (corresponding top= 5× 10 −4 ) the logical error rates are less than 10 −15 for all the three codes. We can also derive an upper bound on the expected logical error rate of an [[n;k;d]] code. Since all errors of weight up tot= ⌊ d−1 2 ⌋ can be corrected, we pessimistically assume that any error of higher weight would lead to a logical error. The approximate logical error probability isP n (p)=∑ n w=t+1 n w p w (1−p) n−w . At eective error ratep e = 0:007, we getP 89 (P 23 (p e ))= 1×10 −16 ,P 127 (P 23 (p e ))= 2:5×10 −19 , andP 255 (P 23 (p e ))= 7×10 −24 , which agree to the simulation results. We see that the [[5865; 143; 105]] code stands out 90 0.0010 0.0020 0.0030 0.0015 10 -23 10 -19 10 -15 10 -11 10 -7 0.001 p Block error rate Figure 3.9: The logical error rate of the memory blocks for the [[2047,23,77]] code (blue), [[2921,57,77]] code (red) and [[5865,143,105]] code (green) versus physical error ratep. The number of samples for each point is up to 4×10 8 . The dashed lines are from extrapolation of linear tting. because of its high code rate and extremely low logical error rate, making it a very promising code in practice. 3.7.3 [[15;1;3]] Reed-Muller Code For the processor block, we need a CSS code that allows a transversal non-Cliord gate such as the concatenated [[15; 1; 3]] shortened Reed-Muller code [77] or the 3D gauge color code [18]. Here we simulate the concatenated [[15; 1; 3]] code, which allows a transversal T gate. This code has the ability to correct almost all bit- ip errors when p e is small. The concatenated [[15; 1; 3]] code is constructed from the truncated classical Reed-Muller code. Classical Reed-Muller codes are weakly self-dual codes with simple and good structural properties [94]. A Reed-Muller code has two parameters r;m, 0 ≤ r ≤ m, 91 and is denoted by RM(r;m). This code is of length 2 m and r is called its order. Let C =RM(1; 4) with parameters [16; 5; 8]. Consider the following (m+ 1) 2 m -tuples: v 0 = 1 = [ 1111 ⋯ 1111 1111 ⋯ 1111 ]; v 1 = [ 0101 ⋯ 0101 0101 ⋯ 0101 ]; v 2 = [ 0011 ⋯ 0011 0011 ⋯ 0011 ]; ⋮ ⋮ ⋱ ⋮ ⋮ ⋱ ⋮ v m = [ 0000 ⋯ 0000 1111 ⋯ 1111 ]; C is generated by v 0 , v 1 , v 2 , v 3 and v 4 . The codewords of C have weight divisible by 8. Let C ′ be the code of C shortened by deleting the rst bit. Then its codewords have weight either 0 or 7 mod 8. Let v ′ j be the punctured generator by deleting the rst bit of v j , and then C ′ is a [15; 5; 7] code generated by v ′ 0 , v ′ 1 , v ′ 2 , v ′ 3 and v ′ 4 . Let C ′ 0 be the even subcode of C ′ generated by v ′ 1 , v ′ 2 , v ′ 3 , v ′ 4 . Note that v 1 , v 2 , v 3 and v 4 have leading bit 0, but v 0 has a leading bit 1. C ′ 0 has parameters [15; 4; 8] and its codewords have weight divisible by 8. The [[15; 1; 3]] quantum code is encoded as following: S0⟩ L = Q u∈C ′ 0 Su⟩; and S1⟩ L = Q u∈C ′ 0 Su+v ′ 0 ⟩: (3.22) The corresponding parity check matrix for the [[15; 1; 3]] code is ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 000000011111111 000111100001111 011001100110011 101010101010101 000000011111111 000111100001111 011001100110011 101010101010101 000000000001111 000000000110011 000000001010101 000001100000011 000010100000101 001000100010001 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ; 92 which is asymmetric between X and Z. Then T ⊗n Sx⟩ L = Q u∈C ′ 0 T ⊗n Su+xv ′ 0 ⟩= Q u∈C ′ 0 e −i wt(u+xv ′ 0 ) 4 Su+xv ′ 0 ⟩ =e −i 7x 4 Q u∈C ′ 0 Su+xv ′ 0 ⟩=e −i 7x 4 Sx⟩ L ; (3.23) which implements the logical T gate. The third equality follows because for any v ∈C ′ 0 , wt(v) = 0 mod 8, while for any v ∈ C ′ 0 +v 0 , wt(v) = 7 mod 8. This code has a 0.00100 0.00050 0.00200 0.00030 0.00150 0.00070 10 -14 10 -11 10 -8 10 -5 0.01 p Block error rate Figure 3.10: The logical error rate of the logical T gate performed on the concatenated [[15; 1; 3]] code of two levels (blue) and three levels (red) using up to 3× 10 7 samples for each point. The green point represents a numerical upper bound for three levels when the physical error rate is p= 7× 10 −4 . The dashed lines are from extrapolation of linear tting. particularly large ability to correct almost all bit- ip errors when p e is small. Monte- Carlo simulation of two and three level concatenation of [[15; 1; 3]] using soft decision decoder has been shown in Fig. 3.10. The simulation of 3-level concatenation using the soft-decision decoder is quite time consuming. It was implemented on the Titan supercomputer at Oak Ridge National Lab. We use 10 6 samples for p e = 0:04, 10 7 samples for p e = 0:03, 3× 10 7 samples for p e = 0:02 and 3× 10 7 sample for p e = 0:015. 93 There exists an optimal ecient soft-decision decoder for concatenated codes [110], which diagnoses the error syndromes using a message-passing algorithm [92]. We per- formed Monte-Carlo simulations of the concatenated [[15; 1; 3]] code of two and three levels with the soft-decision decoder. The results are plotted in Fig. 3.10. Note that for three levels of concatenation, the logical error rate drops to less than 2× 10 −12 at p e = 0:007 (by extrapolation). Hence, when the physical error rate is less than 5× 10 −4 , the logical error rate is below 2× 10 −12 for a single round of syndrome extraction and all logical gates. In this case, the error rate for each logical operation is well below 10 −10 , which will allow interesting quantum algorithms that are impossible to run on classical computers. Nor do we have any reason to believe this is optimal: in all likelihood, better codes exist for both the storage and processor blocks. Note that forp e ≤ 0:01, the logical error rate is so small that the number of samples needed is too large for simulation. Even on Titan, direct Monte Carlo simulation is much too expensive. We need to be careful here, since we are using a message passing algorithm, so the error oor eect in classical message passing decoding might potentially occur in the low error rate region. To evaluate the performance in the region p e ≤ 0:01, we can use the following observation. The distance for X errors in the 3-level concatenated [[15; 1; 3]] code ([[3375; 1; 27]]) is 343, which means the code can correct all X error of weight less than 170 since the decoder is optimal. So we just need to focus on Z errors. Dene P L (w;p e ) as the logical error probability after soft decision decoding using parameterp e when Pauli errors of weightw uniformly occur on each qubit. Dene P Z L (w;p e ) as the logical error probability when Z errors of weight w uniformly occur on each qubit. Consider an all-Z error E of weight w < 170 on a set of qubitsS . Let E ′ be arbitrary string of X errors supported byS . If we can correctE, thenE ′ ⋅E can 94 be corrected. Thus, we have P L (w;p e )= 2 3 w P Z L (w;p e ) forw < 170. Then the logical error probability using soft-decision decoder can be bounded as follows: P L (p e )= 3375 Q w=1 P L (w;p e ) 3375 w p w e (1−p e ) 3375−w = 34 Q w=14 3375 w 2 3 w P Z L (w;p e )p w e (1−p e ) 3375−w + 50 Q w=35 3375 w 2 3 w P Z L (w;p e )p w e (1−p e ) 3375−w + 100 Q w=51 3375 w 2 3 w P Z L (w;p e )p w e (1−p e ) 3375−w + Q w>101 3375 w P L (w;p e )p w e (1−p e ) 3375−w ≤ 34 Q w=14 2 3 w 3375 w P Z L (34;p e )p w e (1−p e ) 3375−w + 50 Q w=35 2 3 w 3375 w P Z L (50;p e )p w e (1−p e ) 3375−w + 100 Q w=51 2 3 w 3375 w p w e (1−p e ) 3375−w + Q w>100 3375 w p w e (1−p e ) 3375−w (3.24) We evaluateP Z L (34;p e ) andP Z L (50;p e ) forp e = 0:01 and get an upper bound on the logical error probability of 2× 10 −10 . 3.8 Summary and Conclusion We have proposed a scheme for FTQC using large block codes as memory blocks and the concatenated [[15; 1; 3]] Reed-Muller code as processor block. We showed that its logical error rate can be made low enough to implement practically interesting quantum algorithm with reasonable physical error rates. The number of physical qubits required to protect a single logical qubit can be reduced from thousands of qubits to hundreds of qubits or less, and no magic state distillation is needed. It is very likely that good codes, such as quantum LDPC codes [29, 93, 139], may allow memory blocks with even better performance [61]. The memory error rate could be further reduced if we exploited the correlations between eective errors before and after the syndrome extraction circuits. The decoding algorithm for concatenated [[15; 1; 3]] code is optimal. However, it cer- tainly does not rule out other codes, such as gauge color codes [18][79] and triorthogonal codes [22] may outperform it. The decoding of these codes needs to be studied to check whether they are more appropriate for processing block. 95 On the other hand, the use frequency of each clean ancilla state varies dramatically. The ancillas S0⟩ ⊗k L and S+⟩ ⊗k L for syndrome extraction are used much more often than those for specic measurements, teleportation and logical gates. Thus, the distillation protocols should vary for dierent ancilla states to maximize the throughput of ancilla generation. The total resources needed depends strongly on the details of the ancilla distillation protocols, which will be carefully investigated in Chapter 4 based on work [81, 158]. 96 Chapter 4 Distillation of Ancilla States for Steane Syndrome Extraction 4.1 Preliminaries In the FTQC protocol using multiqubit large block codes, as proposed in Chapter 3, various types of clean ancilla are needed. There, multiple logical qubits are encoded into large block codes and do logical gates by measuring logical states and teleporting logical qubits states back and forth between memory block and processor blocks. This can be done assisted by preparing dierent types of ancilla states, transversal circuits and bit-wise qubit measurement [133]. In other word, we need large numbers of clean ancilla states of dierent types, and we use them with high frequency. How to build an ancilla factory capable of producing many ancilla eciently is the main obstacle of this scheme. In this Chapter, we brie y introduce how to overcome this obstacle using a so-called ancilla state distillation protocol. More details can be found in our ongoing work in [81, 158]. Before we go further, we distinguish two dierent terminologies: failures and errors. Let the processes which cause imperfection be called \failures", and the resulting imper- fections in the state of the qubits be called \errors". A basic assumption here is that failures occurring in dierent places and time in quantum circuit are independent. In order for the FTQC protocol in [27] to work, those ancillas should contain no correlated errors, so that correlated errors will not pass from ancilla to data block, introducing uncorrectable error. On the other side, if both Z and X ancilla are free of correlated errors, single-shot syndrome extraction and logical states measurements can be valid, as 97 shown in [27], which is not only fault-tolerant, but also ecient. In order to make our statement more precise, here we give the a more formal denition of uncorrelated errors, following Ref. [135]: Denition 2. An error E is uncorrelated if there exists some small value p, such that Pr(p)=a s p s S s≤ wt(E) for wt(E)≤t; s>t for wt(E)>t; (4.1) and we dene E to be correlated otherwise. Here wt(E) is the weight of E. The coecient a s should not be unreasonably large. However, when an approximation of desired ancilla is prepared by any means (e.g., through encoding circuits), there are typically many locations in the preparation where a single circuit failure can result in a high-weight error in the output state, making it a correlated error. Here, we assume the worst case: the preparation leaves an error of any non-zero weight with probability proportional to p. As a result, the ancilla states themselves must be checked for errors. Previously, verication of ancilla state S0⟩ L has been studied for codes like Steane code and Golay code [135, 136, 37, 39, 104]. The complexity of verifying prepared ancillas against errors grows quickly as the code distance increases. For large codes, verication of encoded S0⟩ L and S+⟩ L is accomplished by using additional identically prepared auxiliary ancillas. Errors from the initial ancilla are copied onto the auxiliary ancillas and then the auxiliary ancillas are measured. If measurements suggest the presence of an error, since we cannot be certain exactly where the errors are, no error correction procedure can be implemented, and all of the ancillas are discarded and the entire process restarts. Otherwise, the ancilla is accepted and may be used for error correction with high probability that it contains only uncorrelated error. Even more, the auxilliary ancillas may also contain correlated errors. These errors can spread to the initial ancilla and invalidate the verication. Thus the auxiliary ancillas must also be checked for errors by more auxiliary ancillas before used to check initial. These recursive 98 verications will nally involve many encoded ancillas, and dominates the overall resource overhead cost of error correction. Steane tried to verify the ancilla by checking the sign of syndromes and logical oper- ators. He showed that it can be done in a single round [135] verication based on post-selection by carefully choosing the verication circuit. In Ref. [104], the permuta- tion symmetry of Golay code has been investigated. In particular, by considering many dierent preparation circuits, they observed that certain circuits give favorable combi- nations of correlated errors, and thus require fewer rounds of verication, substantially reducing overhead. In FTQC scheme proposed in [27], we need to verify not only S0 L ⟩, but also a vari- ety dierent kinds of ancillas. We also hope that these ancilla can be produced with extremely high eciency, since they may dominate the resource overhead of the scheme. Some ancilla can be obtained when we have S0⟩ L in hand, Nevertheless, some ancilla states are not easy to obtain. For simplicity of analysis, suppose the quantum codes we are considering are [[n 1 ;k 1 ;d 1 ]] and [[n 2 ;k 2 ;d 2 ]] CSS codes dened by classical codes C 1 ⊆C 1 andC 2 ⊆C 2 , respectively. For the purpose of universal FTQC, a minimal set of dierent kinds of clean ancilla states are required [27], including: 1. S0⟩ L (S+⟩ L ) for Steane's syndrome extraction, which are simultaneous 0 eigenstate of stabilizer generators and logical Z i s ( X i s). 2. S0 1 ;:::;+ j ; 0 j+1 :::0 k 1 ⟩ L (S+ 1 ;:::; 0 j ;+ j+1 :::+ k 1 ⟩ L ) states to measure logical X j (logical Z j ). This state has all logical qubits in S0⟩ (S+⟩), but one in S+⟩ (S0⟩). it is simulta- neous 0 eigenstates of all logical Z j∏ i≠j X i ( Z i∏ i≠j Z i ) and all other. We denote such state S X j ⟩ L (S Z j ⟩ L ). 3. Entangled state between i th and j th logical qubits in the same block: S + ⟩ L = 1 √ 2 (S0 i 0 j ⟩ L + S1 i 1 j ⟩ L ), which is the simultaneous 0 eigenstate of logical X i X j and Z i Z j . We denote this state V Z x i Z x j X x i X x j h L or V Z z i Z z j X z i X z j h L for X (or Z) ancilla. This logical Bell state can be used to teleport logical qubits from memory blocks to 99 processor blocks and verse vise to implement logical T gate. This can help avoid magic state distillation [24] and hopefully, reduce the overall resource overhead of the computation. 4. Entangled state betweeni th logical qubit ofZ ancilla andj th logical qubits ofX ancilla: S ij ⟩ L = 1 2 (S0 i 0 j ⟩ L +S0 i 1 j ⟩ L +S1 i 0 j ⟩ L −S1 i 1 j ⟩ L ). This ancilla is 0 eigenstate of X i ⊗ Z j and Z i ⊗ X i useful to implement logical phase gate. We denote this state V Z z i X x j X z i Z x j h L It can be easily checked that these ancillas are actually stabilizer states whose stabilizer generators contain either onlyX's or onlyZ's. Besides, they are stabilized by additional logical operators, such as Z i , X i , Z i Z j , X i X j , X i Z j and Z i X j . Uniformly, in this Chap- ter, we call stabilizer generators and these logical operators \stabilizers" of the state. We can generate these states using Cliord encoding circuits, but apparently, this is not fault-tolerant, and errors are highly correlated. In other contexts (e.g., entangle- ment purication [12]) we can get around this by distillation taking a bunch of imperfect states, and then carrying out a protocol to produce a smaller number of better states. Herein we show how classical error correcting codes can be applied to distill these logical ancillas. Once errors are detected, it is likely that we can correct them in most cases, rather than simply discarding the ancillas and starting over again. We hope that this scheme will greatly reduce the resource overhead. 4.2 Ancilla State Distillation using Classical Codes Now we introduce the idea to incorporate a classical code for ancilla distillation, given a large number of noisy ancillas. For each ancilla block prepared independently, if no error occurs, the values of all its stabilizers S i will be +1. The errors in each block can be highly correlated. These errors will ip the values of stabilizers of the states to 1. If we know all values of these stabilizers, we can correct all errors. If failure rate is not too high, we can expect 100 the probability of ipping will also be low (though errors may be correlated). Given a set of ancillas blocks encoded in the same logical state (these blocks are certainly not error free), we can treat the string of values of each stabilizer from dierent blocks as noisy from memoryless binary symmetry channel with parameter O(p). Since between dierent blocks, the errors are independent, noise on each bit is independent. As we will see, we can check the parity of the received string of bits, diagnose the \noise" by classical error correction to obtain values of S i s, for those blocks we try to distill. After that, quantum error correction can be done on those blocks and cleaner ancialla states can be obtained. In this section, we will give detailed distillation protocol for all ancilla needed for FTQC. S0⟩ L and S+⟩ L states.| Suppose the classical code C 3 we are working on is an [n 3 ;k 3 ;d 3 ] binary linear code that can correctt= ⌊ d 3 −1 2 ⌋ errors. Such a code hasr =n 3 −k 3 parity checks and letH c = [I r A] be the parity-check matrix ofC 3 in the systematic form, whereI r is the r×r identity and A is r×k 3 . The rst step is to determine which of the n 3 ancillas originally hadX errors, and where, and correct them. Divide the ancillas up into groups, each containing n 3 blocks. Suppose each block is a [[n 1 ;k 1 ;d 1 ]] CSS code (situation is exactly the same for [[n 2 ;k 2 ;d 2 ]]), containing n 1 −k 1 syndrome bits. For example, letg j z i be the syndrome bit ofi thZ generator,j th block, which can be listed as: g 1 z 1 g 1 z 2 ⋯ g 1 z n 1 −k 1 −1 g 1 z n 1 −k 1 g 2 z 1 g 2 z 2 ⋯ g 2 z n 1 −k 1 −1 g 2 z n 1 −k 1 ⋮ ⋮ ⋱ ⋮ ⋮ g N−1 z 1 g N−1 z 2 ⋯ g N−1 z n 1 −k 1 −1 g N−1 z n 1 −k 1 g N z 1 g N z 2 ⋯ g N z n 1 −k 1 −1 g N z n 1 −k 1 (4.2) For xed z i , each column of g N z i is a n 3 bit string. In addition, S0⟩ L (S+⟩ L ) state of each block are also stabilized by k 1 logical Zs ( X)s. Similarly, we can denote L j z i (L j z i ) as 101 the values of i the logical operator in j th block. In each group, choose the rst r of 0 L % 0 L % 0 L % 0 L M z M z Figure 4.1: The circuit for ancilla distillation by the classical [3; 1; 3] repetition code to distill ancilla free of X errors. The rst two S0⟩ L serve as check bits. the ancillas to hold the parity check, and do transversal CNOTs from the remaining k 3 ancillas onto each of the parity-check ancillas according to the pattern of 1s in the rows of A. If [A] i;j = 1, we apply transversal CNOTs from (r+j)-th ancilla to i-th ancilla. Then measure all the qubits of each of the rst r ancillas in the Z basis. Note that X errors will not propagate from the i-th ancilla to (r+j)-th ancilla. As an example, Fig. 4.1 demonstrates the circuit of distillation by the [3; 1; 3] repetition code. We can then use the classical codes C 3 to gure out values of {g j z i } and { Z j i } ({ X j i }), for all r+ 1 ≤j ≤N. Then for each j, rst use {g j z i } and classical code C 1 to decode X error. Assuming the error decoded is E j (which is a tensor product of X). Then for every logical Z j i , following the rules: 1. If [E j ; Z j i ]= 0 andL j z i = 0, then do nothing, 2. If {E j ; Z j i }= 0 andL j z i = 0, then set E j →E j X j i , 3. If [E j ; Z j i ]= 0 andL j z i = 1, then set E j →E j X j i , 4. {E j ; Z j i }= 0 andL j z i = 1, then , do nothing, for everyi. We can correct updatedE j (or just keep track of them). At the moment we neglect any errors in the CNOTs or measurements used in this circuit. As we will see later, after these process, allX with any weight will be left uncorrected with probability O(p t+1 ). 