About the Authors xiii
List of Acronyms xv
Preface xvii
Acknowledgments xix
Part I From Classical to Quantum Codes 1
1 Introduction 3
1.1 Motivation 3
1.2 Historical Overview 6
1.3 Outline of the Book 17
2 Preliminaries on Quantum Information 21
2.1 Introduction 21
2.2 A Brief Review of Quantum Information 21
2.3 Quantum Information Processing 24
2.4 Quantum Decoherence 28
2.5 No-cloning Theorem 33
2.6 Quantum Entanglement 34
2.7 Quantum Channels 35
2.8 Summary and Conclusions 38
3 From Classical to Quantum Coding 39
3.1 Introduction 39
3.2 A Brief Review of Classical Syndrome-based Decoding 40
3.3 A Brief Review of Quantum Stabilizer Codes 43
3.4 Protecting a Single Qubit: Design Examples 46
3.5 Summary and Conclusions 57
4 Revisiting Classical Syndrome Decoding 59
4.1 Introduction 59
4.2 Look-up Table-based Syndrome Decoding 61
4.3 Trellis-based Syndrome Decoding 62
4.4 Block Syndrome Decoding 70
4.5 Results and Discussion 74
4.6 Summary and Conclusions 79
5 Near-capacity Codes for Entanglement-aided Classical Communication 83
5.1 Introduction 83
5.2 Review of the SD Coding Protocol 84
5.3 Entanglement-assisted Classical Capacity 87
5.4 Bit-based Code Structure 90
5.5 Near-capacity Design 91
5.6 Results and Discussion I 95
5.7 Symbol-based Code Structure 101
5.8 Results and Discussion II 101
5.9 Summary and Conclusion 104
Part II Near-term Quantum Codes 109
6 Quantum Coding Bounds and a Closed-form Approximation of the Minimum Distance Versus Quantum Coding Rate 111
6.1 Introduction 111
6.2 On Classical to Quantum Coding Bounds 111
6.3 Quantum Coding Bounds in the Asymptotical Limit 114
6.4 Quantum Coding Bounds on Finite-length Codes 118
6.5 The Bounds on Entanglement-assisted Quantum Stabilizer Codes 122
6.6 Summary and Conclusions 126
7 Quantum Topological Error Correction Codes: The Classical-to-quantum Isomorphism Perspective 127
7.1 Introduction 127
7.2 Classical Topological Error Correction Codes: Design Examples 127
7.3 Quantum Topological Error Correction Codes: Design Examples 135
7.4 Performance of Quantum Topological Error Correction Codes 141
7.5 Summary and Conclusions 151
8 Protecting Quantum Gates Using Quantum Topological Error Correction Codes 153
8.1 Introduction 153
8.2 Protecting Transversal Gates 154
8.3 Design Examples 159
8.4 Error Model 164
8.5 Simulation Results and Performance Analysis 169
8.6 Conclusions and Future Research 179
9 Universal Decoding of Quantum Stabilizer Codes via Classical Guesswork 181
9.1 Introduction 181
9.2 Decoding Classical FEC Codes via Guesswork 182
9.3 Quantum Stabilizer Codes 184
9.4 Decoding Quantum Stabilizer Codes 185
9.5 Results and Discussion 192
9.6 Conclusions and Future Work 197
Part III Advanced Quantum Codes 201
10 Revisiting the Classical to Quantum Coding Evolution 203
10.1 Introduction 203
10.2 Review of Classical Linear Block Codes 204
10.3 Quantum Stabilizer Codes 206
10.4 Quantum Convolutional Codes 218
10.5 Entanglement-assisted Quantum Codes 221
10.6 Summary and Conclusions 222
11 EXIT-chart Aided Near-hashing-bound Concatenated Quantum Codes 225
11.1 Introduction 225
11.2 Design Objectives 226
11.3 Circuit-based Representation of Stabilizer Codes 228
11.4 Revisiting Concatenated Quantum Codes 234
11.5 EXIT Chart Aided Quantum Code Design 239
11.6 Results and Discussion I 242
11.7 Quantum Irregular Convolutional Codes 248
11.8 Results and Discussion II 252
11.9 Summary and Conclusions 255
12 Near-hashing-bound Quantum Turbo Short-block Codes 257
12.1 Introduction to Iterative Decoding 257
12.2 Quantum Short-block Codes 260
12.3 Quantum Turbo Code Design Using QSBCs 271
12.4 Results and Analysis 274
12.5 Conclusions and Future Research 281
13 EXIT-chart-aided Design of Irregular Multiple-rate Quantum Turbo Block Codes 283
13.1 Introduction 283
13.2 Quantum Short-block Codes 284
13.3 Quantum Turbo Short-block Codes 288
13.4 EXIT-chart Analysis 291
13.5 Multiple-rate Quantum Turbo Short-block Codes 298
13.6 Conclusions 304
14 Quantum Low-density Parity Check Codes 307
14.1 Introduction 307
14.2 Quantum LDPC Code Designs 308
14.3 Iterative Decoding of Quantum LDPC Codes 316
14.4 High-rate QLDPC Codes from Row-circulant Classical LDPCs 323
14.5 Results and Discussions I 326
14.6 Modified Non-binary Decoding 328
14.7 Reweighted BP for Graphs Exhibiting Cycles 335
14.8 Results and Discussions II 336
14.9 Summary and Conclusions 341
15 Summary and Future Research 347
15.1 Summary 347
15.2 Future Research 359
A Construction of Syndrome Former 363
A.1 Convolutional Codes 363
A.2 Turbo Trellis Coded Modulation 365
B Simulation of QLDPC Decoding 367
Glossary 369
References 373
Subject Index 389
Author Index 391
