Zunaira Babar is a Senior Algorithm Engineer at VIAVI Solutions Inc. Daryus Chandra is a Senior Quantum Error Correction Researcher at Photonic Inc. Soon Xin Ng, PhD, is a Full Professor of Telecommunications at the University of Southampton, UK. Lajos Hanzo is a Fellow of the Royal Academy of Engineering and a Foreign Member of the Hungarian Academy of Sciences.

• About the Authors xiiiList of Acronyms xvPreface xviiAcknowledgments xixPart I From Classical to Quantum Codes 11 Introduction 31.1 Motivation 31.2 Historical Overview 61.3 Outline of the Book 172 Preliminaries on Quantum Information 212.1 Introduction 212.2 A Brief Review of Quantum Information 212.3 Quantum Information Processing 242.4 Quantum Decoherence 282.5 No-cloning Theorem 332.6 Quantum Entanglement 342.7 Quantum Channels 352.8 Summary and Conclusions 383 From Classical to Quantum Coding 393.1 Introduction 393.2 A Brief Review of Classical Syndrome-based Decoding 403.3 A Brief Review of Quantum Stabilizer Codes 433.4 Protecting a Single Qubit: Design Examples 463.5 Summary and Conclusions 574 Revisiting Classical Syndrome Decoding 594.1 Introduction 594.2 Look-up Table-based Syndrome Decoding 614.3 Trellis-based Syndrome Decoding 624.4 Block Syndrome Decoding 704.5 Results and Discussion 744.6 Summary and Conclusions 795 Near-capacity Codes for Entanglement-aided Classical Communication 835.1 Introduction 835.2 Review of the SD Coding Protocol 845.3 Entanglement-assisted Classical Capacity 875.4 Bit-based Code Structure 905.5 Near-capacity Design 915.6 Results and Discussion I 955.7 Symbol-based Code Structure 1015.8 Results and Discussion II 1015.9 Summary and Conclusion 104Part II Near-term Quantum Codes 1096 Quantum Coding Bounds and a Closed-form Approximation of the Minimum Distance Versus Quantum Coding Rate 1116.1 Introduction 1116.2 On Classical to Quantum Coding Bounds 1116.3 Quantum Coding Bounds in the Asymptotical Limit 1146.4 Quantum Coding Bounds on Finite-length Codes 1186.5 The Bounds on Entanglement-assisted Quantum Stabilizer Codes 1226.6 Summary and Conclusions 1267 Quantum Topological Error Correction Codes: The Classical-to-quantum Isomorphism Perspective 1277.1 Introduction 1277.2 Classical Topological Error Correction Codes: Design Examples 1277.3 Quantum Topological Error Correction Codes: Design Examples 1357.4 Performance of Quantum Topological Error Correction Codes 1417.5 Summary and Conclusions 1518 Protecting Quantum Gates Using Quantum Topological Error Correction Codes 1538.1 Introduction 1538.2 Protecting Transversal Gates 1548.3 Design Examples 1598.4 Error Model 1648.5 Simulation Results and Performance Analysis 1698.6 Conclusions and Future Research 1799 Universal Decoding of Quantum Stabilizer Codes via Classical Guesswork 1819.1 Introduction 1819.2 Decoding Classical FEC Codes via Guesswork 1829.3 Quantum Stabilizer Codes 1849.4 Decoding Quantum Stabilizer Codes 1859.5 Results and Discussion 1929.6 Conclusions and Future Work 197Part III Advanced Quantum Codes 20110 Revisiting the Classical to Quantum Coding Evolution 20310.1 Introduction 20310.2 Review of Classical Linear Block Codes 20410.3 Quantum Stabilizer Codes 20610.4 Quantum Convolutional Codes 21810.5 Entanglement-assisted Quantum Codes 22110.6 Summary and Conclusions 22211 EXIT-chart Aided Near-hashing-bound Concatenated Quantum Codes 22511.1 Introduction 22511.2 Design Objectives 22611.3 Circuit-based Representation of Stabilizer Codes 22811.4 Revisiting Concatenated Quantum Codes 23411.5 EXIT Chart Aided Quantum Code Design 23911.6 Results and Discussion I 24211.7 Quantum Irregular Convolutional Codes 24811.8 Results and Discussion II 25211.9 Summary and Conclusions 25512 Near-hashing-bound Quantum Turbo Short-block Codes 25712.1 Introduction to Iterative Decoding 25712.2 Quantum Short-block Codes 26012.3 Quantum Turbo Code Design Using QSBCs 27112.4 Results and Analysis 27412.5 Conclusions and Future Research 28113 EXIT-chart-aided Design of Irregular Multiple-rate Quantum Turbo Block Codes 28313.1 Introduction 28313.2 Quantum Short-block Codes 28413.3 Quantum Turbo Short-block Codes 28813.4 EXIT-chart Analysis 29113.5 Multiple-rate Quantum Turbo Short-block Codes 29813.6 Conclusions 30414 Quantum Low-density Parity Check Codes 30714.1 Introduction 30714.2 Quantum LDPC Code Designs 30814.3 Iterative Decoding of Quantum LDPC Codes 31614.4 High-rate QLDPC Codes from Row-circulant Classical LDPCs 32314.5 Results and Discussions I 32614.6 Modified Non-binary Decoding 32814.7 Reweighted BP for Graphs Exhibiting Cycles 33514.8 Results and Discussions II 33614.9 Summary and Conclusions 34115 Summary and Future Research 34715.1 Summary 34715.2 Future Research 359A Construction of Syndrome Former 363A.1 Convolutional Codes 363A.2 Turbo Trellis Coded Modulation 365B Simulation of QLDPC Decoding 367Glossary 369References 373Subject Index 389Author Index 391

Zunaira Babar is a Senior Algorithm Engineer at VIAVI Solutions Inc. Daryus Chandra is a Senior Quantum Error Correction Researcher at Photonic Inc. Soon Xin Ng, PhD, is a Full Professor of Telecommunications at the University of Southampton, UK. Lajos Hanzo is a Fellow of the Royal Academy of Engineering and a Foreign Member of the Hungarian Academy of Sciences.