102 0 L % 0 L % 0 L % 0 L % 0 L % 0 L % M z 0 L % 0 L % 0 L % M x M z M z M z M z M z M x 0 L Figure 4.2: The circuit for ancilla distillation by the classical [3; 1; 3] repetition code to distill ancilla free from X and Z errors. After the rst round, the remaining k 3 ancillas from the group will have lower rates ofX errors than they started with, but Z errors on the target qubits can propagate via the CNOTs back onto the these k 3 blocks, and they are still correlated. Even worse, the rate of Z errors on the remaining k 3 ancillas will increase to (m+ 1)p, where m=r is the number of parity checks each qubit is included in (or the number of 1s in each column of the matrix A). The rate of X errors drops from p to cp t+1 for all weight, where c is some constant that depends on the details of the codes. If p is not too big, the rate of Z errors has grown roughly by a constant factor, while the rate of X errors has been substantially reduced. So now, of our original large number of ancillas, we retain a fraction k 3 ~n 3 , which have much lower X error rates and somewhat larger Z error rates. We again divide them up into groups of n 3 . It is important that ancillas that were grouped together in the rst round are not grouped together in the second round, because their errors are correlated. Similarly, we can use the same code C 3 to distill Z errors, except that all the qubits of each of the rst r ancillas are measured in the X basis. Since the error rate for Z is now higher, one may consider to use another 103 classical code with lower eciency but better error correction ability. An example of whole distillation procedure using [3; 1; 3] repetition code is shown in Fig. 4.2. S Z i ⟩ L forZ ancilla and S X i ⟩ L forX ancilla.| The same distillation circuit of Fig. 4.2 can be used to to distill S Z i ⟩ L (or S X i ⟩ L ) state. Still, there's is some dierence from the distillation of S0⟩ L . For S X i ⟩ L , we use classical code to check values of all Z stabilizers generators and Z j (j ≠i) in the rst round and X errors can be removed. Values of X stabilizer generators and X i are checked at second round to remove Z errors. ² i j i j L Z Z X X M z M z M z M z ² i j i j L Z Z X X ² i j i j L Z Z X X i j i j L Z Z X X Figure 4.3: The circuit for S + ⟩ ancilla distillation by the classical [3; 1; 3] repetition code to distill ancilla free of X errors. V Z x i Z x j X x i X x j h L V Z z i Z z j X z i X z j h L for X (Z) ancillas .| We may also need to distill states from a pair of blocks from time to time. In our case, two blocks can either be of the same type (both are memory blocks) or two dierent types (one memory block and one processor block) of quantum codes. It is worthy mentioning here that in this case, the two blocks are of the same type of ancilla (X or Z). So, the distillation circuit is the same for S0⟩ L , if we regarded the two blocks as a unied large block code. For example, we can simultaneously remove X errors from both blocks, as the circuit shown in Fig. 4.3. We can check the values of Z stabilizer generators for both blocks. We also need to check Z q for all q. In the second round, X stabilizer generators value of product X i X j are checked to remove Z errors. The process to distill V Z z i Z z j X z i X z j h L for Z ancillas is similar. 104 ² i j i j L X Z Z X M x M z M z M x ² i j i j L X Z Z X ² i j i j L X Z Z X ² i j i j L X Z Z X i j i j L X Z Z X Figure 4.4: The circuit for S ij ⟩ L ancilla distillation by the classical [3; 1; 3] repetition code to distill X ancilla from X errors and Z ancilla from Z errors. V Z z i X x j X z i Z x j h L state.| This is an entangled state between i-th logical qubit of Z ancilla and j-th logical qubits of X ancilla. Again, we need to distill the ancilla blocks in pair. But the situation is dierent from the previous case, since we need to check the value of X i Z j and Z i X j . We destroy the state if we try to simultaneously remove X errors from both ancillas or Z errors from both ancillas. Instead, in the rst round of distillation, we removeX errors from theX ancilla andZ errors fromZ ancilla, as shown in Fig. 4.4. The same method can be also used to distill more general entangled states such as V Z z i X x j X x k X z i Z x j I x k h L ; which is an entangled state between i-th logical qubit of Z ancilla and j-th and k-th logical qubits in X ancilla. U Z z i Y x i X z i Z x i g L state.| This entangled state between thei−th logical qubit of theZ ancilla and theith logical qubit of theX ancilla is used to measure logicalY i on the data block. If the CSS codes [[n 1 ;k 1 ;d 1 ]] and [[n 2 ;k 2 ;d 2 ]] satises following conditions: 1. X and Z stabilizer generators are symmetric; 2. weight of stabilizer generators is doubly even, 3. weight of Z i is odd, so that X i can be supported by exactly same set of qubits. Then, we can apply bit-wiseS gate on distilled U Z z i X x i X z i Z x i g L state, following a Z i onX ancilla if necessary, to obtain U Z z i Y x i X z i Z x i g L . The reason is as follows: the bit-wiseS gate will preserve 105 the stabilizer group for a code with above properties. On the other hand, we can always choose X i which has the same support as Z i , since they anticommute with each other. All other logical qubits will not be aected by bit-wise S since they are in +1 eigenstate of Z j (j ≠i). Similarly, we can use same trick to obtain clean U Z z i Y x i X x k X z i Z x i I x k g L , if X j and X k do not have overlap. Such a state is used to implement a logical S gate on i-th data qubit. 106 Part IV Ecient Fault-tolerant Quantum Computation: Part II 107 Chapter 5 Fault-tolerant Holonomic Quantum Computation in Stabilizer Codes 5.1 Introduction In Chapter 3, we studied FTQC schemes based on multi-qubit large block codes to reduce the resource overhead. We can see that the error rates on each individual physical qubit (including preparation errors, gate errors and measurement errors) greatly aect the logical error rate. The logical error rate decreases quickly when physical error rate reduces. Similarly, if physical rate is very small, one may not need to extract syndromes and do error correction very frequently (which, as we have seen, often itself introduces many additional errors and increases the resource overhead), and at the same time, achieve the same logical error rate with only small sized codes. In this Chapter and next few Chapters, we exploit a special method to reduce the error rate on each physical qubits. Most errors in quantum system are caused by interaction between qubits and a thermal bath. Even at extremely low temperatures, the thermal bath usually has a crucial eect on the system. Actually, one major obstacle in quantum technology is to struggle with thermodynamics. The reason classical computers do not need active error correction is that classical bits are protected by an energy gap. This energy gap suppresses the error rate exponentially with the size of the gap. In other words, errors 108 suer an energy penalty to ip from 0 to 1 and verse vice. One may expect that the same idea can be exploited to the region of quantum computation. However, there are some additional complexities there. First, one not only needs to suppress the rate of bit ips, but also of phase ips. Second, one not only hopes to protect quantum information, but also wants to manipulate the protected information. One method to take advantage of the thermal gap is to use the adiabatic quantum computing (AQC) [44, 43] model instead of the standard quantum circuit model, which slowly drags the ground state of the system to the nal Hamiltonian, whose ground state encodes the solution of the problem. AQC would take advantage of the energy gap between the ground state and other excited states to suppress thermal noise when evolution is very slow [69, 3]. While considerable work has been done in [69, 85], a fault-tolerant theory for AQC is still lacking. The system's minimal energy gap, which determines the time scale of evolution, scales as an inverse polynomial in the problem size [127, 97], so that the temperature must be lowered polynomially to prevent thermal excitation. Another method is to combine fault-tolerant techniques and HQC, which was studied in Ref. [102, 103], where the system Hamiltonian is an element of the stabilizer group or gauge group. Single qubit or two-qubit unitary operations are realized through con- tinuously deforming the the system Hamiltonian. During this process, the path in the parameter space forms an open loop and results in the desired unitary transformation. After a sequence of such elementary operations, a closed-loop holonomy is obtained in the code space. However, this approach does not protect quantum information from low weight thermal noises, which usually occur with high probability, since there is no energy gap between the code space and error spaces corresponding to these errors. On the other hand, the degeneracy of the ground space in such schemes is usually vulnerable to local perturbation. This will potentially cause stochastic phase errors. 109 A third method is the beautiful idea of topological quantum computation (TQC) rst introduced by Kitaev [73], where excited states of system Hamiltonian behave like par- ticles with exotic statistics, called anyons. By adiabatically braiding anyons around one another in space-time, it induces a unitary operation that depends only on the topology of the anyon world lines. Remarkably, some systems can support non-Abelian anyons, perform universal quantum computation on information encoded in the label space of the anyons [78], while being protected by an energy gap independent of the system size. Unlike HQC, TQC is immune to the eect of small perturbations, since quantum infor- mation is stored and processed nonlocally, so that the splitting of the degenerate ground space will decrease exponentially with the system size [23]. However, this topological pro- tection does not completely eliminate the need for active error correction. The energy gap can protect information only to a certain extent, and unwanted anyons could be created if the computation time is long enough. Besides, unwanted anyons may be gen- erated during the process of creation, fusion and imperfect adiabatic motion of anyons, and they may not be detectable. One must measure anyon occupations to determine when and where unwanted anyons are created [152], but this is usually dicult in most TQC models (like fractional quantum Hall systems). In this Chapter, we introduce our work in Ref. [156], which tries to overcome the shortcomings of the rst two methods mentioned above. In Chapter 6, we will try to combine the best features of all the architectures mentioned above and avoid their weakness. First, we show an equivalence relation between fault-tolerant circuits and fault-tolerant adiabatic processes in the case where quantum information is encoded in a code space, which is also the ground space of a system Hamiltonian. Based on this, we present an alternative way to systematically construct a fault-tolerant HQC process that takes advantage of the energy gap between the ground space and other excitation states. Unlike AQC, this gap does not change with the problem size, and we know the exact value of the gap during the process, which greatly enhances the ability to prevent 110 low-weight thermal excitation. With a lower error rate at the physical level of the fault- tolerant scheme, it may help to reduce the number of qubits needed and the frequency of error detection and error correction. The structure of this Chapter is as follows. In Sec. 5.2.1, we review the geometrical setting of the holonomic problem. In Sec. 5.2.2, basic ideas of fault-tolerant quantum computing is reviewed and generalized. We connect fault-tolerant techniques and HQC in Sec. 5.3. In Sec. 5.3.1, we describe our method to construct an adiabatic process from a fault-tolerant circuit to implement encoded unitary operations. Then in Sec. 5.3.2, we prove that our method of constructing encoded unitary operations is fault-tolerant, and discuss how it can realize universal fault-tolerant quantum computation. Several examples are given in Sec. 5.4. In Sec. 5.4.1, we show how our scheme works on the simplest 3-qubit repetition code. A less trivial example, of the encoded CNOT gate for the Steane code, is given in Sec. 5.4.2. We summarize our results and conclude in Sec. 5.5. 5.2 Preliminaries 5.2.1 HQC: A Geometrical Formalism We introduce a more abstract geometric setting of holonomic problem which is useful for our purpose. We focus on the ground space for simplicity, however, the formalism is general and can be applied to any eigenspace of system Hamiltonian. Suppose we have a family of Hamiltonians acting on the Hilbert space C N , and the ground state of each Hamiltonian isK-fold degenerate (K <N). The natural mathemat- ical setting to describe this system is the principal bundle (S N;K (C);G N;K (C);; U(K)), which consists of the Stiefel manifold S N;K (C), the Grassmann manifold G N;K (C), the projection map ∶ S N;K (C)→ G N;K (C), and the unitary structure group U(K). We will explain the meaning of these mathematical objects in details below. 111 The Stiefel manifold is dened as: S N;K (C)= {V ∈M(N;K;C)SV V =I K }; (5.1) where M(N;K;C) is the set of N ×K complex matrices and I K is the K−dimensional unit matrix. Physically, each column of V ∈S N;K (C) can be regarded as a normalized state inC N , andV can be viewed as an orthonormal set of K basis of the ground space of Hamiltonian: V = S' 1 ⟩;S' 2 ⟩;:::;S' K ⟩: (5.2) Note that we have freedom to transfer from one orthnormal basis of to another through unitary transformation, we can dene a unitary group U(K) that acts onS N;K (C) from the right: S N;K (C)× U(K)→S N;K (C); (V;h)↦Vh; (5.3) by the matrix product ofV andh. V andVh can be regarded as two dierent orthonor- mal basis corresponding to the same ground space. During the adiabatic evolution, the ground space of the Hamiltonian will change. The ground space can be represented as a K-dimensional hyperplane in C N . So we introduce the Grassmann manifold inC N : G N;K (C)= {P ∈M(N;N;C)SP 2 =P;P =P; TrP =K}; (5.4) whereP is a projection operator onto the hyperplane inC N , and the condition TrP =K indicates that the dimension of the hyperplane is K. In our scenario, P ∈G N;K (C) can be regarded as the projector onto the K-dimensional ground space of the Hamiltonian. The relationship between the orthonormal basis V and ground space P can be seen as follows. We dene the projection map ∶S N;K (C)→G N;K (C) as ∶V ↦P ∶=VV : (5.5) 112 The corresponding ground space projector can be obtained when the orthonormal basis is given. We can see that the basis V and basis Vh with h ∈ U(K) belong to the same ground space, since (Vh)= (Vh)(Vh) =Vhh V =VV =(V ): (5.6) For the purpose of the Chapter, we want to transform the ground space adiabatically during the procedure. To formulate such a process, we need also dene the left action of the unitary group U(N) on both S N;K (C) and G N;K (C) by the matrix product: U(N)×S N;K (C)→S N;K (C); (g;V )↦gV; (5.7) and U(N)×G N;K (C)→G N;K (C); (g;P)↦gPg : (5.8) It is easy to check that (gV )=g(V )g . This action is transitive: there is a g ∈ U(N) for any V;V ′ ∈ S N;K (C) such that V ′ = gV . There is also a g ∈ U(N) for any P;P ′ ∈ G N;K (C) such that P ′ =gPg . So this action is sucient to describe any ground space transformation. We can further study the topological structure of S N;K (C) and G N;K (C) for com- pleteness. For each point V in S N;K (C), we can dene an isotropy group: I S (V )= {g ∈ U(N)SgV =V }; (5.9) which is isomorphic to U(N −K) for all V ∈ S N;K (C). Similarly, we can dene an isotropy group for each P ∈G N;K (C): I G (P)= {g ∈ U(N)SgPg =P}; (5.10) 113 which is isomorphic to U(K)× U(N −K) for all P ∈ G N;K (C). Thus, S N;K (C) ≅ U(N)~U(N −K) and G N;K (C)≅ U(N)~U(K)× U(N −K) [99]. The canonical connection form onS N;K (C) is dened as a u(K)-valued one-form on G N;K (C): A=V (P) dV (P); (5.11) which is a generalization of the WZ connection in Eq. (1.27). This is the unique connec- tion that is invariant under the transformation in Eq. (5.3): ~ A=h V (P) d(V (P)h)=h Ah+h dh: (5.12) We apply this formalism to the system dynamic of HQC. The state vector S (t)⟩∈C N evolves according to the Schr odinger equation: i d dt S (t)⟩=H(t)S (t)⟩: (5.13) The Hamiltonian has a spectral decomposition, H(t)= L Q l=0 " l (t)P l (t); (5.14) with projection operators P l (t). Therefore, the set of energy eigenvalues (" 0 (t);:::;" L (t)) and orthogonal projectors (P 0 (t);:::;P l (t)) encodes the information of the control parameters of the system. For the ground space, we write P 0 (t) as P(t) for simplicity. Suppose the degeneracy K = Tr{P(t)} is constant. For all t, there exists V (t) ∈S N;K (C) such that P(t) =V (t)V (t). By the adiabatic approximation, we can substitute for S (t)⟩∈C N a reduced state vector (t)∈C K : S (t)⟩=V (t)(t): (5.15) 114 Since H(t)S (t)⟩=" 0 (t)S (t)⟩, the Schr odinger equation (5.13) becomes d dt +V dV dt (t)=" 0 (t)V (t)(t); (5.16) and the solution can be represented formally as (t)=e −i ∫ t 0 " 0 ()d P exp− S V dV (0): (5.17) Therefore, (t) can be written S (t)⟩=e −i ∫ t 0 " 0 ()d V (t)P exp− S V dV V (0)S (0)⟩: (5.18) In particular, if the system comes back to its initial point, asP(T)=P(0), the holonomy ∈ U(K) is dened as =V (0)V (T)P exp− S V dV ; (5.19) and the nal state is S (T)⟩=e −i ∫ t 0 " 0 ()d V (0)(0): (5.20) According to the formula above, an operation ∈ U(K) is applied to the ground space. If the condition V ⋅ dV dt = 0; (5.21) is satised for all t, the curve V (t) in S N;K (C) is called a horizontal lift of the curve P(t)=(V (t)) in G N;K (C).Then the holonomy (5.19) is greatly simplied to =V (0)⋅V (T)∈ U(K): (5.22) 115 For closed-loop HQC, given a desired unitary operationU op ∈ U(K) and a xed initial point P(0)∈G N;K (C), we want to nd a loop P(t)∈G N;K (C) with base points P(0)= P(T) whose horizontal lift V (t) ∈ S N;K (C) produces holonomy = U op according to Eq. (5.22). For open-loop adiabatic code deformation, Eq. (5.18) is general to obtain the state evolution when the adiabatic condition is satised. A visualization of a horizontal lift is shown in Fig. 5.1. { holonomy horizontal lift general curve G N,k Figure 5.1: Horizontal lift as a specied curve in S N;K (C) whose projection is P(t). The initial condition V (0) becomes V (T), which is generally dierent from V (0). The dierence is the holonomy. Note that for a given , there exist innitely many paths in Grassmannian G N;K (C). Given a path, to nd the holonomy is easy. However, the inverse problem| given a holonomy, to nd the the proper path |is in general not trivial at all. In the rest of the Chapter, we will discuss how to nd a proper path to realize a certain holonomy in the code space, and thus perform an encoded quantum gate operation. Without loss of generality, we can always restrict ourselves to the case such thatP(t) has the form: P(t)=U(t; 0)P(0)U (t; 0)=U(t; 0)v 0 v 0 U (t; 0); (5.23) 116 for some smooth U(t; 0) ∈ U(N) according to Eq. (5.8). Note here, U(t; 0) should be chosen such that in general, at any time t, U(t+;t)P(t)U (t+;t)≠P(t); (5.24) for some neighborhood of t. In other word, U(t) must not be in the isotropy group of P(t). This condition can also stated as @ @ U(t+;t)S =0 ;P(t)≠ 0: (5.25) The case where Eq. (5.25) equals 0 is allowed only at a nite number of points in [0;T]. The horizontal curve should satisfy the following set of equations: V ⋅ dV dt = 0; P(t)=V (t)V (t)=U(t; 0)v 0 v 0 U (t; 0): (5.26) The general solution to these equations can be written as: V (t)=U(t; 0)v 0 h(t; 0) (5.27) for some h(t; 0)∈ U(K). Substituting Eq. (5.27) into Eq. (5.26) we get: _ h(t; 0)=−v 0 U (t; 0) _ U(t; 0)v 0 h(t; 0); (5.28) which completely determines the h(t), horizontal lift, and state evolution for a given adiabatic process. 5.2.2 Stabilizer Codes and Fault-Tolerant Quantum Computation In this section, we slightly reformulate stabilizer formalism and denition of fault- tolerance, such that it is more easy to use for our purpose. A quantum error-correcting 117 code is formally dened as a subspace C of some larger Hilbert space. A necessary and sucient condition for a set of errors {E i } to be correctable is [100, 84]: PE i E j P = ij P; ∀i;j; (5.29) for some Hermitian matrix . Here P is the projector onto C. Since any linear combi- nation of {E i } is also correctable, we dene E = Span{E i } (5.30) to be a correctable error set for codeC. The codes we are interested in are the stabilizer codes [60]. We brie y review the formalism of stabilizer codes. LetG n be the Pauli group acting onn qubits. An Abelian subgroup S of G n is called a stabilizer group if −I ∉S. The stabilizer group denes a subspace of the n-qubit Hilbert space by C = {S ⟩∶SS ⟩= S ⟩ for all S ∈S}: (5.31) This C is called the code space. C is nonzero since −I ∉ S. A state in C is called a codeword. This subspace is the simultaneous +1 eigenspace of the stabilizer group. If the subspace has dimension 2 k (k logical qubits), the stabilizer code can be specied by n−k commuting stabilizer generators, which are elements of G n . The group S can be represented by these stabilizer generators: S = ⟨{S j }⟩. All stabilizer codes can be characterized by three parameters [[n;k;d]], where d is the minimum distance of the code, which is equal to the minimum weight of all nontrivial elements in the normalizer group ofS. With the use of stabilizer codes, it is possible to build a quantum processor that is fault-tolerant [100, 84, 60]. A quantum information processor is called fault-tolerant if the information is encoded in a quantum error-correcting code at all times during the 118 procedure, and a failure at any point in the procedure can only propagate to a small number of qubits, so that error correction can remove the errors. It has been shown that fault-tolerant computation is possible on any stabilizer code [84, 60] for some error model. Typically, there are three elementary quantum \gadgets": encoded state prepa- ration, encoded unitary operations and encoded state measurement. Through enlarging or concatenating the fault-tolerant gadgets, a computation can achieve arbitrary accu- racy, if the error rate is low enough [84]. Encoded Cliord unitary operations play a key role in fault-tolerant computation, since for most proposed schemes of fault-tolerant quantum computation, like concatenation of the Steane code [100], C4 code [75] or sur- face code [52], we can prepare encoded non-Cliord magic states using techniques like state distillation, which can be implemented by encoded Cliord unitary operations. So we will focus on encoded Cliord operations and their holonomic implementation. . . . . . . g 1 g 2 g p - 1 g p Figure 5.2: A logical unitary quantum operation is realized as a series of quantum gates from a universal set of gates in the circuit model. In the standard circuit model, an encoded unitary operation can be realized by a series of quantum gates, say p gates chosen from a universal set of gates as shown in Fig. 5.2. Commonly, the universal set of gatesU 1 = {Hadamard; CNOT;S;~8} is used to describe the circuit, which is good for certain fault-tolerant schemes. Here, we choose another universal set: U 2 = R x = exp−i 4 X;R zz = expi 4 Z⊗Z;S;~8; (5.32) which proves to be much more suitable for our adiabatic scheme. Errors can occur anytime during the process, both between and during gate operations. Noisy gates are 119 always equivalent to a perfect gate followed by an error operator, so we can just focus on the errors occurring between the gates. If an error E q ∈E occurs between gates q− 1 and q, it will propagate to E q′ = p M l=q g p+q−l ⋅E q ⋅ ⎛ ⎝ p M l=q g p+q−l ⎞ ⎠ : (5.33) If a circuit is fault-tolerant, we can suppose thatE q′ would be still in the same correctable error setE. According to this observation, we give a special version of Def. 1 for a fault- tolerant circuit on codeC: Denition 3. Given a code C and a particular correctable error set E for this code, a circuitG that realizes an encoded unitary operation is called a fault-tolerant circuit forC if for any 1≤q <p, U qp =∏ p l=q g p+q−l maps any subset of E to another subset of E. This denition of a fault-tolerant circuit may be, however, too strong. In practice, it may be very dicult to nd such a code and corresponding circuit. For a practical error model, strongly correlated errors happen with much lower probability than weakly correlated or local ones, so we will focus on local errors. For example, if our codes are stabilizer codes, the error set E local can be spanned by Pauli operators with weight less than ⌊ d−1 2 ⌋, which occur with relatively high probability. If we limit ourselves to such a high-probability correctable local error set E local , then we get a weaker version of the denition of a fault-tolerant circuit: Denition 4. Given a stabilizer code [[n,k,d]] with a correctable error set E, and there is a high-probability local error setE local ⊂E, a circuitG that realizes an encoded unitary operation is called a fault-tolerant circuit for this code if for any 1≤l<p,U qp =∏ p l=q g p+q−l mapsE local to some subset of E. Remark 1. According to this denition, the encoded fault-tolerant unitary circuit does not necessarily need to be transversal, although the reverse is always true. If a circuit built of gates fromU 1 is fault-tolerant, then we can decompose its gates into gates from 120 U 2 , and the new circuit we obtain is also fault-tolerant. So, in the rest of the Chapter, we assume that the given circuits are composed of gates fromU 2 . Remark 2. We should mention here that in the following discussion, we only consider circuits that contain no~8 gates. In other words, we limit ourselves to Cliord circuits, since non-Cliord circuits will cause tremendous complexity. This restriction will be further discussed in Sec. 5.3.2. Fortunately, fault-tolerant encoded Cliord operations for stabilizer codes are made of Cliord circuits, that do not contain~8 gates. Encoded non-Cliord operations usually do contain ~8 gates. We will not directly implement encoded non-Cliord operations, but instead make use of magic state distillation, so this is not a serious restriction. 5.3 Fault-tolerant Holonomic Quantum Computation To combine the advantages of holonomic quantum computation with fault-tolerant com- putation techniques, the basic idea is to obtain a holonomy on the code space, which is the ground space of the system Hamiltonian, during an adiabatic evolution. One must make sure that the encoded quantum information is protected by a suitable error- correcting code throughout the Hamiltonian deformation. For simplicity, we assume that error correction is applied at the end of the cyclic adiabatic evolution. However, this may not necessarily be true in practice. We require that an error occurring during the deformation be correctable at the end: Proposition 1. Given a code C with a correctable error set E, suppose the initial state is S (0)⟩ ∈ C, and the deformation of the Hamiltonian is adiabatic. Then, in general, each eigenspace will undergo some transformation. Given a desired encoded operation (in our case, a holonomy) g on the code space, suppose a series of errors {E t i } occur at times t i during the evolution. Then {E t i } is correctable only if the nal state S (T)⟩∝ E f g S (0)⟩, for some E f ∈E. 121 In the case when {E t i } is empty, the statement is obvious. The fault-tolerance of this process is well dened in the case when {E t i } just has one element, say E t 1 . Following the spirit of fault-tolerant quantum computation in the circuit model, we dene fault- tolerance for an adiabatic process: Denition 5. Given a code C, dened by the ground space of an initial Hamiltonian with a correctable error setE, a desired encoded operation (holonomy) g , and an initial state S (0)⟩ ∈C, the corresponding cyclic adiabatic process is called fault-tolerant if any E t ∈E local occurring at time t leads to a nal state E f a S (0)⟩ for some E f ∈E. Unlike AQC, we need to measure the stabilizer generators and do error correction after a single or multiple cycles of encoded operations. At those points, we turn o the Hamiltonian and apply a standard error correction procedure. If this scheme is robust to low-weight thermal noise, and also evolves slowly enough that the adiabatic error is well below the threshold (which we will examine in some detail in Sec. 5.3.2), then the frequency of error recovery operations can be greatly reduced. We will show how to construct a fault-tolerant adiabatic process to do a holonomic encoded quantum unitary operation starting from a fault-tolerant circuit, and we will prove that such a process is fault-tolerant by Def. 5. 5.3.1 Scheme Given a stabilizer code with stabilizer groupS for n qubits (2 n =N), we set the system Hamiltonian at the very beginning to be H(0)=−Q j S j : (5.34) Thus the code space is the ground space of the Hamiltonian with dimension K = 2 k . We deform the Hamiltonian as follows: H(t)=Q j C j (t)S j (t)=Q j C j (t)U(t; 0)S j U (t; 0); (5.35) 122 with S j (t) = U(t; 0)S j U (t; 0), and [S i (t);S j (t)] = 0 for all i, j. C j (t) ∈ [−1; 0] is the weight ofS j (t) which is assumed to be controllable. The {S j (t)} can be viewed as a set of generators of an Abelian group, such as a stabilizer group. The Hamiltonian also has a spectral decomposition H(t)=Q s " s (t)P s (t): (5.36) Here, the {P s (t)} are projectors onto the simultaneous eigenspace of all the S j (t), with eigenvalues: " s (t)=Q j C j (t)s j ; (5.37) where the labels s j =±1 form a vector: s= {s 1 ;s 2 :::s n−k }: (5.38) When the Hamiltonian changes, as shown previously in Eq. (5.8), the ground space will also evolve. This denes a time-dependent codeC t . LetP 0 (t)=U(t; 0)P 0 (0)U (t; 0) be the projector onto the ground space of the Hamiltonian H(t) such that s j = 1 for all j. We emphasize that U(t; 0) should be chosen such that @ @ U(t+;t)S =0 ;P s (t)≠ 0 for all s; (5.39) except for a nite set points t, so that the deformation procedure is non-trivial for all eigenspaces. This method will work only if the adiabatic condition for each eigenspace P s is sat- ised, so that each eigenspace undergoes some non-trivial holonomy during the cyclic 123 evolution, in case an error takes the system to P s during the process. The standard adiabatic condition [96] can be reformulated for the eigenspace {P s }: ∥P s (t) @ @t H(t)P s (t)∥ 1 K " s (t)−" s (t) 2 ≈ 0; for any ≠: (5.40) This must hold for all t∈ [0;T], where∥⋅∥ 1 is the trace norm (∥A∥ 1 = Tr √ A A). For Hamiltonians of the form in Eq. (5.35), it is very likely that dierent P s (t)'s share the same eigenvalues so the adiabatic condition would not be directly satised. We will show later a systematic way to engineer the deformation procedure so that each eigenspace P s (t) satises this condition during the adiabatic process. In addition, each eigenspace should undergo the same holonomy to satisfy Prop. 1. Let's see how it works. Dene: U (t; 0) _ U(t; 0)=iQ(t; 0); (5.41) whereQ(t; 0) is Hermitian. In order to obtain the same holonomy for eachP s , according to Eq. (5.28), P s (0)Q(t; 0)P s (0) should be related to P 0 (0)Q(t; 0)P 0 (0) in some way. If we can makeP s (0)Q(t; 0)P s (0) either equal to the zero matrix or proportional to P s (0) for all s, then the character of the horizontal lift of P s (t) is completely determined by U(t; 0). Now we are ready to describe the scheme to construct a fault-tolerant adiabatic process for a holonomic unitary operation, starting from a fault-tolerant circuit. First, we divide the time of evolution [0;T] into p segments. The lth segment is [t l−1 ;t l ], and we set t 0 = 0 and t p =T . Given a fault-tolerant circuit G that realizes an encoded operation g =∏ p l=1 g p−l+1 , we can follow the steps listed below: 1. Set l= 1 and t 0 = 0. 2. Check the number ofS j (t l−1 ) such that [S j (t l−1 );g l ]≠ 0. If this number is odd, go to step 3, else go to step 4. 124 3. For the lth time segment [t l−1 ;t l ], choose a unitary operator U l (t;t l−1 ) = g f l (t) l , with f l ∶ [t l−1 ;t l ] → [0; 1] a monotonic smooth function with boundary condi- tions f(t l−1 ) = 0 and f(t l ) = 1. We deform the Hamiltonian such that H(t) = U l (t;t l−1 )H(t l−1 )U l (t;t l−1 ) in the interval [t l−1 ;t l ]. All S j (t l−1 ) are replaced at t l by S j (t l )=g l S j (t l−1 )g l , and H(t l )=−∑ j S j (t l ). Then go to step 5. 4. We need an additional operation to break the degeneracy in this case, in order that the adiabatic condition be satised for all P s (t). From those S j (t) such that [S j (t l−1 );g l ] ≠ 0, we arbitrarily select one element, say S b (t l−1 ), and change the Hamiltonian to H(t ′ l−1 ) = H(t l−1 )+C b S b (t l−1 ). C b is a constant between 0 and 1; we will choose it to be 0.5. This procedure can be done arbitrarily fast, and it will not aect a state entirely contained in any P s , so we can just set t ′ l−1 =t l−1 . We choose U l (t;t l−1 ) =g f l (t) l in this case, where f l ∶ [t l−1 ;t ′ l ]→ [0; 1] is a monotonic smooth function with boundary conditions f l (t l−1 )= 0 and f l (t ′ l )= 1. At time t ′ l , the Hamiltonian becomes H(t ′ l ) = −∑ j S j (t ′ l )+C b S b (t ′ l ) with S j (t ′ l ) = g l S j (t l−1 )g l . Then we remove the additional term in the Hamiltonian, leaving H(t l )=−∑ j S j (t l ), whereS j (t l )=S j (t ′ l ). Again, this can be done arbitrarily fast, so, we can set t ′ l =t l . Go to step 5. 5. If l=p, the process is nished. Else, set l=l+ 1 and go to step 2. First, we will prove that in the case where no error happens during the adiabatic evolution this process indeed gives an encoded operation g on the code space. For a circuit G, we dene a setT (G) = {Z m }⋃{X m }⋃{Z m 1 ⋅Z m 2 }, where m ranges over all qubits inG and m 1 , m 2 range over all pairs of qubits shared by any two-qubit gates in G. Theorem 1. Given a fault-tolerant circuit G dened for a stabilizer code C, with E ⊇ E local ⊃T (G), then following the steps listed above we can perform a holonomic encoded operation g =∏ p l=1 g p−l+1 for the code space P 0 . 125 Proof. It is easy to check that there is always a nite energy gap between P 0 (t) and any otherP s (t) during the process. So if we choose the time scale properly,P 0 (t) can always satisfy the adiabatic condition. Consider the qth step of the implementation. If the qth gate is single qubit gate, it acts on some qubit m. If it is a two qubit gate, it acts on a pair of qubits m 1 and m 2 . According to Eq. (5.41), for the qth step, we dene i ~ Q(t;t q−1 )=P 0 (t q−1 )U q (t;t q−1 ) _ U q (t;t q−1 )P 0 (t q−1 ): (5.42) Assume we are in step 3 (the argument for step 4 is the same with a trivial modication). U q (t;t q−1 ) is chosen to be g fq(t) q , where g q is one of the gates fromU 2 . U q (t;t q ) can be represented explicitly for this set of gates as follows: U R x m q (t;t q−1 )= exp−i 4 f q (t)X m ; U Zm 1 Zm 2 q (t;t q−1 )= expi 4 f q (t)Z m 1 Z m 2 ; U Sm q (t;t q−1 )= exp−i 4 f q (t)Z m ; U 8m q (t;t q−1 )= exp−i 8 f q (t)Z m : (5.43) Dene the code C q−1 as the ground space of H(t q−1 ), i.e., the space projected onto by P 0 (t q−1 ), and assume V 0 (t q−1 ) to be the horizontal lift of P 0 (t q−1 ) = V 0 (t q−1 )V 0 (t q−1 ). According to Def. 3, ∏ p l=q g p+q−l maps E local to a subset of E, which is the correctable error set of our codeC, so it's easy to check thatE local is a correctable error set for code C q−1 , dened by P 0 (t q−1 ). SinceT (G)⊂E local , according to Eq. (5.29), we could have: ~ Q R x m (t;t q−1 )=−P 0 (t q−1 ) 4 _ f q (t)X m P 0 (t q−1 )= 1 (t)P 0 (t q−1 ); ~ Q Zm 1 Zm 2 (t;t q−1 )=P 0 (t q−1 ) 4 _ f q (t)Z m 1 Z m 2 P 0 (t q−1 )= 2 (t)P 0 (t q−1 ); ~ Q Sm (t;t q−1 )=−P 0 (t q−1 ) 4 _ f q (t)Z m P 0 (t q−1 )= 3 (t)P 0 (t q−1 ); ~ Q 8m (t;t q−1 )=−P 0 (t q−1 ) 8 _ f q (t)Z m P 0 (t q−1 )= 4 (t)P 0 (t q−1 ): (5.44) 126 It is easy to see that r (t), r =1, 2, 3, 4 are all real. It is necessary to check that, for each step, Eq. (5.39) is satised. We have P 0 (t)=U q (t;t q−1 )P 0 (t q−1 )U q (t;t q−1 ): (5.45) We will just show the case where anR x gate is applied at theqth stage; the calculations for other gates are just the same. @ @ U Xm q (t+;t)S =0 ;P 0 (t)=−i 4 _ f q (t)X m ;P 0 (t) =−i 4 U Xm q (t;t q−1 ) _ f q (t)X m ;P 0 (t q−1 )U Xm q (t;t q−1 ): (5.46) We multiply P 0 (t q−1 ) by [X m ;P 0 (t q−1 )] and have P 0 (t q−1 )[X m ;P 0 (t q−1 )]=P 0 (t q−1 )(X m P 0 (t q−1 )−P 0 (t q−1 )X m ) =−P 0 (t q−1 ) 4 1 (t) _ f q (t) I+X m ≠ 0: (5.47) So, [X m ;P 0 (t q−1 )] ≠ 0, if X m ∉ ⟨S j (t q−1 )⟩. This is indeed true in this case, since for a well-dened circuit, X m ∉ ⟨S j (t q−1 )⟩. Otherwise, R x m would have no eect at the qth stage. Then we have @ @ U Xm q (t+;t)S =0 ;P 0 (t)≠ 0; (5.48) when _ f q (t)≠ 0. For any ~ Q r (t;t q−1 ), from Eq. (5.28) we get @ @t h(t;t q−1 )=−iV 0 (t q−1 ) ~ Q r (t;t q−1 )V 0 (t q−1 )h(t;t q−1 )=−i r (t)h(t;t q−1 ): (5.49) The solution of this equation is: h(t;t q−1 )∝h(t q−1 ;t q−1 )=I K : (5.50) 127 So the horizontal lift during [t q−1 ;t q ] is completely determined by U q (t;t q−1 ) up to an unimportant global phase: V 0 (t)∝U q (t;t q−1 )V 0 (t q−1 ): (5.51) At the end of this step, V 0 (t q )∝g q V 0 (t q−1 ). From Eq. (5.18), we could obtain the nal state for a given initial state (0)∈C: (T)∝V (T)V 0 (0) (0) = p M l=1 g p−l+1 V 0 (0)V 0 (0) (0) = g P 0 (0) (0)= g (0); (5.52) which is the encoded operation we desired. Note that the nal Hamiltonian H f = ∑ s " s g P s (0) g ≠H i , our evolution is cyclic only for ground space. Remark 3. Theorem. 1 solves the problem stated in Sec. 5.2.1 to nd a proper path for given holonomy. Note that the requirement of a fault-tolerant circuit in the imple- mentation is crucial here. If it is not satised, the horizontal lift of P 0 (t) may not be completely determined by U q at each step. The condition thatT (G) ⊂E local is not a very strong restriction. Indeed, it is always satised by stabilizer codes with d ≥ 5, and will generally be satised if we start with a fault-tolerant construction. Also note that in principle this theorem is not restricted to Cliord circuits that contain no ~8 gates. In practice, it is dicult to build the corresponding Hamiltonians constructed in our procedure, because they are hard to represent. The following theorem will show that, in order to make this process fault-tolerant, Cliord circuits are sucient and preferred. 5.3.2 Fault-Tolerance of the Scheme In this section, we will discuss the fault-tolerance of the steps to realize holonomic quan- tum computation as presented above. 128 Theorem 2. Suppose we are given a fault-tolerant circuitG dened for a stabilizer code C with E ⊇E local ⊃T (G). If G doesn't contain ~8, then by following the steps of the scheme listed in Sec. 5.3.1, we will get a fault-tolerant cyclic adiabatic process by the meaning of Def. 5. Proof. Without loss of generality, we assume an error happens at time t q (the extension of the proof to any time t is trivial). Let P 0 (t q )=V 0 (t q )V 0 (t q ) be the projector for the code C tq . Since the circuit we follow is fault-tolerant, ∏ p l=q+1 g p+q−l+1 maps E local to a subset of E, which is a correctable error set of our nal code C (since the evolution is cyclic). It is easy to check thatE local is a correctable error set for codeC tq . Assuming that E tq ∈E local is the error that happens at time t q , it can be represented as E tq =∑ c E tq where {E tq } is a nite set of operators that spansE local . {E tq } can always be chosen to satisfy the following error-correction condition: P 0 (t q )E tq E tq P 0 (t q )=d P 0 (t q ); (5.53) where d is a diagonal matrix whose elements are either one or zero. Those E tq with d = 0 have no eect on the code space. We can always pick K ′ = 2 n−k operators from {E tq } with d = 1 to form a set {E tq K ′ } . We can then construct another set of correctable errors with linear combination of element in E tq K ′ : F tq = K ′ Q =1 E tq R ; (5.54) with some unitary matrixR (such a unitary matrix always exists and is not unique) such that: F tq P 0 (t q )F tq =P s (t q ); for all : (5.55) It is easy to verify that {F tq } still satises the error correction condition: P 0 (t q )F tq F tq P 0 (t q )=d P 0 (t q ): (5.56) 129 Now,E tq can be represented by: E tq =∑ c ′ F tq . As long as we can correct eachF tq , we can correctE tq . So we consider these errors individually. If no error happens, according to the Theorem. 