• About the Authors xiiiList of Acronyms xvPreface xviiAcknowledgments xixPart I From Classical to Quantum Codes 11 Introduction 31.1 Motivation 31.2 Historical Overview 61.3 Outline of the Book 172 Preliminaries on Quantum Information 212.1 Introduction 212.2 A Brief Review of Quantum Information 212.3 Quantum Information Processing 242.4 Quantum Decoherence 282.5 No-cloning Theorem 332.6 Quantum Entanglement 342.7 Quantum Channels 352.8 Summary and Conclusions 383 From Classical to Quantum Coding 393.1 Introduction 393.2 A Brief Review of Classical Syndrome-based Decoding 403.3 A Brief Review of Quantum Stabilizer Codes 433.4 Protecting a Single Qubit: Design Examples 463.5 Summary and Conclusions 574 Revisiting Classical Syndrome Decoding 594.1 Introduction 594.2 Look-up Table-based Syndrome Decoding 614.3 Trellis-based Syndrome Decoding 624.4 Block Syndrome Decoding 704.5 Results and Discussion 744.6 Summary and Conclusions 795 Near-capacity Codes for Entanglement-aided Classical Communication 835.1 Introduction 835.2 Review of the SD Coding Protocol 845.3 Entanglement-assisted Classical Capacity 875.4 Bit-based Code Structure 905.5 Near-capacity Design 915.6 Results and Discussion I 955.7 Symbol-based Code Structure 1015.8 Results and Discussion II 1015.9 Summary and Conclusion 104Part II Near-term Quantum Codes 1096 Quantum Coding Bounds and a Closed-form Approximation of the Minimum Distance Versus Quantum Coding Rate 1116.1 Introduction 1116.2 On Classical to Quantum Coding Bounds 1116.3 Quantum Coding Bounds in the Asymptotical Limit 1146.4 Quantum Coding Bounds on Finite-length Codes 1186.5 The Bounds on Entanglement-assisted Quantum Stabilizer Codes 1226.6 Summary and Conclusions 1267 Quantum Topological Error Correction Codes: The Classical-to-quantum Isomorphism Perspective 1277.1 Introduction 1277.2 Classical Topological Error Correction Codes: Design Examples 1277.3 Quantum Topological Error Correction Codes: Design Examples 1357.4 Performance of Quantum Topological Error Correction Codes 1417.5 Summary and Conclusions 1518 Protecting Quantum Gates Using Quantum Topological Error Correction Codes 1538.1 Introduction 1538.2 Protecting Transversal Gates 1548.3 Design Examples 1598.4 Error Model 1648.5 Simulation Results and Performance Analysis 1698.6 Conclusions and Future Research 1799 Universal Decoding of Quantum Stabilizer Codes via Classical Guesswork 1819.1 Introduction 1819.2 Decoding Classical FEC Codes via Guesswork 1829.3 Quantum Stabilizer Codes 1849.4 Decoding Quantum Stabilizer Codes 1859.5 Results and Discussion 1929.6 Conclusions and Future Work 197Part III Advanced Quantum Codes 20110 Revisiting the Classical to Quantum Coding Evolution 20310.1 Introduction 20310.2 Review of Classical Linear Block Codes 20410.3 Quantum Stabilizer Codes 20610.4 Quantum Convolutional Codes 21810.5 Entanglement-assisted Quantum Codes 22110.6 Summary and Conclusions 22211 EXIT-chart Aided Near-hashing-bound Concatenated Quantum Codes 22511.1 Introduction 22511.2 Design Objectives 22611.3 Circuit-based Representation of Stabilizer Codes 22811.4 Revisiting Concatenated Quantum Codes 23411.5 EXIT Chart Aided Quantum Code Design 23911.6 Results and Discussion I 24211.7 Quantum Irregular Convolutional Codes 24811.8 Results and Discussion II 25211.9 Summary and Conclusions 25512 Near-hashing-bound Quantum Turbo Short-block Codes 25712.1 Introduction to Iterative Decoding 25712.2 Quantum Short-block Codes 26012.3 Quantum Turbo Code Design Using QSBCs 27112.4 Results and Analysis 27412.5 Conclusions and Future Research 28113 EXIT-chart-aided Design of Irregular Multiple-rate Quantum Turbo Block Codes 28313.1 Introduction 28313.2 Quantum Short-block Codes 28413.3 Quantum Turbo Short-block Codes 28813.4 EXIT-chart Analysis 29113.5 Multiple-rate Quantum Turbo Short-block Codes 29813.6 Conclusions 30414 Quantum Low-density Parity Check Codes 30714.1 Introduction 30714.2 Quantum LDPC Code Designs 30814.3 Iterative Decoding of Quantum LDPC Codes 31614.4 High-rate QLDPC Codes from Row-circulant Classical LDPCs 32314.5 Results and Discussions I 32614.6 Modified Non-binary Decoding 32814.7 Reweighted BP for Graphs Exhibiting Cycles 33514.8 Results and Discussions II 33614.9 Summary and Conclusions 34115 Summary and Future Research 34715.1 Summary 34715.2 Future Research 359A Construction of Syndrome Former 363A.1 Convolutional Codes 363A.2 Turbo Trellis Coded Modulation 365B Simulation of QLDPC Decoding 367Glossary 369References 373Subject Index 389Author Index 391