1, the horizontal lift ofP 0 (t) isV 0 (t)∝U(t; 0)V 0 (0), and the nal state is (T)∝ g (0); (5.57) for (0)∈C. The state after an error F tq occurs is (t q )∝F tq V 0 (t q )V 0 (0) (0): (5.58) According to Eq. (5.55), F tq V 0 (t q ) represents an orthonormal frame of P s (t q ). Next, we prove for P s (t), t >t q , that the adiabatic condition Eq. (5.40) is satised by this scheme. We need consider only a single time segment [t q ;t q+1 ]. Again, we just treat the case ofg q+1 =R x m gate; the arguments for the other gates are exactly the same, since the generators of these gates are all Pauli operators. For any ≠, P s (t) @H(t) @t P s (t)= −i 4 _ f q+1 (t)⋅U Xm q+1 (t;t q )P s (t q )X m H(t q )P s (t q ) −P s (t q )H(t q )X m P s (t q )U Xm q+1 (t;t q ); (5.59) where P sr (t)=U Xm q+1 (t;t q )P sr (t q )U Xm q+1 (t;t q ); (5.60) for r = or . Dene the index set I = {1; 2;:::n−k}, and two sets A = {j ∈ IS[S j (t q );g q+1 ]≠ 0} andB =IA . We have P sr (t q )= n−k M j=1 I+s r j S j (t q ) 2 = M j∈A I+s r j S j (t q ) 2 ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ P A sr (tq) ⋅ M j ′ ∈B I+s r j ′ S j ′(t q ) 2 ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ P B sr (tq) : (5.61) 130 G is composed of gates fromU 2 , whose elements are in the normalizer of G n , so at any stage q, S j (t q )∈G n . So we have P s (t q )X m H(t q )P s (t q ) =" s (t q )X mM j∈A I−s j S j (t q ) 2 P B s (t q )P B s (t q )P A s (t q ): (5.62) Similarly, we have P s (t q )H(t q )X m P s (t q ) =" s (t q )P A s (t q )P B s (t q )P B s (t q )M j∈A I−s j S j (t q ) 2 X m : (5.63) If s j ≠ s j for any j ∈B, then Eq. (5.62) and Eq. (5.63) would be zero, and the adiabatic condition is automatically satised. For those s such that s j = s j for all j ∈B, we have P s (t q )X m H(t q )P s (t q ) =" s (t q )X mM j∈A I−s j S j (t q ) 2 I+s j S j (t q ) 2 P B s (t q ); (5.64) and P s (t q )H(t q )X m P s (t q ) =" s (t q )P B s (t q )M j∈A I+s j S j (t q ) 2 I−s j S j (t q ) 2 X m : (5.65) The above two expressions are nonzero only if s j =−s j for all j ∈A . Therefore, there is only one such that P s _ H(t)P s ≠ 0 and hence needs further calculation. For that specic s , we have a simple relation: X m P s (t q )X m =P s (t q ): (5.66) 131 We obtain ZP s (t) _ H(t)P s (t)Z 1 = 4 _ f q+1 (t)Z" s (t q )X m P s (t q )−" s (t q )P s (t q )X m Z 1 = 4 _ f q+1 (t)K⋅T" s (t q )−" s (t q )T: (5.67) So the LHS of Eq. (5.40) reduces to _ f q+1 (t) 4T" s (t q )−" s (t q )T : (5.68) If we are in Step 3, since SA S is odd, we have S" s (t q )−" s (t q )S= T− Q j∈A 2s j T≥ 2; (5.69) and if we are in Step 4, because of our operation to break the degeneracy by setting C b = 0:5, we have S" s (t q )−" s (t q )S= U− Q j∈A j≠b 2s j −s b U≥ 1: (5.70) If 4 _ f q+1 (t) ≪ 1 is satised, which is always possible, then P s (t) satises the adiabatic condition for time segment [t q ;t q+1 ]. The same argument can be applied to the rest of the time segments to show that the adiabatic condition can always be satised by choosing appropriate functions f(t). Now, we can use the evolution equation Eq. (5.18): (T)=V s (T)V 0 (t q )F tq F tq V 0 (t q )V 0 (0) (0) =V s (T)V 0 (t q )V 0 (t q ) ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ I K V 0 (t q )F tq F tq V 0 (t q )V 0 (t q )V 0 (t q ) ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ I K V 0 (0) (0) =V s (T)V 0 (t q )d V 0 (t q )V 0 (t q )V 0 (t q )V 0 (0) (0) =V s (T)V 0 (0) (0); : (5.71) 132 where the third equality follows from Eq. (5.56). Here, V s (t) is dened as V s (t)=U(t;t q )F tq V 0 (t q )h(t;t q ) for t>t q ; (5.72) as the horizontal lift of P s (t) given initial condition V s (t q )=F tq V 0 (t q ). Again, we just focus on one time segment t∈ [t q ;t q+1 ] and the single gate R x m , since the rest are just the same: ~ Q R x m (t;t q )=P s (t q )Q R x m (t;t q )P s (t q ) =− 4 _ f q (t)P s (t q )X m P s (t q )= 0: (5.73) So h(t;t q )=I K , and according to Eq. 5.28, V s (t)=U(t;t q )F tq V 0 (t q ) (5.74) is the only horizontal lift of P s (t) for t>t q . Following Eq. (5.71), we have (T)=U(T;t q )F tq V 0 (t q )V 0 (0) (0)=F T U(T;t q )V 0 (t q )V 0 (0) (0) =F T U(T; 0)V 0 (0)V 0 (0) (0)=F T g (0); (5.75) where F T is dened as U(T;t q )F tq U (T;t q ), with U(T;t q ) =∏ p l=q+1 g p+q−l+1 . Since the circuit we follow is fault-tolerant, F T ∈E. Taking dynamic phases into account, if E tq occurs, the nal state should be (T)=Q c ′ exp−i S T tq " s (t)dtF T g (0); (5.76) so the adiabatic process we propose here is fault-tolerant by the meaning of Def. 5. Remark 4. We now discuss some details of the adiabatic theorem and its application to our scheme. The traditional version of the adiabatic theorem stated in [96] guarantees 133 that the adiabatic approximation is satised with precision ≤ 2 during the adiabatic evolution, if the condition sup t∈[0;T] ∥P s (t) @ @t H(t)P s (t)∥ 1 inf t∈[0;T] K " s (t)−" s (t) 2 ≤; for any ≠; (5.77) is satised (note thatK = 2 k is the dimension of the code space), which is equivalent to: sup q;t∈[0;T] _ f q (t) 4 ≤ (5.78) in our case. However, it is known that this statement is neither sucient nor necessary, and under certain conditions on the Hamiltonian, we can obtain better results [64, 86]. According to Ref. [86], for a HamiltonianH(#)(#=t~T ) that is analytic near [0; 1] in the complex plane, with the absolute value of the imaginary part of the nearest pole being , and the rstN ≥ 1 derivatives at boundaries equal to zero, i.e., H (l) (0)=H (l) (1)= 0 for l≤N , if we set T = e N 2 d 3 min ; (5.79) where = sup #∈[0;1] ∥ dH~d#∥ ∞ ; (5.80) (∥ ⋅∥ ∞ is standard operator norm, and d min is the minimum spectral gap) then the adiabatic approximation error satises ad ≤ (N + 1) +1 e −N ; (5.81) or equivalently, ad ≲ (cT + 1) +1 e −cT ; (5.82) with c = d 3 e 2 . This means that we can decrease the adiabatic error exponentially with evolution time T , which is proportional to N . Applying this theorem to our piecewise 134 adiabatic evolution, for the qth segment, we set T q =t q −t q−1 to be e N 2 q , where q is dened on theqth time segment, andf q (t) is chosen such that a) the boundary condition mentioned above is satised, and b)H(#) is analytic near [0; 1]. The adiabatic error for an encoded unitary operation composed of p gates can therefore be bounded by ad ≲p⋅ sup 1≤q≤p (c q T q + 1) +1 e −cqTq : (5.83) During the adiabatic process, the energy gap between the ground space and any other eigenspace is lower-bounded by 1, and this does not decrease with the size of the problem or the number of levels of code concatenation. Again, we assume the qubits are independently coupled to the thermal environment and the corresponding thermal errors are local and low-weight during certain period of evolution time. Those low- weight thermal excitations will cause transitions from the ground space to higher energy excited spaces. Their rate will decrease exponentially with the existence of an energy gap [3], thermal ∼O(exp(−d min )), while Eq. (5.79) shows that the time needed to nish the process grows inversely as the cube of the minimum gap. The system error can be bounded by the sum of these two errors [85]: S < thermal + ad : (5.84) So, qualitatively, we can make both the adiabatic error and thermal excitation exponen- tially small with ecient overhead in processing time. (However see the discussion in Sec. 5.5 for possible limitations of this argument.) Theorem 2 builds an equivalence relation between a fault-tolerant encoded Cliord circuit and a fault-tolerant adiabatic process that gives the same encoded unitary oper- ation. For most stabilizer codes (e.g., Steane code, the surface code, or the C4 code), encoded operations in the Cliord group can all be realized by such fault-tolerant circuits. Standard techniques, like magic state injection and distillation, can realize fault-tolerant 135 encoded non-Cliord gates like the encoded ~8 and encoded Tooli gates. Magic state distillation can be implemented by fault-tolerant encoded unitary gates from the Cliord group. Thus, this holonomic scheme is universal for fault-tolerant quantum computation. 5.4 Examples In this section, we apply the procedure developed above to construct adiabatic processes for specic codes. For pedagogical purposes, our rst example realizes the encoded X for the simplest 3-qubit repetition code. Our second example is the 7 qubit Steane code. This example is of practical importance because through concatenation of this simple code, fault-tolerant quantum computation can be achieved when the error rate is lower than the threshold. 5.4.1 3-qubit Repetition Code There are two stabilizer generators for this code, as shown below: 1 2 3 S 1 Z Z I S 2 I Z Z The encoded X operator for this code is X = X 1 X 2 X 3 . The encoded X gate can be performed by a circuit like X = R x 1 R x 1 R x 2 R x 2 R x 3 R x 3 , so the process takes 6 steps. The initial Hamiltonian is H(0)=−Z 1 Z 2 −Z 2 Z 3 : (5.85) For the rst step of the adiabatic process, we have [X 1 ;Z 1 Z 2 ] ≠ 0 for t ∈ [0;t 1 ]. So the Hamiltonian during that interval is H(t)=− cosf 1 (t) 2 Z 1 Z 2 + sinf 1 (t) 2 Y 1 Z 2 −Z 2 Z 3 ; (5.86) 136 with H(t 1 )=Y 1 Z 2 −Z 2 Z 3 . In the second step, for t∈ [t 1 ;t 2 ], the Hamiltonian is: H(t)= cosf 2 (t) 2 Y 1 Z 2 + sinf 2 (t) 2 Z 1 Z 2 −Z 2 Z 3 ; (5.87) withH(t 2 )=Z 1 Z 2 −Z 2 Z 3 . In the third step, we see that [X 2 ;Z 1 Z 2 ]≠ 0 and [X 2 ;Z 2 Z 3 ]≠ 0, which implies that there might exist aP s that doesn't satisfy the adiabatic condition. Actually, as shown on the left side of Fig. 5.3, we have [P s={1;−1} (t 2 )+P s={−1:1} (t 2 );X 2 ]= {s 1 =1, s 2 =1} {s 1 = 1, s 2 = -1} {s 1 = -1, s 2 = 1} {s 1 = -1, s 2 = -1} {s 1 = -1, s 2 = -1} {s 1 = -1, s 2 = 1} {s 1 = 1, s 2 = -1} {s 1 =1, s 2 =1} Figure 5.3: The variation of the energy diagram at the beginning of the third and fourth step to break the degeneracy of space {s 1 = 1;s 2 =−1} and {s 1 =−1;s 2 = 1} . 0, which means both P s={1;−1} (t 2 ) and P s={−1:1} (t 2 ) will not satisfy the adiabatic condi- tion during the evolution if we do not break this degeneracy. Following the scheme in Sec. 5.3.1, we alter the Hamiltonian att 2 to beH(t 2 )=Z 1 Z 2 −0:5Z 2 Z 3 . The correspond- ing energy diagram is shown on the right side of Fig. 5.3. Then we vary the Hamiltonian in the following way for t∈ [t 2 ;t 3 ]: H(t)= cosf 3 (t) 2 Z 1 Z 2 − sinf 3 (t) 2 Z 1 Y 2 − 0:5 cosf 3 (t) 2 Z 2 Z 3 + 0:5 sinf 3 (t) 2 Y 2 Z 3 ; (5.88) 137 withH(t 3 )=−Z 1 Y 2 +Y 2 Z 3 . We continue to break the degeneracy in the fourth step, since X 2 again does not commute with both P s={1;−1} (t 3 ) and P s={1;−1} (t 3 ). So for t∈ [t 3 ;t 4 ], the Hamiltonian is: H(t)=− cosf 4 (t) 2 Z 1 Y 2 − sinf 4 (t) 2 Z 1 Z 2 + 0:5 cosf 4 (t) 2 Y 2 Z 3 + 0:5 sinf 4 (t) 2 Z 2 Z 3 ; (5.89) withH(t 4 )=−Z 1 Z 2 +0:5Z 2 Z 3 , which can then be restored toH(t 4 )=−Z 1 Z 2 +Z 2 Z 3 . The fth and sixth steps are just like the rst and the second steps. The nal Hamiltonian is H(T =t 6 ) =−Z 1 Z 2 −Z 2 Z 3 , which is equal to the initial Hamiltonian, and we obtain our geometric encoded X operation. Remark 5. The encoded Z operator for this code is Z =Z 1 Z 2 Z 3 . However, we cannot use our scheme to build the adiabatic process according to this circuit, since Z 1 ,Z 2 and Z 3 all commute with the initial HamiltonianH(0). We can see that for this simple code, there doesn't exist anE that includes {Z 1 ;Z 2 ;Z 3 }, so the conditions for both Theorem 1 and Theorem 2 are not satised. 5.4.2 The Steane Code Fault-tolerant quantum computation can be realized through concatenation of Steane code. In our scheme, we only apply our scheme at the bottom level of concatenation. For higher levels, encoded unitary operations and error correction are done in the usual way. By doing so, we keep the constant energy gap between the ground space and error spaces and thus maintain the ability to suppress low weight thermal errors, and we can bound the weight of terms in the system Hamiltonian. One set of generators of the stabilizer group of the Steane code is listed below: 138 1 2 3 4 5 6 7 S 1 X I X X I X I S 2 I X X I I X X S 3 X I X I X I X S 4 Z I Z Z I Z I S 5 I Z Z I I Z Z S 6 Z I Z I Z I Z The circuits for the encoded Hadmard, encoded S and encoded X and Z gates are all bit-wise transversal, and thus naturally fault-tolerant. Note that each Hadmard can be decomposed into SR x S up to a global phase. The geometric realizations of such gates are similar to that of the encoded X for the 3-qubit repetition code shown above. So in this section we focus on the CNOT gate. 5.4.2.1 CNOT Gate For the Steane code, the fault-tolerant encoded CNOT 1→2 (control qubit encoded in block 1, target qubit encoded in block 2) can be realized transversally between two blocks of qubits. We illustrate our scheme for one pair of qubits from the two blocks, all the other operations are the same. The ini- tial Hamiltonian H(0) can be written as −∑ 7 j=1 S 1 j +S 2 j , where S i j is the jth generator for the ith block. Each physical CNOT can be decomposed into R zz S S R x S S S R x S up to a global phase. As we can see, there are nine gates in the circuit. The transformation of the rst and last four single qubit gates has been discussed before. We will just show the Hamiltonian during time interval [t 4 ;t 5 ] when the two-qubit gate is performed. 139 The Hamiltonian at time t 4 can be shown to be: H(t 4 )=− 7 Q j=1 S 1 j −S 2 2 −S 2 5 −Z 2 1 X 2 3 X 2 4 X 2 6 −Z 2 1 X 2 3 X 2 5 X 2 7 −Y 2 1 Z 2 3 Z 2 4 Z 2 6 −Y 2 1 Z 2 3 Z 2 5 Z 2 7 : (5.90) For an R zz gate acting on qubit 1 in both blocks, we have [Z 1 1 Z 2 1 ;X 1 1 X 1 3 X 1 4 X 1 6 ] ≠ 0, [Z 1 1 Z 2 1 ;X 1 1 X 1 3 X 1 5 X 1 7 ] ≠ 0, [Z 1 1 Z 2 1 ;Y 2 1 Z 2 3 Z 2 4 Z 2 6 ] ≠ 0 and [Z 1 1 Z 2 1 ;Y 2 1 Z 2 3 Z 2 5 Z 2 7 ] ≠ 0 for t ∈ [t 4 ;t 5 ], so the Hamiltonian during this interval can be chosen to be: H(t)= −Q j≠1 j≠3 S 1 j −S 2 2 −S 2 5 −Z 2 1 X 2 3 X 2 4 X 2 6 −Z 2 1 X 2 3 X 2 5 X 2 7 −cosf5(t) 2 X 1 1 X 1 3 X 1 4 X 1 6 +sinf5(t) 2 Y 1 1 X 1 3 X 1 4 X 1 6 Z 2 1 −cosf5(t) 2 X 1 1 X 1 3 X 1 5 X 1 7 +sinf5(t) 2 Y 1 1 X 1 3 X 1 5 X 1 7 Z 2 1 −cosf5(t) 2 Y 2 1 Z 2 3 Z 2 4 Z 2 6 −sinf5(t) 2 Z 1 1 X 2 1 Z 2 3 Z 2 4 Z 2 6 −0:5cosf5(t) 2 Y 2 1 Z 2 3 Z 2 5 Z 2 7 −0:5sinf5(t) 2 Z 1 1 X 2 1 Z 2 3 Z 2 5 Z 2 7 ; (5.91) with H(t 5 )=−Q j≠1 j≠3 S 1 j −S 2 2 −S 2 5 −Z 2 1 X 2 3 X 2 4 X 2 6 −Z 2 1 X 2 3 X 2 5 X 2 7 +Y 1 1 X 1 3 X 1 4 X 1 6 Z 2 1 +Y 1 1 X 1 3 X 1 5 X 1 7 Z 2 1 −Z 1 1 X 2 1 Z 2 3 Z 2 4 Z 2 6 −Z 1 1 X 2 1 Z 2 3 Z 2 5 Z 2 7 : (5.92) After all nine gates have been performed, the Hamiltonian will be: H(T 1 )=−Q j≠1 j≠3 (S 1 j +S 2 j ) −X 1 1 X 1 3 X 1 4 X 1 6 X 2 1 −X 1 1 X 1 3 X 1 5 X 1 7 X 2 1 −Z 1 1 Z 2 1 Z 2 3 Z 2 4 Z 2 6 −Z 1 1 Z 2 1 Z 2 3 Z 2 5 Z 2 7 : (5.93) 140 After repeating this procedure on all 7 pairs of qubits, the nal Hamiltonian will be: H(T)=−X 1 1 X 1 3 X 1 4 X 1 6 X 2 1 X 2 3 X 2 4 X 2 6 −X 1 2 X 1 3 X 1 6 X 1 7 X 2 2 X 2 3 X 2 6 X 2 7 −X 1 1 X 1 3 X 1 5 X 1 7 X 2 1 X 2 3 X 2 5 X 2 7 −Z 1 1 Z 1 3 Z 1 4 Z 1 6 −Z 1 2 Z 1 3 Z 1 6 Z 1 7 −Z 1 1 Z 1 3 Z 1 5 Z 1 7 −X 2 1 X 2 3 X 2 4 X 2 6 −X 2 2 X 2 3 X 2 6 X 2 7 −X 2 1 X 2 3 X 2 5 X 2 7 −Z 1 1 Z 1 3 Z 1 4 Z 1 6 Z 2 1 Z 2 3 Z 2 4 Z 2 6 −Z 1 2 Z 1 3 Z 1 6 Z 1 7 Z 2 2 Z 2 3 Z 2 6 Z 2 7 −Z 1 1 Z 1 3 Z 1 5 Z 1 7 Z 2 1 Z 2 3 Z 2 5 Z 2 7 ; (5.94) which is equal to H(0). Although the nal Hamiltonian equals to the initial Hamilto- nian, the maximum weight of its elements has doubled, which is not good for practical implementation. Although recent results have shown how to map such Hamiltonians to more physically reasonable two-body interactions [148, 72, 101], it is still important to decrease the maximum weight of the Hamiltonian terms. 5.4.2.2 Lowering the Weight of the Hamiltonian The maximum weight of the terms in the nal Hamiltonian given by Eq. (5.94) is 8, com- pared to the 4 for the initial Hamiltonian. This problem may not exist for a fault-tolerant scheme based on the surface code [52], since during the process of code deformation, the weight of the stabilizer generators is always bounded by 4. However, even for the Steane code, the weight of Hamiltonian can be lowered. Here we setU L = CNOT 12 . After the rst three transversal CNOTs, the Hamiltonian would become: H(T 3 )=−X 1 1 X 1 3 X 1 4 X 1 6 X 2 1 X 2 3 −X 1 2 X 1 3 X 1 6 X 1 7 X 2 2 X 2 3 −X 1 1 X 1 3 X 1 5 X 1 7 X 2 1 X 2 3 −Z 1 1 Z 1 3 Z 1 4 Z 1 6 −Z 1 2 Z 1 3 Z 1 6 Z 1 7 −Z 1 1 Z 1 3 Z 1 5 Z 1 7 −X 2 1 X 2 3 X 2 4 X 2 6 −X 2 2 X 2 3 X 2 6 X 2 7 −X 2 1 X 2 3 X 2 5 X 2 7 −Z 1 1 Z 1 3 Z 2 1 Z 2 3 Z 2 4 Z 2 6 −Z 1 2 Z 1 3 Z 2 2 Z 2 3 Z 2 6 Z 2 7 −Z 1 1 Z 1 3 Z 2 1 Z 2 3 Z 2 5 Z 2 7 : (5.95) The maximum weight of any term is 6. This Hamiltonian can be transited the following form: H ′ (T 3 )=−X 1 1 X 1 3 X 1 4 X 1 6 X 2 4 X 2 6 −X 1 2 X 1 3 X 1 6 X 1 7 X 2 6 X 2 7 −X 1 1 X 1 3 X 1 5 X 1 7 X 2 5 X 2 7 −Z 1 1 Z 1 3 Z 1 4 Z 1 6 −Z 1 2 Z 1 3 Z 1 6 Z 1 7 −Z 1 1 Z 1 3 Z 1 5 Z 1 7 −X 2 1 X 2 3 X 2 4 X 2 6 −X 2 2 X 2 3 X 2 6 X 2 7 −X 2 1 X 2 3 X 2 5 X 2 7 −Z 1 4 Z 1 6 Z 2 1 Z 2 3 Z 2 4 Z 2 6 −Z 1 6 Z 1 7 Z 2 2 Z 2 3 Z 2 6 Z 2 7 −Z 1 5 Z 1 7 Z 2 1 Z 2 3 Z 2 5 Z 2 7 ; (5.96) which again has maximum weight of 6. Note that the transition between these two Hamiltonian representations can be arbitrarily fast or adiabatically, since they commute 141 with each other. During the transition, the energy gap always exists. Then, if we perform the remaining four transversal CNOTs by an adiabatic process, the nal Hamiltonian will return to the initial one represented by H(T)=−∑ 7 j=1 (S 1 j +S 2 j ). During the whole process, the maximum weight of terms in the Hamiltonian is reduced from 8 to 6. The weight does not increase when the codes are concatenated, since we just apply our scheme to the bottom level of concatenation and do higher levels operations in their usual way. So we can realize HQC fault-tolerantly with maximum Hamiltonian weight 6 while keeping the constant energy gap. For a small code like the Steane code, this may be the best we can do. For larger block codes, especially for topological codes like the surface code, as we will see in Chapter 6, the weight of the Hamiltonian terms during the adiabatic process can be well bounded by a small constant. 5.5 Summary and Conclusion We have described a scheme for fault-tolerant HQC on a stabilizer code, which takes advantage of a constant energy gap during the process as well as of the intrinsic resilience of HQC. We've shown that from a fault-tolerant circuit without ~8 gates, we can sys- tematically construct a fault-tolerant adiabatic process that implements the very same encoded unitary operation as the original circuit, with information encoded in the ground state of the system Hamiltonian. As long as we can realize holonomic versions of gates in the Cliord group, we can implement fault-tolerant universal quantum computation by using magic state distillation. Holonomic single-qubit operation has been recently realized on trapped single 40 Ca + ion system through adiabatic evolution [140]. Theoretical work on non-adiabatic non- abelian HQC has also been proposed [128], and corresponding experiments have recently been realized in superconducting qubits [1], NMR [45] and diamond nitrogen-vacancy centres [159]. Applying our strategy to actual physical systems will need certain tech- niques, like quantum gadgets [72, 101] or the digital quantum simulator [148], to build 142 the many-body interactions. If the system Hamiltonian is built in one of these eective ways, rather than being fundamental, it may dramatically change the local error model we have assumed. This eect needs further investigation. Our fault-tolerant HQC scheme diers from adiabatic gate teleportation (AGT) [8], and the scheme in Refs. [102, 103], in the following ways: 1. Instead of focusing on single qubit unitary operations or two-qubit unitary operations, our scheme obtains holonomy through directly dragging the ground space (code space) of the system, whose path in the Grassmann manifold forms a closed loop. 2. During the adiabatic procedure, the energy spectrum of system basically remains the same, and there always exists an energy gap between the code space and the excited spaces, so that information is protected from low weight thermal excitation by an energy gap. There are several advantages over other schemes here. We can reduce the low-weight error rate at the bottom level of code concatenation due to the existence of an energy gap, so that the frequency of error correction procedure can be greatly reduced at bottom physical level. The measurement of stabilizer generators and subsequent error correction can themselves introduce more errors, and this is one of the reasons why thresholds for current fault-tolerant schemes are so low. Moreover, the number of physical qubits needed in a fault-tolerant scheme is strongly dependent on the error rate at the physical level. Error rates substantially smaller than the threshold allow much smaller numbers of physical qubits. Eventually, our scheme may reduce the resource overhead needed to do fault-tolerant quantum computation. On the other hand, our scheme seems naturally compatible with Hamiltonian- protected quantum memories [150], and has the potential to do fault-tolerant compu- tation based on those kind of memories, especially those with self-correcting ability [65, 33, 66, 26] No dynamical method seems capable of manipulating the topological degrees of freedom encoded in the ground space of these memory in the presence of a system Hamiltonian, as far as we know, since they would introduce terms that do not commute with the system Hamiltonian. However, our method of locally deforming 143 these Hamiltonians could potentially do quantum computation on such systems. We conjecture that during this kind of local deformation procedure, the system will keep its self-correction capability in a thermal environment while quantum computation is implemented. Further investigation of this idea is the main topic of Chapter 6. We note that this method to construct fault-tolerant HQC is basically a serial proce- dure from gate to gate. For circuits with large depth, we could investigate the possibility of parallel operation. As we will see, This generalization is quite nature in the case of surface codes. 144 Chapter 6 Fault-tolerant Holonomic Quantum Computation in Surface Codes 6.1 Preliminaries As we have seen in Chapter 5, fault-tolerant HQC in general stabilizer codes usually suers several problems, which includes: 1) the weight of the Hamiltonian during the evolution can be large (even not bounded by a constant) and not local; 2) for a system of small code, it is dicult to maintain the degeneracy of the ground space, which is easily broken by even small perturbations, causing unavoidable phase errors; 3) paral- lelization of dierent logical operations may be dicult. In this Chapter, we apply try to overcome all these diculties by applying the idea of HQC to FTQC scheme using surface codes [73]. A combination of ideas from TQC and QEC gives schemes of active error correction architecture based on topological QEC codes, especially the surface codes [113, 115] and color codes [82], using code deformation [21, 17]. In this approach, one works directly with the quantum error correcting code used in TQC, without introducing a Hamiltonian to protect quantum information with energy gap [38]. In the case of surface codes, one truncates it by turning o some stabilizer generators in a region to create a hole or defect. Rather than encoding information in the label space of anyons in TQC, each hole can be viewed as an encoded qubit. Via a sequence of measurements, the boundary of holes can 145 be deformed. One can then braid holes by using suitable deformations to perform logical operations between logical qubits associated with the holes. Because of its tolerance of local errors [38], scalable structure and high threshold (0:57%) [53, 52], surface codes have attracted a great deal of attention, and impressive experimental progress in this direction has been made recently with superconducting qubits [10, 71, 35]. In this Chapter, we focus on surface codes with a stabilizer Hamiltonian turned on to form a topological quantum memory [38, 150] on a single 2D lattice, to protect quantum information encoded in the degenerate ground space from both thermal errors and perturbations. We explicitly construct all processes needed to do universal HQC based on the surface code, by adiabatically deforming this gapped Hamiltonian. By adiabatically braiding dierent types of holes on the surface, one performs a topologically protected non-Abelian geometric logical CNOT gate. Throughout the entire information processing procedure, including logical state initialization, logical state measurement, logical gates, state injection and distillation, quantum information is protected from local thermal excitations by a constant energy gap, and the weight of the Hamiltonian terms is local and bounded by 4 during the whole adiabatic code deformation process. To deal with unwanted excitations caused by errors (creation of anyons) during the adiabatic code deformation, we analyze errors propagation, and give conditions when turning o the stabilizer Hamiltonian is needed to do syndrome measurement and error correction. It can be shown that with gap protection the frequency of error correction and the physical resources needed can be greatly reduced. We conclude that the computation procedures are scalable, and that the scheme is fault tolerant. The main results of this Chapter can be found in Ref. [157]. Scheme with similar idea applied on both surface codes and color codes can be found in Ref. [30]. Some results in this Chapter may overlap with those in Chapter 5, but with dierence in the specied scenario of surface codes. 146 6.2 Surface Codes A good introduction to the surface code can be found in Refs. [53, 52]. In this section, we follow Ref. [52] and give a brief review to establish our notation. Surface codes can be viewed as a special kind of stabilizer codes dened on a 2D square lattice. In this Chapter, we implement the surface code on a two-dimensional L×L lattice, with qubits on the edges of the lattice, as shown in Fig. 6.1 for L= 8. The stabilizer generators of surface codes are two dierent kinds of operators: X s =M i∈s x i ; Z p =M i∈p z i ; (6.1) that represents vertices (X s ) and plaquette operators (Z p ) on the square lattice. Besides stabilizer generators inside the lattice, there are also ones on the boundaries for each lattice. Typically, for each surface code, there are two kinds of boundaries: X boundaries and Z boundaries. X boundaries comprise three-body X s operators on the boundary of lattice, while Z boundaries comprise three-body Z p operators, as in the boundaries shown in Fig. 6.1. In general, a lattice with two X boundaries and two Z boundaries has 2L 2 − 2L+ 1 qubits and 2L 2 − 2L stabilizer generators, and encodes 2 degrees of freedom to form a logical qubit. The corresponding logical operators are given by Z L =∏ k∈lz z k and X L =∏ k∈lx x k where l z and l x are chains of qubits that support z and x operators all the way across the lattice (see Fig. 6.1 for an example). Not shown in Fig. 6.1 are additional syndrome qubits for each plaquette and vertex, that enable one to check the sign of the associated stabilizer generator, as shown in Fig. 6.2. Inside the surface, each syndrome qubit contacts four data qubits and performs four-qubit joint measurement. On the boundaries, each syndrome qubit contacts only three data qubits and performs a three-qubit joint measurement. The corresponding quantum circuit for one stabilizer generator measurement of the Z p and X s operators are 147 X L Z L Figure 6.1: A surface code based on an 8×8 lattice with 113 physical qubits on the edges. This code contains 1 logical qubit and has distance d = L = 8, where d is the distance of the code. The four-body (or three-body) plaquette stabilizer generator (Z p ) and vertex stabilizer generator (X s ) are indicated as cyan and yellow plaquettes, respectively inside the lattice (or on the boundary). A particular choice of logical operators X L and Z L is shown. A number of qubits are aected by x (red dots) or z (purple dots) errors, leading to excited Z p operators (or m anyons) and X s operators (or e anyons). Measuring these operators yields the positions of the excited vertices and plaquettes but reveals no information about the actual physical errors which cause them. A minimum- weight matching error correction procedure applies x and z to the qubits marked by the larger red and purple circles. While the z errors are annihilated properly (up to a trivial loop of multiplication of Z p operators), the red pair underneath is connected by a topologically non-trivial path across the surface. This introduces a logical error in the state to be protected. S0⟩ M Z 1 ● 2 ● 3 ● 4 ● and 148 1 z s 2 z s 3 z s ( a ) ( b ) 2 x s 3 x s 4 x s 1 x s 4 z s Figure 6.2: Four-body plaquette operatorZ p (a) and vertex operatorX s (b) as stabilizer generators of surface code inside the lattice. The black dot in the center of the plaquettes are syndrome qubits used to do stabilizer measurement. S0⟩ H ● ● ● ● H M Z 1 2 3 4 6.3 Sketch of the Scheme In this scheme, we always regard all physical qubits on the lattice as a single big stabilizer code. We assume that the qubits independently and weakly interact with a thermal bath in the Markovian approximation. The corresponding thermal errors are local and low- weight during a certain period of evolution. Those low-weight thermal excitations will cause transitions from the ground space to excited spaces. Their rate should decrease as thermal ∼ exp(−c min ), where min is the minimum spectral gap of the system, is the inverse of temperature, and c is a constant depending on the coupling strength between system and thermal bath [4]. This is true even when the Hamiltonian is not static and changes slowly, so long as the system is weakly coupled to the thermal bath [3]. The goal is to do the whole quantum computation fault-tolerantly, while the code space is protected by an energy gap of the stabilizer Hamiltonian that exponentially suppresses errors at low temperature throughout the information processing procedure. According to Def. 1, the fault-tolerance of a procedure can be regarded as a property of the procedure itself regardless of the error model of the system. Before we go deeper, we must impose some requirements to follow in the rest of the Chapter: 149 1. Procedures like logical state preparation, logical state measurement, encoded gate operations, state injection and state distillation should be done when the system Hamiltonian is \turned on", so that a constant energy gap protects the information and the error rate for each procedure is low. 2. All procedures should be done fault-tolerantly according to Def. 1, whether adia- batic or not. 3. Syndrome measurements and error correction should be done before uncorrectable errors happen. 4. Syndrome measurements and error correction should be done as seldom as possible, since they are in general not compatible with the system Hamiltonian and we must turn o the Hamiltonian before doing them. Besides, the syndrome measurement procedure itself is quite expensive. The frequency of error correction is expected to be low if all procedures are gap-protected. 5. A threshold theorem should exist, in the sense that if the error rate is below the threshold, the computation can be made arbitrarily long by suitably increasing the lattice size. 6. It is possible to measure x and z of single physical qubits in certain circumstances even when the Hamiltonian is turned on. 7. Maximum weight of the Hamiltonian terms should be low, and the Hamiltonian should be geometrically local. 8. All procedures should be done in a single lattice. Requirements 1−4 are crucial to our main objective of reducing the physical resources and 5 guarantees that arbitrarily large-scale computation can be done. Requirement 6 is physically reasonable, and we will see its importance in Sec. 6.4. Requirement 7 comes from the fact that in real experiments, high weight and nonlocal Hamiltonians 150 are dicult or impossible. Requirement 8 is technical rather than fundamental, since it simplies the the computation architecture. 6.3.1 Adiabatic Processes In most cases, adiabatic processes can be used to simultaneously fulll most of the requirements above. For the purpose of encoding and measuring logical qubits, we will show that these can be done by open-loop adiabatic processes (for logical measurement, we also need qubit measurement), while the logical CNOT can be done by a closed-loop adiabatic processes to get a holonomy on the code space. Both such processes can be described by Eq. (1.29). In this and the following sections, we will focus on a special kind of adiabatic evolution that turns out to be particularly useful. In addition, we will discuss how it can be used to analyze propagation of potential errors and parallelism of the processes. Assume the total number of qubits on the lattice isn= 2L 2 −2L+1, so the dimension of the Hilbert space is N = 2 n . The number of logical qubits in our scheme may change over time, since we can create defects on the lattice to create logical qubits. However, we assume that when an adiabatic process is applied, the dimension of code space is xed. This can be realized by isospectral deformation of the Hamiltonian. Denote the number of logical qubits encoded in the ground space by k. Assume that at time t 0 , the initial Hamiltonian can be written as H(t 0 )=− n−k Q j=1 JS j ; (6.2) where the {S j } are a set of stabilizer generators of the surface code at time t 0 and that ⟨S j ⟩ forms the stabilizer group S. Consider the following way to adiabatically deform the Hamiltonian isospectrally: H(t)=− n−k Q j=1 JS j (t)=− n−k Q j=1 JU(t;t 0 )S j (t 0 )U (t;t 0 ); (6.3) 151 with S j (t) = U(t;t 0 )S j U (t;t 0 ) and [S i (t);S j (t)] = 0 for all i, j. The {S j (t)} can be viewed as a set of generators of an Abelian group, like the stabilizer group. The Hamiltonian also has a spectral decomposition: H(t)=Q s " s P s (t): (6.4) Here, the {P s (t)} are projectors onto the simultaneous eigenspaces of all the S j (t), with eigenvalues: " s =−JQ j s j ; (6.5) where the labels s j =±1 form a vector: s= {s 1 ;s 2 ;:::s n−k }: (6.6) The ground space evolves with the system Hamiltonian. This denes a time-dependent code spaceC t . LetP 0 (t)=U(t;t 0 )P 0 (t 0 )U (t;t 0 ) be the projector onto the ground space of H(t), such that s j = 1 for all j. We emphasize that U(t+;t) should be chosen such that @ @ U(t+;t)S =0 ; P s (t)≠ 0 for all s; (6.7) for any time t, so that the deformation procedure is nontrivial for all eigenspaces. In other word, U(t+;t) should not belong to the isotropy group of P s (t) for small values of . Like we have discussed in Chapter 5, the adiabatic condition must hold for each eigenspaceP s , so that each eigenspace undergoes nontrivial evolution under the adiabatic process, in case an error excites the system to P s during the process. The standard 152 adiabatic condition [96] for any eigenspace {P s } is stated in Eq. 5.40. Here, we presented it again for clarity: ∥P s (t) @ @t H(t)P s (t)∥ 1 K " s (t)−" s (t) 2 ≈ 0; for any ≠: (6.8) Here, K is the degeneracy of each P s . This must hold for all t ∈ [t 0 ;t p ], where∥⋅∥ 1 is the trace norm ∥A∥ 1 = Tr √ A A. It is very likely that for a Hamiltonian of the form Eq. (6.3), severalP s (t) ′ s will share the same eigenenergy, so that the adiabatic condition cannot be directly satised. Fortunately, for the surface code, we will show later that there is a natural way to cope with this problem, so that each P s (t) can satisfy the adiabatic condition during the adiabatic code deformation. As shown in Chapter 5, a closed loop adiabatic logical gate operation can be built from a fault-tolerant circuit of the corresponding stabilizer code. However, for the surface code, we in general don't know the exact fault-tolerant circuit for encoded gate operations. Moreover, we wish to do encoding and logical state measurement with gap protection, so the result in Chapter 5 cannot be applied here directly. Instead, in this Chapter, we consider a special kind of quantum circuitG composed of a sequence of gate operations {g 1 ;g 2 :::g p } giving the unitary operation p =∏ p l=1 g l . Here, g l = expi 4 Q l for some Hermitian operator Q q ∈ G n , where G n is the Pauli group acting on n qubits. For simplicity, when we talk about a \circuit" in the rest of this Chapter, we means the circuit of this type. We divide the information processing time [t 0 ;t p ] intop small steps and represent the qth time segment as [t q−1 ;t q ]. Now, set the unitary operator U q (t;t q−1 )= expif q (t)Q q ; (6.9) 153 for t ∈ [t q−1 ;t q ] and let f q ∶ [t q−1 ;t q ]→ [0;~4] be a monotonic smooth function with boundary conditions f q (t q−1 ) = 0 and f q (t q ) =~4. For each time segment [t q−1 ;t q ], we adiabatically deform the Hamiltonian: H(t;t q−1 )=U q (t;t q−1 )H(t q−1 )U q (t;t q−1 ); t∈ [t q−1 ;t q ] (6.10) and assume [Q q ;H(t q−1 )]≠ 0 so that Eq. (6.7) is satised. A state in the ground space will evolve as described by the following lemma: Lemma 1. (State Evolution) Consider a circuit composed of gates {g q } and an initial state S (t 0 )⟩ ∈ C(t 0 ), with H(t 0 ) = −∑ j JS j . We apply a sequence of Hamiltonian deformations as in Eq. (6.10), for 1 ≤q ≤p. Then, under the adiabatic approximation, the nal state will be: S (t p )⟩=e −i" 0 (tp−t 0 ) p M l=1 g l S (t 0 )⟩=e −i" 0 (tp−t 0 ) p S (t 0 )⟩: (6.11) Proof. See Appendix 6.7.1. In the case of a many-body system like the surface code, it is dicult to follow the change of the state in code space since it is hard to represent the state. One normally uses the stabilizer formalism (Heisenberg picture) to track the change of the logical Z L and X L operators during the process. The following theorem is a direct consequence of Lemma 1: Theorem 2. Suppose the initial state S (t 0 )⟩ is in the code space of a stabilizer code with generators {S j } and logical operators {X i L ;Z i L }, and that H(t 0 )=−∑ j JS j . Under the adiabatic Hamiltonian deformation described in Eq. (6.10) for 1 ≤q ≤p, the logical operators will map to X i L → p X i L p ;Z i L → p Z i L p , and the system Hamiltonian will become H(t p )=−∑ j JS ′ j =−∑ j J p S j p . If the process is cyclic for the ground space, which means p P 0 (t 0 ) p =P 0 (t 0 ), then p can be viewed as an encoded gate operation, and we have following conclusion: 154 Corollary 1. If p ∈N(S)S, where N(S) is the normalizer ofS in U(N), then p is a closed-loop holonomic operation under the adiabatic process. Remark 6. These results build a relationship between the special kind of circuitsG we are interested in and the corresponding adiabatic process. If we can nd a circuit in G giving a particular unitary, then we can translate it to an adiabatic process. However, in general, the weight of the Hamiltonian terms changes with time, and it is quite possible that during the adiabatic process, the Hamiltonian terms will become both nonlocal and high weight. Fortunately, as we will see, in the case of surface codes this can be avoided. 6.3.2 Error Propagation Although in the process described by Eq. (6.3), the ground space is protected by a constant energy gap 2J, the lifetime is aboute 2cJ in the presence of a thermal bath. This lifetime doesn't grow with the lattice size L, so the the thermal gap does not guarantee fault-tolerance. We still need to do active error correction to make the computation time arbitrarily long. We must analyze how an error caused by thermal excitation will propagate during the adiabatic process to choose the proper circuit from G and design the subsequent error correction procedure. Without loss of generality, we assume that an error E tq happens at time t q (q ≤l). Since any error operator E tq on an n-qubit system can be decomposed into a sum of Pauli operatorsE tq =∑ c F , it is sucient to analyze Pauli errors. We have following lemma: Lemma 2. (Error Propagation) If an error E tq =∑ c F q (F q ∈G n ) happens at time t q in the procedure described by Eq. (6.10), and there is an odd number of stabilizer generators S j (t r ) such that [Q r ;S j (t r−1 )]≠ 0 for all times 1≤r ≤p, then S (t p )⟩=Q c e −i"s (tp−tq) F pq p M l=1 g l S (t 0 )⟩ (6.12) where F pq =U pq F q (U pq ) withU pq =∏ p l=q+1 g l . 155 Proof. See Appendix 6.7.2. Lemma 2 gives the condition that the error will just propagate to some other error under the expected unitary evolution. The condition that at each step r the number of S j (t r−1 ) such that [Q r ;S j (t r−1 )] ≠ 0 should be odd is crucial. In general, an error will excite the ground space to another eigenspace P s , which will usually share the same energy with some other eigenspaces, so that the adiabatic condition will not hold. This condition guarantees that even when this is the case, the degenerate eigenspaces will still satisfy the adiabatic condition Eq. (6.8) and adiabatic evolution will not fail. Also, note that if p is a logical gate operator, although for the ground space P 0 the process is a cyclic evolution, e.g, p P 0 (t 0 ) p =P 0 (t 0 ), this is not true for the other eigenspaces. In general, p P s (t 0 ) p ≠P s (t 0 ) for s ≠ 0. This means that after an error excites the ground space P 0 to P s , the adiabatic process becomes open loop for P s . 6.3.3 Parallelism of Adiabatic Operation The method described in the previous sections is basically a serial operation, meaning that we need to adiabatically deform the Hamiltonian according to the gates in the circuitG step by step. However, for a large scale QC on a lattice (not only the surface code), we expect that many operations can be done in parallel, so that operations which commute with each other can be done simultaneously. Here we give the condition for those operations to parallelize. Lemma 3. (Parallelism) Suppose that at time t q , S (t q )⟩ is in the ground space C(t q ). Dene C Qr = {j S 1 ≤ j ≤ n−k;{S j (t q );Q r } = 0}. Suppose the set of operators P q = {Q r Sq+ 1≤r ≤q+M} satises the following conditions: 1. [Q r ;Q m ]= 0, for any Q r ;Q m ∈P q , 2. C Qr ⋂C Qm =∅ for any Q r ;Q m ∈P q , 3. SC Qr S is odd for all Q r ∈P q . 156 and set U q+1 (t;t q ) =∏ q+M r=q+1 exp(if(t)Q r ) with f(t) =f q+1 (t) for t ∈ [t q ;t q+1 ]. Assume the Hamiltonian changes adiabatically as H(t) = U q+1 (t;t q )H(t q )U q+1 (t;t q ). Then we have: 1. The state at time t q+1 will be: S (t q+1 )⟩=e −i" 0 (t q+1 −tq) ⎛ ⎝ q+M M l=q+1 g l ⎞ ⎠ S (t q )⟩: (6.13) 2. If an error E tq =∑ c F q (F q ∈G n ) occurs at time t q , then the state at time t q+1 will be: S (t q+1 )⟩=Q c e −i"s (t q+1 −tq) F q+1;q ⎛ ⎝ q+M M l=q+1 g l ⎞ ⎠ S (t q )⟩; (6.14) where F q+1;q =U q+1;q F q U q+1;q withU q+1;q =∏ q+M r=q+1 g r . Proof. See Appendix 6.7.3. Lemma 3 suggests that it is possible to do M steps of the adiabatic transformation described in Lemma 1 in one step, and gives the conditions for the adiabatic evolution to still be valid when errors occur. This property is extremely important. Since we need to apply our scheme to surface codes of large size, operations applied simultaneously on dierent parts of the surface can greatly improve the eciency of computation. 6.4 HQC in Surface Codes We are ready to show how to do QC fault-tolerantly by adiabatically deforming the stabilizer Hamiltonian of the surface code. As mentioned in the previous section, our goal is that all the procedures, including state preparation, ancilla preparation, logical gate operations and logical state measurements, be implemented fault-tolerantly with constant energy gap protection. In the next few subsections, we discuss how to construct these procedures, and discuss error propagation and error detection in detail. 157 State measurement is a special case worth more discussion here. At the end in the computation, when we want to read all of the data in the logical qubits, we can just turn o the Hamiltonian and measure everything. However, during the computation, when the stabilizer Hamiltonian exists, we still may need to measure logical qubits from time to time, so that actions conditioned on those classical measurement outcomes of logical qubit can be applied. We must put some restrictions on the kinds of measurements we can do that are compatible with the existence of the stabilizer Hamiltonian. The rst requirement is that the observable O we want to measure should commute with the Hamiltonian: [H;O]= 0: (6.15) This requirement guarantees that if a state encoding quantum information is in one of the eigenspaces P s before the measurement, then after the projective measurement, the state will still be in P s . If Eq. (6.15) is not satised, the measurement will lead to excitations out of the eigenspace. The second requirement is that the observable should be geometrically local, so that the measurement procedure will not introduce non-local interactions. Note that when the Hamiltonian is turned on, we do not do X s orZ p stabilizer measurements even though they commute with the system Hamiltonian and are local. The reason for this is that to projectively measure these many-body observables, we would need to introduce CNOT gates and syndrome qubits, which are not compatible with the system Hamiltonian. So in our scheme, syndrome measurements are always done when the system Hamiltonian is turned o. However, as stated in requirement 6 in the previous section, we do allow single physical qubit measurements as long as they commute with the system Hamiltonian. Errors can happen during the single qubit measurement process. There are two kinds of measurement errors. The rst kind is that, instead of an ideal measurement, some quantum process occurs during the measurement process which is equivalent to one of the following circuits: 158 S ⟩ x M Z S ⟩ z M X for z measurement and x measurement, respectively. The second kind of error can be regarded as a software error: even though the measurement is perfect, some classical noise corrupts the measurement result and we get the wrong outcome. This can be modeled by the circuits S ⟩ M Z X S ⟩ M X X In this Chapter, we assume we can completely overcome errors of the second kind, and focus only on the rst kind of errors. Finally, note that in the process of computation, we are frequently required to do logicalX L and logicalZ L gates. We do not necessarily implement these gates physically; rather, we can simply keep a record of it, and apply X L and Z L to that logical qubit in \software", as described in Secs. IX and XVI.A of Ref. [52]. 6.4.1 Creation of S+⟩ (S0⟩) State for X (Z)-cut Double Qubit Before computation begins, we assume the system is already prepared with the eigen- values of all stabilizer generators equal to +1. This can be done by several methods. One of them is preparing all qubits in the S0⟩ state and then measuring all X s stabilizer generators and resetting their eigenvalues to +1. After that, we turn on the stabilizer Hamiltonian: H(t 0 )=−JQ i X s i −JQ j Z p j : (6.16) There are two types of initialization procedures. The rst is the creation of a S+⟩ (S0⟩) state for a X (Z)-cut and second is the creation a S+⟩ (S0⟩) state for Z (X)-cut qubit. Here, we give an example of preparing a S+⟩ state forX-cut double logical qubit; theZ-cut case is similar. We will see that if we can do the rst type of preparation fault-tolerantly, 159 ( b ) ( a ) X s 2 X s 1 Figure 6.3: Creation of S+⟩ for X-cut double qubit. System Hamiltonians before and after are shown in (a) and (b) respectively. Colored squares indicate that the corre- sponding X s (yellow) and Z p (cyan) stabilizer generators are turned on. we can do the second type fault-tolerantly as well, as will be shown in Sec. 6.4.4. Suppose initially the state of the system is shown in panel (a) of Fig. 6.3 with a fully stabilized array, and the stabilizer Hamiltonian terms in this area are all turned on. Turning o the X s 1 and X s 2 terms and makes the Hamiltonian: H(t 1 )=−JQ i≠1;2 X s i −JQ j Z p j : (6.17) This will make the state S+ X DL ⟩= S+ X SL ⟩ 1 S+ X SL ⟩ 2 . This process can be done either adiabat- ically or instantaneously. If errors occur, they will leave nonzero syndromes for future correction, and no errors will be propagated when theX s 1 andX s 2 terms are turned o. We can see that the distance for x errors is restricted by 4, no matter how far the pair of holes are separated. To increase the error protection ability of x errors, we need to enlarge the size of the holes. We will describe in detail the adiabatic procedure to enlarge the holes with gap protection in Sec. 6.4.2. Also note that all state preparations of this type are done right after the initialization of the whole surface, such that X s 1 andX s 2 are known to be +1 for certain. During the computation, X s 1 and X s 2 can be ipped to −1 before they are turned o, and we have no way to know their values except by doing syndrome measurement, which we try to avoid. So all qubits needed in the computation are prepared at the beginning. 160 6.4.2 Enlarging the Hole After holes are created, we need to enlarge the size of the hole to improve the ability to correct z ( x ) errors forZ (X)-cut double qubits. In this section, we will show how to enlarge the hole adiabatically with gap protection. First, we will assume that no error occurs on any qubits during the process. Then we will analyze how errors propagate, and the fault-tolerance of the process. Since this is the rst example where we apply the results of Sec. 6.3, we will follow the state transformations based on stabilizer formalism in detail. 6.4.2.1 Scheme (a) (b) (c) 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 8 9 10 Z p1 Z p2 Z p3 Z p4 Z p3 Z L X L 1 2 3 4 5 6 7 8 9 10 Z ' L X' L Z" L X' ' L Z p4 7 Figure 6.4: Enlarging a hole of an X-cut logical double qubit adiabatically. Colored squares indicate that the corresponding X s (yellow) and Z p (cyan) stabilizer generators are turned on. The yellow qubits in (b) and (c) indicate that x for that qubit is turned on in the Hamiltonian. Adiabatic evolution between (a) and (b) maps X L to X ′ L and Z L toZ ′ L . Similarly, adiabatic evolution between (b) and (c) maps X ′ L toX ′′ L andZ ′ L to Z ′′ L . 161 L 1 (t 0 ) Z s 1 L 2 (t 0 ) X L S 1 (t 0 ) Z s 2 S 2 (t 0 ) Z s 3 S 3 (t 0 ) Z s 4 g 1 ⇒ L 1 (t 1 ) Z s 1 Z s 2 L 2 (t 1 ) X L S 1 (t 1 ) x 1 S 2 (t 1 ) Z s 3 S 3 (t 1 ) Z s 4 g 2 ⇒ L 1 (t 2 ) Z s 1 Z s 2 Z s 3 L 2 (t 2 ) X L S 1 (t 2 ) x 1 S 2 (t 2 ) x 2 S 3 (t 2 ) Z s 4 g 3 ⇒ L 1 (t 3 ) Z s 1 Z s 2 Z s 3 Z s 4 L 2 (t 3 ) X L S 1 (t 3 ) x 1 S 2 (t 3 ) x 2 S 3 (t 3 ) x 3 Table 6.1: The related transformation of stabilizer generators {S i } and logical operators {L i } of a Z-cut qubit in Fig. 6.4 is shown under gate operation {g i }. Consider the case of a Z-cut qubit, the situation for X-cut qubits is similar. Right after the creation of the pair of holes, we rst expand one of the two holes vertically down and then horizontally right, as shown in Fig. 6.4. Following the spirit of Sec. 6.3, consider a circuit G composed of three gates of the form g l = expi 4 Q l , where Q l are dened as: Q 1 = y 1 z 2 z 5 z 6 ; Q 2 = y 2 z 3 z 7 z 8 ; Q 3 = y 4 z 3 z 9 z 10 : (6.18) In this case , the state is stabilized by Z s 2 , Z s 3 , Z s 4 (and other stabilizer generators) with logical operators X L and Z s 1 . The transformation of the stabilizer generators and logical operators under G is listed in Table. 6.1. We can see that the circuit G maps logical operator X L and Z L to X ′′ L and Z ′′ L in panel (c) of Fig. 6.4, and also maps the system Hamiltonian in panel (a) to the ones shown in panel (c). Now we transform this procedure to an adiabatic one that gives the same state evolution following Theorem 2. Set U l (t;t l−1 ) = expi~4f l (t)Q l for time segment 162 t ∈ [t l−1 ;t l ], and adiabatically deform the Hamiltonian as in Eq. (6.10). Note that Q 1 only anticommutes with theZ p 2 term in the system Hamiltonian, which guarantees that even if errors occur, the adiabatic evolution is still valid (Lemma 2). The situation is the same for Q 2 and Q 3 . We rst consider the adiabatic transformation generated by U 1 (t;t 0 ): H(t)=−J cos[f 1 (t)]Z p 2 −J sin[f 1 (t)] x 1 −JQ j≠1;2 Z p j −JQ i X s i ; (6.19) for t∈ [t 0 ;t 1 ], with H(t 1 )=−J x 1 −JQ j≠1;2 Z p j −JQ i X s i : (6.20) At this time, qubit 1 is in the state S+⟩. For U 2 and U 3 , we see that Q 2 commutes with Q 3 , while Q 2 only anticommutes with Z p 3 , and Q 3 only anticommutes with Z p 4 . According to Lemma 3, the adiabatic procedures generated by U 2 and U 3 can be done simultaneously with the same state transformation as if done serially. The corresponding Hamiltonian deformation is H(t)=−J x 1 −J cos[f 2 (t)]Z p 3 −J sin[f 2 (t)] x 2 −J cos[f 2 (t)]Z p 4 −J sin[f 2 (t)] x 4 −J Q j≠1;2;3;4 Z p j −JQ i X s i (6.21) for t∈ [t 1 ;t 2 ], with H(t 2 )=−J x 1 −J x 2 −J x 4 −J Q j≠1;2;3;4 Z p j −JQ i X s i ; (6.22) with qubits 1, 2, 3, and 4 all in the state S+⟩, while they are all protected from z errors by the energy gap. This procedure can be generalized to obtain arbitrarily large square hole with distance equal to the perimeter d (assuming d is a multiple of 4). We rst adiabatically expand 163 d~4 times vertically down to form a long strip like that in panel (b) of Fig. 6.4, and then adiabatically expanding horizontally right parallel d~4 times as in panel (c). In all, we need about d~2 time steps of adiabatic evolution. 6.4.2.2 Error Propagation Even though the ground space is protected by an energy gap, there is still a nonzero probability that thermal excitations will occur at nite temperature. In this section, we apply the the result of Lemma 2 to study the propagation of these errors. If errors occur outside the hole or inside the hole, they will not be aected by the adiabatic process at all. However, if errors occur on the boundary of the hole before the adiabatic process, they may potentially propagate during the adiabatic procedure and cause uncorrectable logical errors. Consider the case in Fig. 6.5. Before expanding the hole vertically down, 1 2 3 4 5 6 7 8 9 1 0 ( a ) ( b ) ( c ) ( d ) 11 1 2 Figure 6.5: Error propagation during an adiabatic process to enlarge a hole of anX-cut logical qubit. Colored squares indicate that the corresponding X s , x (yellow) and Z p , z (cyan) operators are turned on. The purple circle around a qubit indicates a z error occurs on that qubit. (a) A z error occurs on qubit 1. (b) Eective errors after the adiabatic process. (c) An additional z error occurs on qubit 4. (d) Eective errors after the adiabatic procedure to enlarge the hole will cause a logical error after decoding. 164 assume a z error occurs on qubit 1. Then according to Lemma 2, the z error will propagate to x 1 z 2 z 5 z 6 , as shown in panel (b). The eective errors are z 2 z 5 z 6 , since x 1 has no eect because state of qubit 1 is S+⟩. However, if another z error occurs on qubit 4, as shown in panel (c), then after expanding horizontally rightward, we get eective errors z 5 z 6 z 7 z 8 z 9 z 10 , which occupy majority of the qubits around the hole. If the minimum-weight error correction is taken, it will close the path by applying z 11 z 12 and cause a logical Z error. So in general, this procedure is not fault-tolerant by the meaning of Def. 1. However, we can get around this problem by the following observation: if before the hole expansion, the system is prepared in the S0 Z DL ⟩, then a logical Z error has no eect on the state. The situation is the same for the S+ X DL ⟩ state for anX-cut double qubit. Fortunately, as we will see later, in this scheme we only need to expand aZ-cut hole after creation a S0 Z DL ⟩ state andX-cut hole after creation a S+ X DL ⟩ state, so the non fault-tolerance of this procedure can be overcome. 6.4.3 Moving Logical Qubits We now turn to the realization of logical gate operations in surface codes, like logical CNOT,S, Hadamard andT gates. An element way to do these logical gates is by adia- batically moving the holes around each other on a single 2D lattice. In this section, we focus on the details of hole movement by adiabatically deforming the system Hamilto- nian. We start with a scheme free of errors at rst and then discuss the corresponding error propagation and fault-tolerance. 6.4.3.1 Scheme We focus on theZ-cut qubit in this section, the method for theX-cut is similar. Consider a Z-cut qubit hole as shown in Fig. 6.6. Initially, the system Hamiltonian is H(t 0 )=−J 8 Q i=5 x i −J 14 Q i=12 x i −J 4 Q j=1 Z p j +H rest ; (6.23) 165 where H rest represents terms which are not altered in this process but are shown in (a) (b) (c) (d) Z p1 Z p2 Z p3 Z p4 Z p5 Z p6 Z p7 Z p8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 10 11 9 10 11 12 13 14 12 13 14 1 2 3 4 9 10 11 5 6 7 8 12 13 14 1 2 3 4 5 6 7 8 9 10 11 X L Z L X' L Z' L Z' L X' L X'' L Z'' L Figure 6.6: Adiabatic process for moving a Z-cut logical qubit hole horizontally right. Colored squares indicate that the corresponding X s , x (yellow) and Z p , z (cyan) operator are turned on. Logical operators of the qubit are X L and Z L in (a). An adiabatic process between (a) and (b) maps X L to X ′ L and Z L to Z ′ L . Similarly, an adiabatic process between (c) and (d) maps X ′ L to X ′′ L and Z ′ L to Z ′′ L . Fig. 6.6. We start with a circuitG composed of gates {g l } generated by {Q l }. For illus- tration purposes, we divide them into two groups. We rst expand the hole horizontally right as shown from panel (a) to panel (b), and then we shrink the hole rightward, as shown from panel (c) to panel (d). Consider the expansion procedure generated by: Q l =i x l Z p l ; 1≤l≤ 4; (6.24) 166 and the corresponding unitary transformations of the Hamiltonian U l = exp(if l (t)Q l ), forl from 1 to 4. We can see that eachQ l anticommutes only withZ p l , so we can apply the adiabatic procedures generated by Q 1 , Q 2 , Q 3 , Q 4 simultaneously H(t)=−J 4 Q j=1 cos[f 1 (t)]Z p j + sin[f 1 (t)] x j −J 8 Q i=5 x i −J 14 Q i=12 x i +H rest ; (6.25) for t∈ [t 0 ;t 1 ], and obtain H(t 1 )=−J 4 Q i=1 x i −J 8 Q i=5 x i −J 14 Q i=12 x i +H rest (6.26) at time t 1 as shown in panel (b). At this time, all qubits inside the hole are set to the S+⟩ state. To contract the hole, rightward, we follow the circuit generated by Q l , Q l =iZ p l x l ; 5≤l≤ 8; (6.27) and the corresponding unitary transformation of the Hamiltonian U l = exp(if l (t)Q l ). We need to be a little careful here, since Q l here anticommutes with two terms in the Hamiltonian. For example,i x 5 Z p 5 anticommutes with both x 5 and x 12 . To get around this, we turn o the terms −J x 12 , −J x 13 , −J x 14 in the above equation, and turn on −J x 9 , −J x 10 , −J x 11 instead. We can see that this procedure doesn't change the state of the system and can be done either adiabatically or instantaneously, making the Hamiltonian to be: H ′ (t 1 )=−J 11 Q i=1 x i +H rest : (6.28) Q l now anticommutes with just one stabilizer generator (which is x l ). Like the expansion process, we can adiabatically deform the Hamiltonian: H(t)=−J 8 Q j=5 cos[f 2 (t)] x j + sin[f 2 (t)]Z p j −J 4 Q i=1 x i −J 11 Q i=9 x i +H rest ; (6.29) 167 for t∈ [t 1 ;t 2 ], and obtain H(t)=−J 8 Q j=5 Z p j −J 4 Q i=1 x i −J 11 Q i=9 x i +H rest ; (6.30) which completes a full cycle of hole movement and leaves us ready for the next cycle of Hamiltonian deformation. The original ground space will be mapped to the one with a hole sitting one unit rightward of the original one (see panel (d)), and X L and Z L will be mapped to X ′′ L and Z ′′ L following Theorem. 2. Remark 7. Note that the two steps of the adiabatic expansion and contraction of the hole can be combined into one step, if we turn o − x 12 , − x 13 , − x 14 while turning on − x 9 , − x 10 and − x 11 at the beginning. So we need just one time step to adiabatically deform the system Hamiltonian to move a hole by one unit. 6.4.3.2 Error Propagation and Fault Tolerance Like the case of hole enlargement, there's chance that thermal errors will cause an excita- tion. Errors outside or inside the holes will not be propagated by the process. However, if errors occur on the boundary of the hole before moving, they may potentially propa- gate to uncorrectable logical errors. Consider the case in Fig. 6.7 for a 2 units movement rightward. Before expanding the hole horizontally right, assume z errors occurs on qubit 1 and qubit 2, as shown in panel (a). They will be propagated to: z 1 z 2 ↦ z 3 z 4 z 5 z 6 z 7 z 8 z 9 z 10 z 11 z 12 ; (6.31) by the subsequent adiabatic operation, as shown in panel (b). If we keep expanding the hole rightward, the errors will occupy more than half of the qubits on the perimeter of 168 ( a ) ( b ) ( c ) ( d ) 1 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Figure 6.7: Error propagation during adiabatic process to move a hole of an X-cut logical double qubit horizontally right. Colored squares and qubits indicate that the corresponding X s , x (yellow) and Z p , z (cyan) operators are turned on. The purple circle around qubit indicates a z error occurs on that qubit and a red one indicates a x error occurs. (a) z errors occur on qubit 1 and 2. (b) Eective errors caused by z 1 and z 2 after adiabatic process. (c) x errors occur on qubit 1 and 2. (d) Eective errors caused by x 1 and x 2 after adiabatic process. the hole, and cause a logical Z error after later decoding. Similarly, if x errors occur on qubit 1 and qubit 2, the eective errors after the adiabatic procedure will be x 1 x 2 ↦ z 1 z 2 z 3 z 4 z 5 z 6 z 7 z 8 z 9 z 10 z 11 z 12 ; (6.32) as shown in panel (d). In general, the adiabatic procedure to move the hole on its own is not fault-tolerant, since the circuitG we follow to build the adiabatic procedure is not a fault-tolerant one, and the results from Chapter 5 cannot be used here directly. Fortunately, we can still make this process fault-tolerant. Errors that occur on the boundary of the hole, like qubit 1 and qubit 2 in this example, can be detected after 169 each step of hole movement by measuring the qubits inside the hole after the expansion, since they are correlated, as shown in Fig. 6.7. In this case, we will do x measurement on qubit 3, 4, 8, 9, 13 and 14, when we are in panel (b). If any of these measurements give −1, it indicates that errors (which could be x or z ) occurred on the boundary's right side before the hole expansion, and we need to turn o the system Hamiltonian and do a full cycle of syndrome measurement and error correction before they become uncorrectable. A z error happens on the boundary with probability about exp(−4cJ) per time step, while x happens on the boundary with probability about exp(−2cJ), so the probability that we must do a full cycle of error correction during hole movement is low. In practice, measurements themselves involve errors whose eect was discussed earlier in this section. Here, we need to check the probability that the measurement outcomes cause us to make a wrong decision about error correction. As an example, if a z error occurs on qubit 1 in panel (a), qubit 3 and 4 in panel (b) will not be protected by an energy gap, and we assume that the probability of a wrong measurement outcome in these cases is p each time step. Meanwhile, if a x error occurs on qubit 1 in panel (c), qubit 3 and 4 in panel (d) are protected by an energy gap 4J. Fortunately, we d / 4 3 d / 8 d / 4 (a) (b) Measure these qubits . . . Figure 6.8: Scheme to fault-tolerantly detect errors occurring on the boundary. (a) Before the movement, an error occurs on the boundary. (b) After expanding the hole d~8 units rightward, the error propagates to a strip of errors. We measure all qubits in the dashed box and determine if the corresponding error happened on the boundary based on the majority vote of the measurement outcomes of each row of qubits. 170 can make the uncorrectable error rate arbitrarily small by growing the lattice size and hole size, using majority vote. Consider a square hole with perimeter d as shown in panel (a) of Fig. 6.8. Now we expand the hole d~8 units rightward and measure x in the dashed area of panel (b). The number d~8 is chosen so that error detection can be applied before an error can propagate to an uncorrectable error. If an error occurs on the boundary of the hole before moving, it will corrupt an entire row of qubits in the dashed area of panel (b) in Fig. 6.8. So, for each row of qubits, we do a majority vote based on the measurement outcomes to determine whether an error happened on the boundary. For any row, if more than half of the measurement outcomes are −1, we infer that an corresponding error occurred at the boundary of the hole before moving, and therefore error correction must be applied. Let E be the event that errors happened on the boundary before movement, and let D be the event that we decide to do decoding and error correction based on the majority vote. Then the probability that such errors occurred on the boundary and is not detected is roughly P L =P( D;E)=P( DSE)P(E)∼O d 4 p ⌊ d 16 ⌋+1 e −4cJ : (6.33) Here d~4 indicates that misidentication can occur on any of d~4 rows. This gives a rough bound on the probability of logical errors during the d~8 unit hole movement. On the other hand, the probability that no error occurred on the boundary, but we do an unnecessary decoding can be estimated as P U =P(D; E)=P(DS E)P( E)∼O d 4 p ⌊ d 16 ⌋ e −4cJ : (6.34) We can see that both P L and P U can be made arbitrarily small with the growth of hole size, and thus the adiabatic movement process can be rendered fault-tolerant. Remark 8. We only analyzed the error propagation for the case of hole expansion. It is worth noting that for the procedure to adiabatically contract the hole, errors occurring 171 on the boundary of the hole will not accumulate to uncorrectable logical errors, and thus can be left for future error correction. 6.4.4 Creation of S0⟩ (S+⟩) State for X (Z)-cut Double Qubit The second type of logical state initialization is to prepare the S0⟩ state for an X-cut qubit or S+⟩ state for a Z-cut qubit. We show an example for an X-cut qubit in detail. For a Z-cut qubit, the procedure is similar. This can be done using a logical Hadamard after initializing the S+⟩ state for X-cut qubit. However, we have not shown how to perform a logical Hadamard yet, and it is also extremely useful to directly initialize the S0⟩ state for anX-cut qubit, as we will see in next few sections. Suppose we have created a S+⟩ state for an X-cut qubit with two holes attached to each other, as shown in panel (a) of Fig. 6.9. The logical Z operator in this case can be X L 2 X L 1 1 2 3 X L 2 X L 1 1 2 3 ( a ) ( b ) Z L Figure 6.9: Creation of S0⟩ state for X-cut double qubit. (a) Create a S+ X DL ⟩ with two holes attached to each other. Measure z 1 , z 2 and z 3 and do majority vote to determine whether S0 X DL ⟩ or S1 X DL ⟩ is prepared. (b) Move two holes apart to increase error correction ability of z errors of the logical qubit. Note that both z and x errors on qubits between two holes during the adiabatic movement have no uncorrectable eect on logical state S0 X DL ⟩ or S1 X DL ⟩ and can be left for future error correction. z 1 , z 2 or z 3 , and they all commute with the system Hamiltonian. If we measure any 172 one of them, we can prepare the logical state S0⟩ or S1⟩. Either one is useful as long as we know which state it is for certain. If any z errors occur on these qubits, it will have no eect, and any single x errors on these qubits suers an energy penalty of 4J and leavesZ p operators nearby ipped and correctable by a future error correction procedure. However, when a x happens on these qubits, it will give an incorrect measurement outcome, and will aect any future operations conditioned on whether the state is S0⟩ or S1⟩. This can also be resolved by measuring z on all three qubits and taking the majority vote to determine the measurement outcome. This procedure can be extended to the square hole with perimeter d, where there are d~4 qubits shared by two holes. Note that the rst measurement error is suppressed by the energy penalty, and occurs with probability exp(−4cJ), while the subsequent measurement errors may not suer an energy penalty. We assume that the probability to obtain a wrong measurement result is p each time step. The probability that we prepare a S0 X DL ⟩ (S1 X DL ⟩) state with an erroneous measurement −1 (+1) can be estimated to be: P L ∼O(p ⌊ d 8 ⌋+1 e −4cJ ); (6.35) which decreases rapidly with the growth of the hole size, and can be made arbitrarily small. After the measurement, we separate the two holes by distance d, as illustrated in panel (b) of Fig. 6.9 for a single time step of movement. It takes about d~2 time steps in total to move the pairs of holes apart by distance d if the two holes move simultaneously. Any x and z errors on qubit 1, 2, 3 will not propagate to uncorrectable errors during the movement. The hole movement process can be done adiabatically and fault-tolerantly with gap protection, as described in the previous section. Thus, the whole state preparation process can be made fault-tolerant. 173 6.4.5 Logical Z (X) Measurement for X (Z)-cut Double Qubit Like the case of initialization, there are two types of measurement procedures. The rst is measuring in the Z (X) basis for an X (Z)-cut qubit while the second is measuring in the Z (X) basis for a Z (X)-cut qubit. The rst type of measurement is essentially the reverse process of creating the state S0⟩ (S+⟩) for anX (Z)-cut qubit. For anX-cut qubit shown in Fig. 6.9, we rst move two holes that are initiallyd units apart together to contact each other, and then measure z on all qubits shared by the two holes and take a majority vote of the outcomes. After that, we separate the two holes back to their original positions. Note that unlike traditional measurement-based QC on the surface code, this measurement is non-destructive and we do not annihilate the holes. The measurement procedure can also be viewed as a logical state preparation that will be used in the future computation. The second type of measurement procedure will be discussed in Sec. 6.4.7. 6.4.6 Holonomic Logical CNOT The logical CNOT gate is one of the most important logical operations in the surface code HQC scheme. Based on our results on adiabatic hole movement, we can realize the logical CNOT gate. In this section, we show that by adiabatically braiding one hole around a dierent type of hole, we can get a closed loop holonomy which can be recognized as a logical CNOT. Starting from panel (a) of Figs. 6.10 and 6.11, the adiabatic movement procedure is shown in details from panel (b) to panel (f). In Fig. 6.10, following the discussion in Sec. 6.4.3 and Theorem. 2, X L 1 ⊗I L 2 transforms to X L 1 ⊗X L 2 up to a multiplication by X s stabilizer generators inside the dashed square. We can conclude that X L operators transform in the following way: X L 1 ⊗I L 2 →X L 1 ⊗X L 2 ; I L 1 ⊗X L 2 →I L 1 ⊗X L 2 : (6.36) 174 ( a ) ( b ) ( c ) ( d ) ( e ) ( f ) } X L 1 Z-cut { X-cut X L 2 ’ X = L 1 X X L 1 L 2 Figure 6.10: Adiabatic braiding process of a Z-cut hole (dark blue) around an X-cut hole (orange) . The operator X L 1 has been stretched to multiply a loop of x operators which is equivalent toX L 2 up to multiplication byX s stabilizer generators (yellow) inside the loop, while X L 2 remains the same under the transformation. Similarly, from Fig. 6.11, we can see that I L 1 ⊗Z L 2 transforms to Z L 1 ⊗Z L 2 up to multiplication by Z p stabilizer generators inside the strip. The Z L operators transform as: Z L 1 ⊗I L 2 →Z L 1 ⊗I L 2 ; I L 1 ⊗Z L 2 →Z L 1 ⊗Z L 2 : (6.37) The closed loop adiabatic evolution can be recognized as a closed loop holonomy which gives a logical CNOT with aZ-cut qubit as the control and anX-qubit as the target. It also re ects the topological property of braiding on 2D lattice since local deformation of movement path does not have eects on the state. Note that the fault-tolerance of this operation is guaranteed by the fault-tolerance of adiabatic hole movement. 175 ( a ) ( b ) ( c ) ( d ) ( e ) ( f ) } Z L1 Z-cut { X-cut Z L2 Z L1 ’ Z =Z Z L2 L1 L2 Figure 6.11: Adiabatic braiding process of a Z-cut hole (dark blue) around an X-cut hole (orange). The operator Z L 2 has been stretched to form a strip of z operators, which is equivalent to Z L 1 up to multiplication by Z p stabilizer generators (cyan) inside the strip, while Z L 1 remains the same under the transformation. CNOTs from Z-cut qubits to X-cut qubits are not enough. We need to extend to CNOTs between logical qubits of the same type. ForZ-cut qubits, we have the following circuit: Z-cut control in ● Z-cut control out S0 X DL ⟩ M Z S+ Z DL ⟩ ● Z-cut target out Z-cut target in ● M X which is equivalent toZ (1−M X )~2 L on the target qubit followed by a CNOT, then followed byX (1−M Z )~2 L on the target qubit. Similarly, the CNOT between twoX-cut logical qubits can be built from following circuit: 176 S0 X DL ⟩ X-cut control out X-cut control in M Z S+ Z DL ⟩ ● ● ● M X X-cut target in X-cut target out up to a correction of logicalXs andZs. The last kind of CNOT, with anX-cut qubit as control and aZ-cut as target, can be obtained from the circuit realizing CNOT between Z-cut qubits: S0 X DL ⟩ X-cut control out X-cut control in M Z S+ Z DL ⟩ ● ● ● M X Z-cut target in Z-cut target out Note that for all four dierent logical CNOTs, the building block is the CNOT from Z-cut to X-cut. In addition, we also need to prepare ancillas in logical S0 X DL ⟩ and S+ Z DL ⟩ (which is shown in Sec. 6.4.4), and to do Z measurements of X-cut qubits and X measurements ofZ-cut qubit (as discussed in Sec. 6.4.5). All of these procedures can be done fault-tolerantly, and thus make all kinds of logical CNOT fault-tolerant. 6.4.7 Measurement of Z (X) Basis for Z (X)-cut Double Qubit This type of measurement is necessary when doing state distillation (discussed later). Naively, this process can be done by contracting the size of the hole and doing stabilizer measurements. However, stabilizer measurement is not compatible with the system Hamiltonian. What is worse, we close the hole after the measurement to destroy the logical qubit, and we cannot reuse it later. To avoid these problems, we can use the following circuits for Z and X measurement of Z-cut and X-cut qubits, respectively: S Z DL ⟩ ● S X DL ⟩ S0 X DL ⟩ M Z S+ Z DL ⟩ ● M X 177 These circuits take an ancilla state S0 X DL ⟩ or S+ Z DL ⟩, and a logical CNOT with a Z-cut qubit as the control and an X-cut qubit as the target, which can both be realized fault- tolerantly. Thus, this type of measurement procedure is fault-tolerant. Note that, like the measurement of the rst type in Sec. 6.4.5, this measurement procedure doesn't annihilate the hole after measurement. The ancilla qubits after measurement are eec- tively prepared to S0 X DL ⟩ (or S1 X DL ⟩) fault-tolerantly, which can be used again as ancillas for future computation. 6.4.8 Ancilla Recycling As we have seen so far, to implement dierent types of CNOTs, we need to frequently create and measure logical qubits. Moreover, state distillation procedures also need large number of fresh ancilla qubits and logical state measurements. We have discussed two dierent types of state creation|S0⟩ (S+⟩) forX (Z)-cut and S0⟩ (S+⟩) forZ (X)-cut|and two dierent types of measurement|X (Z) measurement forZ (X)-cut qubit andZ (X) measurement for Z (X)-cut qubit. All can be done fault-tolerantly with constant gap protection, and both kinds of logical state measurement can be made non-destructive, so states after measurement can be reused as ancillas to avoid having to create a new logical qubits. This is particularly important, as we have seen that to create a logical state we need to turn o some X s or Z p operators, whose eigenvalues are uncertain when stabilizer Hamiltonian is turned on. With this ancilla recycling process, we can prepare all logical qubits, data or ancilla, right after we turn on the system Hamiltonian at the very beginning of the computation and never create new logical qubits during the computation. 6.4.9 State Injection As will be seen in Sec. 6.4.11 and 6.4.12, to get the logicalS,T and Hadamard gates, we need to create particular logical ancilla states SY DL ⟩ = 1 √ 2 (S0 DL ⟩+iS1 DL ⟩) and SA DL ⟩ = 1 √ 2 (S0 DL ⟩+e i~4 S1 DL ⟩). However, there's no obvious way to perform arbitrary rotation 178 of logical qubit with large distance and local Hamiltonians transformation. To deal with this problem, we need to create a logical qubit in which the logical Z operator is just one z on single qubit, with the stabilizer Hamiltonian turned on. We focus on X-cut Z L 1 Z L 2 ( a ) ( b ) X L 1 X L 2 X L 1 X L 2 Z L 1 Z L 2 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 X S 1 X S 2 Z p 1 Z p 2 Z p 3 Z p 1 Z p 2 Z p 3 Z p 4 Figure 6.12: State injection for a X-cut qubit. Colored squares indicate that the corresponding X s (yellow) and Z p (cyan) operator is turned on. double qubits. We rst put an existing X-cut qubit into the state S+ X DL ⟩ with the two holes attached to each other, as in panel (a) of Fig. 6.12. This can be done by doing a logical X measurement on an existing X-cut qubit (Sec. 6.4.7) and moving the two holes together. Without loss of generality, assume the state after measurement to be S+ X DL ⟩. Note thatZ L = z 5 is equivalent toZ L 1 Z L 2 up to multiplication byZ p operators, as shown in panel (a), which gives: z 5 =Z p 1 Z p 2 Z p 3 Z L 1 Z L 2 : (6.38) For the S± X DL ⟩ state, the eect of z 5 is z 5 S± X DL ⟩= z 5 S± X SL ⟩ 1 S± X SL ⟩ 2 =Z L 1 Z L 2 S± X SL ⟩ 1 S± X SL ⟩ 2 = S∓ X SL ⟩ 1 S∓ X SL ⟩ 2 : (6.39) Applying a pulse V c =g z 5 for a short time , with Hamiltonian H =H stab +V c ; (6.40) 179 we can see that [V c ;H stab ]= 0, whereH stab is the stabilizer Hamiltonian shown in panel (a). The pulse will not cause a transition from the ground space to another eigenspace of H stab . If is chosen such that g =~2, we have the state evolution: exp−i 2 z 5 S+ X SL ⟩ 1 S+ X SL ⟩ 2 = e −i 2 √ 2 S+ X SL ⟩ 1 S+ X SL ⟩ 2 +S− X SL ⟩ 1 S− X SL ⟩ 2 √ 2 +e i 2 S+ X SL ⟩ 1 S+ X SL ⟩ 2 −S− X SL ⟩ 1 S− X SL ⟩ 2 √ 2 = e −i 2 √ 2 S0 X DL ⟩+e i S1 X DL ⟩; (6.41) which gives the desired state we want to inject. Note that if a x 5 error occurs, it will suer from the energy penalty, and cause theZ p s adjacent to it to be ipped, leaving the syndrome for future error correction. On the other hand, the imprecise control of the pulse V c can aect the state injected and cannot be detected. However, as long as rate of z 5 error is lower than a threshold, logical states SY DL ⟩ and SA DL ⟩ can be obtained with sucient precision by state distillation [117]. Then two holes can be adiabatically separated to distance d to better protect against errors, as illustrated in panel (b) of Fig. 6.12. The process of state injection for aZ-cut qubit is slightly more complicated. We rst inject state the S ⟩ = SY ⟩ or SA⟩ for an X-cut qubit and prepare a Z-cut qubit in state S+⟩ and then we swap the state of these two logical qubits using following circuit: S X DL ⟩ ● S+ X DL ⟩ S+ Z DL ⟩ ● ● S Z DL ⟩ Note that the S+ X DL ⟩ is ready to be reused for state injection, and all process included here can be done fault-tolerantly. 6.4.10 State Distillation The logical ancilla states, SY ⟩ = S0⟩+iS1⟩ and SA⟩ = S0⟩+e i~4 S1⟩ after injection are not good enough in general for the purpose of fault-tolerant QC. Fortunately, they can be 180 distilled to much higher delity [24]. The reversed encoding circuit for 7-qubit Steane code can be used to distill the SY ⟩ state, with seven input logical states approximately equal to SY ⟩ [53] as shown in Fig. 6.13. The output S ⟩ will be closer to the logical SY ⟩ state. Repeating this process multiple times, arbitrarily high delity SY ⟩ states can be S ̃ Y DL ⟩ ● M X S ̃ Y DL ⟩ ● M X S ̃ Y DL ⟩ ● SY DL ⟩ S ̃ Y DL ⟩ ● M X S ̃ Y DL ⟩ M Z S ̃ Y DL ⟩ M Z S ̃ Y DL ⟩ M Z Figure 6.13: Circuits for logical SY ⟩ distillation from imperfect S ̃ Y ⟩ states. obtained exponentially quickly if the original delity of the input states is higher than some threshold [117]. A similar distillation circuit exists for the SA⟩ state, as shown in Fig. 6.14, which is the reverse of the encoding circuit for the [[15; 1; 3]] truncated Reed- Muller code [114, 53]. As before, given a good enough input SA⟩ state, the convergence is rapid. Note that these distillation circuits use CNOTs between the same type of qubits, and both types of logical state measurements described in Sec. 6.4.5 and 6.4.7. If the input states are X-cut qubits, then the logical X measurements are of the second kind, and the states after measurement are S+ X DL ⟩ or S− X DL ⟩, which are ready to be reused to inject SY ⟩ or SA⟩ for future state distillation. The logicalZ measurements are of the rst type, and will prepare logical states S0 X DL ⟩ or S1 X DL ⟩. To recycle these logical qubits to inject new SY ⟩ or SA⟩, we need to reset them to S+ X DL ⟩ or S− X DL ⟩, which can be done by a subsequent logical X measurement: 181 S ̃ A DL ⟩ ● M X S ̃ A DL ⟩ ● M X S ̃ A DL ⟩ ● SA DL ⟩ S ̃ A DL ⟩ ● M X S ̃ A DL ⟩ M Z S ̃ A DL ⟩ M Z S ̃ A DL ⟩ M Z S ̃ A DL ⟩ ● M X S ̃ A DL ⟩ M Z S ̃ A DL ⟩ M Z S ̃ A DL ⟩ M Z S ̃ A DL ⟩ M Z S ̃ A DL ⟩ M Z S ̃ A DL ⟩ M Z S ̃ A DL ⟩ M Z Figure 6.14: Circuits for logical SA⟩ distillation from imperfect S ̃ A⟩ states. S X DL ⟩ M Z S+ X DL ⟩ or S− X DL ⟩ S+ Z DL ⟩ ● M X Note that the ancilla states S+ Z DL ⟩ or S− Z DL ⟩ introduced here after logicalX-measurement can also be reused directly as ancilla for another logical X-measurement. The recycling process for Z-cut qubit inputs is similar. 6.4.11 Logical Phase and T Gates Given the distilled SY ⟩ state, we can implement high quality logical S gates and logical R X L (~2)= exp−i 4 X L gates using the following circuits [53]: 182 SY DL ⟩ ● Z L X L S L S DL ⟩ S DL ⟩ M Z ● SY DL ⟩ Z L X L R X L (~2)S DL ⟩ S DL ⟩ ● M X ● If the measurement outcome is +1, nothing needs to be done; otherwise, do a ZX gate. Note that this ZX gate can be done in \software" rather than physically. The non-Cliord gates play a central role in quantum speedup [60], and are necessary to obtain a universal gate set. For the surface code, the logical T gate is implemented with high quality distilled logical SA⟩ states using this circuit [52]: SA DL ⟩ ● Z L X L S L T L S DL ⟩ S DL ⟩ M Z ● If the logicalZ measurement yields a +1 outcome, the output state is the desired one. If the measurement yields a −1 outcome, the output is X L T L S DL ⟩ and Z L X L S L needs to be applied to getT L . Again, the logicalX andZ gate can be done in classical \software" rather than physically. Details of commuting X L , Z L through S L and T L for classical software control were discussed in Sec.XVI.A of Ref. [52]. As usual, the states after the measurements in these circuits can all be recycled and used as ancillas for logical CNOT gates, state injection and state distillation in future computational steps. 6.4.12 Hadamard In the existing, measurement-based QC on the surface code, a logical Hadamard is realized by rst digging a \moat" around the double logical qubits by measuring single qubits around the double hole to create a logical qubit island. On the \island", a logical Hadamard gate is then realized by a sequence of code deformations through single qubit and stabilizer measurements, and then the \moat" at last is repaired [52]. This version of logical Hadamard is easy and ecient enough in measurement-based QC, but dicult 183 to implement in our system when the stabilizer Hamiltonian is turned on. Instead, the logical Hadamard gate can be done directly: Had=S⋅R X (~2)⋅S: (6.42) Both logical S and R X (~2) are fault-tolerant but heavily rely on the state distillation of logical SY ⟩ state. There is a more ecient way to do a logical Hadamard, as illustrated in Ref. [16], by introducing a nontrivial domain wall on the lattice and moving the holes across the wall. The wall can be created by shifting the geometry of the lattice along a line, as shown in Fig. 6.15. The ve body interaction terms terminating the dislocation are called twists [16]. One can see that the insertion of two twists changes the degeneracy of ground space. This can form an additional logical qubit, which we call gauge qubit F. The corresponding logical operators of this qubit are also shown in Fig. 6.15. z s z s z s xs x s x s ys y s x s x s z s z s x s x s z s z s y s z s y s x s y s y s y s y s y s x s z s x s x s x s x s x s z s z s z s z s z s L 2 L 1 y s Figure 6.15: A dislocation in the geometry of the Hamiltonian produced by shifting the stabilizer generators along a line between two twists. The stabilizer generators corre- sponding to two dierent parallelograms (yellow/cyan and cyan/yellow) and a pentagon (dark gray) are shown on the right side. A pair of anticommuting strings of Pauli oper- ators L 1 (solid red) and L 2 (dashed blue) that commute with all stabilizer generators forms the logical operators of the extra qubitF attached to the pair of twists. 184 If a singleZ (X)-cut hole is adiabatically dragged across the wall, it will change to a X (Z)-cut hole, as shown in Fig. 6.16. However, note that this process can also change the state of F, since it will change logical operators L 1 and L 2 . This eect in general will yield additional entanglement between data qubit and F. However, if we drag the second hole of the logical data qubit across the wall, it will reverse the change caused by the rst hole and leave the state ofF unchanged. In summary, adiabatically moving two holes of a logical qubit across the wall will give a state transformation on the data qubit: S Z DL ⟩→ Had S X DL ⟩; S X DL ⟩→ Had S Z DL ⟩; (6.43) ( a ) ( b ) ( c ) twists Figure 6.16: Adiabatically moving a pairs of holes of a Z-cut qubit (dark blue holes) across a twist on the surface to get a logical Hadamard gate. This process will transform a Z-cut qubit to an X-cut qubit (orange holes). for Z-cut qubits and X-cut qubits. Another problem of this method is that it will change of the type of qubits we are working on. However, we can use an ancilla to swap the data qubit back by the circuit S Z DL ⟩ ● ● S0 Z DL ⟩ or S+ Z DL ⟩ S0 X DL ⟩ or S+ X DL ⟩ ● S X DL ⟩ for a Z-cut qubit, and S X DL ⟩ ● S0 X DL ⟩ or S+ X DL ⟩ S0 Z DL ⟩ or S+ Z DL ⟩ ● ● S Z DL ⟩ 185 for an X-cut qubit. The position of the twists can be xed on the lattice so that they can be used repeatedly for Hadamard gates. 6.5 Fault-tolerance of the Scheme We have described a way to fault-tolerantly implement QC in surface codes with a con- stant energy gap to suppress errors in a thermal environment. Table. 6.2 lists a summary of each procedure. Note that although adiabatic hole enlargement and state injection are not themselves fault-tolerant, they do not aect the fault-tolerance of the whole QC scheme. In addition to gap protection during the computation, fault-tolerance is guaran- teed by performing single qubit and syndrome measurements before errors can propagate to become uncorrectable. We discuss the interval betweens syndrome measurements in Sec. 6.5.1. So far, the error models we considered are induced by weak coupling to a thermal bath. We also need to consider other decoherence channels, which may aects qubits collectively or directly act on logical qubits. In this section, we will discuss two of them: local perturbations and adiabatic errors. In the following sections we show that they can both be exponentially bounded. Process Gap Fault- Dynamics Number of protection tolerance time steps Creation S0⟩ (S+⟩) for Z (X)-cut qubit Yes Yes Adiabatic ∼d~2 Creation S0⟩ (S+⟩) for X (Z)-cut qubit Yes Yes Adiabatic+Measurement ∼d Z (X) measurement for X (Z)-cut qubit Yes Yes Adiabatic+Measurement ∼d Z (X) measurement for Z (X)-cut qubit Yes Yes Adiabatic+Measurement O(d) Hole enlargement Yes No Adiabatic ∼d~2 Hole movement Yes Yes Adiabatic N/A Logical CNOT Yes Yes Adiabatic+Measurement O(d) State injection Yes No Adiabatic+ Pulse control ∼d State distillation Yes Yes Adiabatic+Measurement N/A Logical S, T, Hadamard Yes Yes Adiabatic+Measurement N/A Table 6.2: Summary. 186 6.5.1 Error Correction A proper time period to turn o the system Hamiltonian and do error correction, in the case that there are no errors detected during the adiabatic hole movement process, is crucially important. We assume that syndrome measurement is done every m time steps, and m exp(−2cJ) can be regarded as the error rate on each qubit for every m time steps (m exp(−2cJ) ≪ 1), since all processes necessary for universal QC are protected by a gap of at least 2J. Besides thermal errors accumulating on each qubit, the following types of physical errors can occur in a single syndrome measurement cycle in Sec. 6.2 [52]: 1. x error occurs when a syndrome qubit is initialized to S0⟩, with probability p. 2. The Hardamard gate on syndrome qubit is not perfect. There is extra x , y or z error following the gate, each with probability p~3. 3. Error occurs when a syndrome qubit is measured, with probability p. 4. CNOT gate on syndrome qubit-data qubit CNOT is not perfect, but with following erros: I⊗ x ,I⊗ y ,I⊗ z , x ⊗I, x ⊗ x , x ⊗ y , x ⊗ z , y ⊗I, y ⊗ x , y ⊗ y , y ⊗ z , z ⊗I, z ⊗ x , z ⊗ y or z ⊗ z , each with probability p~15. Note that one needs several cycles of syndrome measurements to establish values of syndrome before actual decoding. Then, the logical error rate of surface code form time steps with active error correction can be roughly estimated as [52] P m L ≈d d! (d e − 1)!d e ! me −2cJ + 7p de ; (6.44) where d e = (d+ 1)~2. A plot of this estimate is shown in Fig. 6.17, for various values of cJ , p and m. We can use these scaling relations to estimate the number of qubits needed to obtain a desired error rate after error correction. Our goal is that the error rate after the whole computer procedure is bounded by some particular value ≪ 1. 187 0.000 0.002 0.004 0.006 0.008 0.010 10 -17 10 -14 10 -11 10 -8 10 -5 0.01 p P L m d=19, cbJ=12, m=10 8 d=19, cbJ=8, m=5´10 4 d=15, cbJ=12, m=10 8 d=15, cbJ=8, m=5´10 4 d=11, cbJ=12, m=10 8 d=11, cbJ=8, m=5´10 4 d=7, cbJ=12, m=10 8 d=7, cbJ=8, m=5´10 4 Figure 6.17: Logical error rate per m time steps for various values of m and d. The dashed lines are for cJ = 8 and solid lines for cJ = 12. The blue (top), green (second top), red (third) and yellow (bottom) lines are for d = 7, d = 11, d = 15 and d = 19, respectively. Denoted by M the product of number of logical operation and the number of logical qubits used in an algorithm. We need to have: P m L ≲ m dM ; (6.45) since each logical operation needs about d time steps in our scheme. For a particular computation like Shor's algorithm implemented on surface codes, M is of the order larger than 10 14 [115]. We can choosep= 0:001 andcJ = 12, which may be achievable in current experiments. Also, set = 0:1,d= 11 andm= 10 8 , then we haveP m L ≈ 10 −8 , which satises the condition of Eq. (6.45). This requires a number of data and measurement qubits n tot = (2d− 1) 2 ≈ 450 to protect a logical qubit, and perform Shor's algorithm with reasonable success probability. We can see that if large cJ is not achievable, one can always choose a code with larger distance and more frequent error correction to compensate for the small cJ. However, if the cJ can increase to 15, we can even reduce d to 7 and n q to about 170, with m = 10 10 and same value of , making it more ecient to build a scalable QC in the near future. 188 6.5.2 Local Perturbation Perturbations will split the degeneracy of the ground space and cause stochastic phase errors between dierent logical states. This is one of the main obstacles to realizing non- Abelian holonomic quantum gates on system with a small number of qubits. However, for surface codes, the splitting of the ground space (and any other error space) caused by local perturbations will decay exponentially with the distance of the surface code, as shown by Kitaev in Ref. [73]. Actually, any system with quantum topological order is in general stable under local perturbations [23]. This might suggest that holonomic QC is more naturally suitable with systems with topological order than systems with small number of qubits. Consider a local perturbation of the general form: V local =−Q j h j ⃗ j −Q j<p J jp (⃗ j ;⃗ p ); (6.46) which includes all one-qubit and two-qubit interactions. The eect of V local only occurs in the d~2−th order of perturbation theory, and the energy splitting vanishes as split ∼OJe −vd~2 ; (6.47) where v = min ij {ln(J~Sh i S); ln(J~YJ ij Y 1 )}, which decreases quickly with growth of the code distance. Consider the case when d = 11, J = 1. To achieve an error rate of order 10 −15 , we must to control the values of Sh i S~J and YJ ij Y 1 ~J so that they are less than 10 −3 , which is practically achievable for current or near future technology. 6.5.3 Adiabatic Error As discussed in Sec. 5.3.2, the adiabatic error for typical processes listed in Table. 6.2 can be bounded by ad ∼Od⋅ sup q (c q T q + 1) +1 e −cqTq ; (6.48) 189 for some set of constant {c q }. Here so T q and f q are for q-th step. In principle, we can make adiabatic process arbitrarily small with careful chosen {T q }, {f q } and boundary condition of {f q }. Note that the thermal error rate decreases exponentially with J, while the during of each adiabatic time segment decreases as the cube of J at xed temperature, so the processing time overhead of an adiabatic process can be small if J is large. Remark 9. We've analyzed that it is possible to use on the order of 10 2 physical qubits to protect a single logical qubit in practical quantum computation with protection by a constant gap enabling fault-tolerant QC in surface codes. This is quite ecient compared to the existing QC scheme in surface codes [52]. However, the assumption here is that the thermal error model is local, and the stabilizer Hamiltonian is fundamental, given by Nature. Such 4-bodyX s andZ p interactions are hard to build directly, and usually needs certain techniques, like quantum gadgets [72, 101], digital quantum simulator [148, 147] , the low energy approximation from Kitaev's honey-comb model [74] or to be generated dynamically [11]. If the Hamiltonian is eective, rather than being fundamental, it may dramatically change the local thermal error model we have assumed, and cause nonlocal errors. This possibility calls for future investigation. 6.6 Summary and Conclusion We have outlined a scheme for fault-tolerant universal HQC based on surface codes, with stabilizer Hamiltonian to protect quantum information encoded in the degenerate ground space, from both thermal errors and small perturbations. We explicitly constructed all necessary processes with energy gap protection and parallel operations. These processes include logical state creation, a logical universal gate set, and logical state measurement. Logical state initialization and measurement are realized by open-loop adiabatic evolu- tion and measurements on single qubits compatible with system Hamiltonian, while the logical CNOT is implemented by a closed-loop holonomic operation. All other logical 190 gates can be implemented using the logical CNOT, logical state preparation, and logical state measurement. It is worth mentioning that if a twist is allowed to exist on the surface, the logical Hadamard can be done much more eciently. Conditions for active error correction are also discussed. The number of physical qubits needed to protect a logical qubit for fault-tolerant QC can reduce to the order of 10 2 , if large coupling constant J and low temperature are achievable in experiment. Applying our scheme to an actual physical system needs local 4-body interactions. Several theoretical proposals have been proposed to build such interactions eectively, which include low energy perturbations [74, 72, 101] of systems with strong two body interactions, and dynamic simulation [148, 147, 11]. As argued in Sec. 6.5, the eect of such eective interaction on local error models needs further study. It is important to nd out under what conditions these eective Hamiltonians behave like the ideal ones in open quantum systems. We concentrated on surface codes in this Chapter, but we hope the methods can be extended to fault-tolerant QC schemes with constant gap protection on other topological codes, including color codes [20, 82] and Turaev-Viro codes [78]. Another interesting question is, could it be possible to do QC fault-tolerantly on an arbitrarily large scale without any active error correction? It has been shown that it is possible to do so with 6D topological color codes [19]. In our scheme on a 2D lattice, if J is very large and the temperature is suciently low (which is certainly a challenging engineering problem), then for practical algorithm, it may not be necessary to do active error correction. It has also been shown that a self-correcting quantum memory to store quantum information for a polynomially (or even exponentially) long time in the lattice size exists, if long range interactions between anyons is allowed [65, 33, 106, 66, 151, 26]. Theoretical work to realize such a long range interaction was also proposed in [107, 11]. Long range interaction can freeze the density of excited anyons on the lattice for such a long time that logical errors are quite unlikely to happen. One may ask whether such interactions can be allowed when we adiabatically deform the stabilizer Hamiltonian in 191 our scheme. One diculty here is that, when enlarging or moving the holes, it is hard to dene the concept of anyons on the boundaries of the holes. How to introduce similar long range interactions during hole movement and enlargement is an interesting problem, and if it is possible, one may be able to implement self-correcting QC on a 2D lattice. 6.7 Appendix: Proof of Lemma 1, 2, 3 We rst prove a lemma which will be used to prove other lemmas: Lemma 4. ∀g q ∈G is in the normalizer of G n . Proof. For any M ∈G n , either [M;Q q ]= 0 or {M;Q q }= 0. In the second case, we have [Q q ;M]= 2Q q M = 2M ′ , with M ′ ∈G n . g q Mg q = expi 4 Q q M exp−i 4 Q q =M +i 4 [Q q ;M]− 2 16⋅ 2! [Q q ;[Q q ;M]]::: = cos(~2)M +i sin(~2)M ′ =iM ′ : (6.49) Further, if M, Q q are Hermitian, M ′ is anti-Hermitian and g q Mg q is Hermitian. 6.7.1 Lemma 1 The deformation of the Hamiltonian is isospectral, so the number of logical qubits encoded in the ground space is constant, say k. The horizontal lift V 0 (t) for P 0 (t) in general can be written as V 0 (t) = U q (t;t q−1 )V 0 (t q−1 )h(t;t q−1 ). From Eq. (5.28), U q (t;t q−1 )@ t U(t;t q−1 )=i@ t f q (t)Q q , @h @t =iV 0 (t q−1 )@ t f q (t)Q q V 0 (t q−1 ) (6.50) 192 for t∈ [t q−1 ;t q ], and V 0 (t q−1 )@ t h(t; 0)V 0 (t q−1 )=iP 0 (t q−1 )@ t f q (t)Q q P 0 (t q−1 ): (6.51) Since S j (t 0 )∈G n for all j, g l ∈G n , for all l. P 0 (t q−1 )= q−1 M l=1 g q P(0) q−1 M l=1 g l = q−1 M l=1 g q n−k M j=0 I+S j (0) 2 q−1 M l=1 g q = n−k M j=1 I+S j (t q−1 ) 2 ; (6.52) whereS j (t q−1 )= ∏ q−1 l=1 g l S j (t 0 )∏ q−1 l=1 g l is inG n because {g q } are all in the normalizer of G n (Lemma. 4). Since [Q q ;H(t q−1 )] ≠ 0, so there exists at least one S j (t q−1 ) such that {Q q ;S j (t q−1 )} = 0. According to Eq. (6.51), V 0 (t q−1 )@ t h(t; 0)V 0 (t q−1 ) = 0 and h(t;t q ) = I. Thus V 0 (t) = U q (t;t q−1 )V 0 (t q−1 ) and V 0 (t) = U q (t;t q−1 )∏ q−1 l=1 g l V 0 (t 0 ). From Eq. (5.18). S (t)⟩=e −i" 0 (t−t q−1 ) U q (t;t q−1 ) q−1 M l=1 g l S (t 0 )⟩: (6.53) Setting q =p and t=t p , we get S (t p )⟩=e −i" 0 (tp−t q−1 ) p S (t 0 )⟩: (6.54) 6.7.2 Lemma 2 First, we show that for any ≠, the adiabatic condition for P s and P s is satised. We have S j (t l ) ∈ G n according to Lemma. 4 for 1 ≤ l ≤ q. Consider the time segment [t q ;t q+1 ] rst. Dene the index setI = {1; 2;:::;n−k} to be the number of terms in 193 the Hamiltonian H(t q ) with setsA = {j ∈IS{S j (t q );F }= 0},B =IA ,C Q l = {j ∈ IS{S j (t q );Q l }= 0} andD Q l =IC Q l . Since F ∈G n , F q P 0 (t q )(F q ) =M j∈A I+S j (t q ) 2 M j ′ ∈B I−S ′ j (t q ) 2 = M m∈C Q q+1 I+s m S m (t q ) 2 ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ P C s ⋅ M m ′ ∈D Q q+1 I+s m ′ S m ′(t q ) 2 ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ P D s =P s (t q ): (6.55) Here, P C s and P D s are short for P C Q q+1 s and P D Q q+1 s . For any ≠, P s (t) @H(t) @t P s (t)= −i@ t f q (t)U q+1 (t;t q )P s (t q )Q q+1 H(t q )P s (t q )− P s (t q )H(t q )Q q+1 P s (t q )U q+1 (t;t q ); (6.56) whereU q+1 (t;t q )= exp(if q+1 (t)Q q+1 ). We examine the two terms in the square brackets: P s (t q )Q q+1 H(t q )P s (t q ) =" s (t q )Q q+1 M m∈C Q q+1 I−s m S m (t q ) 2 P D s (t q )P C s (t q )P D s (t q ); (6.57) and P s (t q )H(t q )Q q+1 P s (t q ) =" s (t q )P C s (t q )P D s (t q )P D s (t q ) M m∈C Q q+1 I−s m S m (t q ) 2 Q q+1 : (6.58) For those s such that s m ≠ s m for any m ∈ D Q q+1 , Eq. (6.56) will be zero, and the adiabatic condition will be satised automatically. For those s such that s m = 194 s m for all m ∈ D Q q+1 , it's easy to check the above two expression are not equal to zero only if s m = −s m for all m ∈C Q q+1 . Therefore, there is only one such that P s (t)@ t H(t)P s (t)≠ 0 and hence that needs further checking. For that specic , we have a simple relation: Q q+1 P s (t q )Q q+1 =P s (t q ); (6.59) and ZP s (t) @H(t) @t P s (t q )Z 1 =@ t f q+1 (t)T" s (t q )−" s (t q )T⋅ZP s (t q )Q q+1 Z 1 =K@ t f q+1 (t)T" s (t q )−" s (t q )T: (6.60) The left hand side of Eq. (6.8) reduces to S@ t f q+1 (t)S T" s (t q )−" s (t q )T ; (6.61) since SC Q q+1 S is odd. We have T" s (t q )−" s (t q )T= U Q m∈C Q q+1 2s m U≥ 2: (6.62) If @ t f q+1 (t) ≪ 1 is satised (which is always possible by setting appropriate controls), then P s (t) satises the adiabatic condition for time segment t ∈ [t q ;t q+1 ]. The same argument can be applied to the time segmentsl>q to show that the adiabatic condition can be satised between P s (t) and P s (t) for any . According to Eq. (5.18), S (t p )⟩∝V s (t p )(F q V 0 (t q )) F q V 0 (t q )V 0 (0)S (0)⟩ =V s (t p )V 0 (t q )V 0 (t q )V 0 (0)S (0)⟩ =V s (t p )V 0 (0)S (0)⟩; (6.63) 195 where V s (t) is dened as V s (t)=U(t;t q )F q V 0 (t q )h(t;t q ); t>t q ; (6.64) and is the horizontal lift of P s (t) given the initial condition F V 0 (t q ). From the same argument in the proof of Lemma 1, we get S (t p )⟩=Q c e −i"s (tp−tq) ⎛ ⎝ p M l=q+1 g l ⎞ ⎠ F q S (t q )⟩ =Q c e −i"s (tp−tq) F pq p M l=1 g l S (t 0 )⟩: (6.65) 6.7.3 Lemma 3 For part 1, according to condition 1, U q+1 (t;t q )= exp ⎛ ⎝ i q+M Q r=q+1 f(t)Q r ⎞ ⎠ ; (6.66) for t∈ [t q ;t q+1 ]. From the procedure in the proof of Lemma 1, @h @t =if(t)V s (t q )P s (t q ) ⎛ ⎝ Q Qr∈Pq Q r ⎞ ⎠ P s (t q )V s (t q )= 0; (6.67) according to Q r ∈G n and S (t)⟩=e −i" 0 (t−t q−1 ) U q+1 (t;t q )S (t q )⟩: (6.68) When t=t q+1 , when f(t q+1 )=~4, and S (t q+1 )⟩=e −i" 0 (t q+1 −tq) ⎛ ⎝ q+M M l=q+1 g l ⎞ ⎠ S (t q )⟩; (6.69) under the adiabatic approximation. 196 For part 2, supposeF q takes the system from the ground space toP s . Then for any ≠, P s (t) @H(t) @t P s (t)= i(@ t f(t))U q+1 (t;t q ) Q Qr∈Pq P s (t q )Q r H(t q )P s (t q )− P s (t q )H(t q )Q r P s (t q )U q+1 (t;t q ): (6.70) By the same argument as in the proof of Lemma 2, for each Q r , there is only one r such that P s (t q )Q r H(t q )P s r (t q ) and P s (t q )H(t q )Q r P s r (t q ) do not equal 0. Since C Qr ⋂C Qm =∅ for any Q r ;Q m ∈P q , r ≠ m when r ≠m. Then, for any such r , ZP s (t) @H(t) @t P s r (t)Z 1 =K@ t f(t) T " s (t q )−" s r (t q )T: (6.71) Since SC Qr S is odd, then S" s (t q )−" s r (t q )S ≥ 2, the adiabatic condition Eq. (6.8) holds for arbitrary , and we get S (t q+1 )⟩=Q c e −i"s (t q+1 −tq) F q+1;q ⎛ ⎝ q+M M l=q+1 g l ⎞ ⎠ S (t q )⟩: (6.72) 197 Chapter 7 Epilogue In this thesis, we have studied: 1) the quantum computation system using clouds of cold atoms trapped on atomic chips, which is robust to inhomogenous coupling, imperfect controls and frequency detuning; 2) using highly ecient quantum error correcting code to implement FTQC, which suggests a great reduction of resource overhead; 3) taking advantage of energy gap to reduce rate of thermal error occurring on each qubits, which may reduce the number of qubits by an order if temperature is low or coupling is strong. In Part III, we brie y discussed a protocol to eciently distill various ancilla states needed using classical code, assuming distillation process is perfect. However, the non- perfect distillation circuits may greatly aect the quality of output ancillas. Meanwhile, it will increase the overhead of the distillation protocol. All these eects needs carefully study. A pipeline may be introduced to increase the distillation throughput, whose structure can be dierent for dierent ancilla states depending on their frequency of use. The resource overhead of whole FTQC scheme in this Part is highly determined by the details of distillation protocol, pipeline design and the codes chosen for memory and processor, which is a our future research topic. According to Chapter 6, it seems encouraging to introduce long range interaction between excited anyons, which can be compatible with our adiabatic code deformation protocol. Intuitively, the adiabatic process doing computation should not change the underlying mechanism to freeze the density of thermal anyons. If so, one may actu- ally implement self-correcting quantum computation in 2D lattice in nite temperature without strong coupling. However, it is worth noting that it may not lead to true self- correction [83]|the life time of the state may saturate if a perturbation (no matter how small) exists when realizing long range anyon interaction eectively. Still, an interesting 198 question is how much benet one can obtain from such long range interaction. It is possible that the lifetime of the state can be suciently long to implement an interesting enough quantum algorithm at high temperature without any active error correction. Since the rst threshold theorem were proposed, almost twenty years has passed. Real world quantum computation is still in its infancy. Slow experimental progress in quantum computation speaks volumes about how dicult it is to build a quantum computer. The diculty here is to simultaneous satisfy two crucial requirements: 1) extremely low error rate for each device; 2) perfect controlling of large scale of such quantum devices, which is formidably daunting. 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Abstract (if available)

Abstract The threshold theorem indicates that if errors are all local and their rates are below a certain threshold, it is possible to implement large scale quantum computation with arbitrarily small error based on active quantum error correction (QEC). However, most fault-tolerant quantum computation (FTQC) schemes require enormous overhead to achieve this rate by increasing either the concatenation levels for concatenated codes or the distance (hence the size) for topological codes. As a result, a logical qubit is usually encoded in more than thousands of physical qubits, even when the error rate is below the threshold. In this thesis, we try to reduce the resource overhead of FTQC. We first study a quantum computation scheme using clouds of atoms trapped on atomic chips. It suggests that this scheme is robust to noise to some extent and has potential to be scalable. Secondly, we propose a FTQC scheme of encoding many logical qubits into large code blocks. The error performance of such scheme is numerically studied. Preparation of various ancilla states necessary for computation is also explored. Thirdly, we explore the idea of introducing energy gap to suppress thermal noise on each qubit. We show that fault-tolerant holonomic quantum computation (HQC) can be implemented in stabilizer codes with the existence of such energy protection. Especially, we studied fault-tolerant HQC in surface codes in detail. This scheme opens the possibility for self-correcting quantum computation in 2D lattice.

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Creator Zheng, Yi-Cong (author)

Core Title Towards efficient fault-tolerant quantum computation

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 08/06/2015

Defense Date 06/12/2015

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

Tag fault-tolerance,OAI-PMH Harvest,quantum computer,quantum error correction

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Brun, Todd A. (committee chair), Lidar, Daniel A. (committee member), Reichardt, Benjamin (committee member)

Creator Email yicong.zheng@gmail.com,yicongzh@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-627948

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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...

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