MEASUREMENT BASED FAULT TOLERANT ERROR CORRECTING QUANTUM CODES ON FOLIATED CLUSTER STATES Andrew Bolt Bsc. Hons. Physics. The University of Queensland A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY AT THE UNIVERSITY OF QUEENSLAND IN 2017 SCHOOL OF MATHEMATICS AND PHYSICS Abstract Quantum communication and computation is an emerging field in quantum mechanics. There are many promising applications including secure communications, simulation of complex quantum sys- tems or quantum algorithms which perform faster than classical methods. The main barrier to realis- ing these possibilities is coping with the inevitable errors induced by noise and experimental imper- fections. Quantum error correction is a process which allows for the storage and retrieval of encoded qubits with high fidelity even in an imperfect environment. Error correction operates by encoding qubit states into a subspace of a larger system of qubits. The presence of errors is detected by per- forming stabilizer measurements. These measurements do not act on the encoding subspace and the measurement results, or syndrome, can be used to infer errors. The decoding problem is to use syndrome information to find the most likely errors which have occurred. This is performed using classical computing resources. Once the errors have been inferred operations can be applied to the system to counteract their effect. While many quantum error correcting codes are known not all are suitable in a practical setting. When choosing a potential code we must consider the following. Does the construction use simple interactions that can be physically realised? How high is the physical qubit to encoded qubit over- head? Does an efficient decoder exist? Is the encoding scheme scalable and do the error correcting properties improve with scaling? All of these properties are desirable for a practical scheme, though some compromise may be required. In this thesis I show how cluster states can be used as a resource for implementing CSS quantum codes. Additionally I show how these cluster codes can be linked together in a layered fashion to create foliated codes. These foliated codes can be used with any constituent CSS code as a basis for fault tolerant error correction. This process is an extension of Raussendorf’s toric code measurement based processing scheme. I first generalise this measurement based scheme to all CSS quantum codes, which is presented in chapter 3. I then present a novel decoding scheme for foliated turbo codes, chapter 4, and present the decoding performance of some turbo code families in chapter 5. These turbo codes are a class of finite rate codes whose overhead and complexity are flexible. Chapter 6 presents the decoding performance of foliated bicycle codes. The low overhead of physical resources to logical qubits of both these code families make these foliated schemes an attractive alternative to the toric code when considering practical measurement based computing designs with large numbers of encoded qubits. Declaration by author This thesis is composed of my original work, and contains no material previously published or written by another person except where due reference has been made in the text. I have clearly stated the contribution by others to jointly-authored works that I have included in my thesis. I have clearly stated the contribution of others to my thesis as a whole, including statistical as- sistance, survey design, data analysis, significant technical procedures, professional editorial advice, and any other original research work used or reported in my thesis. The content of my thesis is the result of work I have carried out since the commencement of my research higher degree candidature and does not include a substantial part of work that has been submitted to qualify for the award of any other degree or diploma in any university or other tertiary institution. I have clearly stated which parts of my thesis, if any, have been submitted to qualify for another award. I acknowledge that an electronic copy of my thesis must be lodged with the University Library and, subject to the policy and procedures of The University of Queensland, the thesis be made available for research and study in accordance with the Copyright Act 1968 unless a period of embargo has been approved by the Dean of the Graduate School. I acknowledge that copyright of all material contained in my thesis resides with the copyright holder(s) of that material. Where appropriate I have obtained copyright permission from the copyright holder to reproduce material in this thesis. Publications During Candidature A. Bolt, G. Duclos-Cianci, D. Poulin and T.M. Stace Foliated Error-Correcting Codes. Phys. Rev. Lett. 117, 070501 (2016) Publications Included in this Thesis None. Contributions by Others in this Thesis No contributions by others. Statement of parts of the thesis submitted to qualify for the award of another degree None. Acknowlegements I would like to thank my adviser Thomas Stace for his guidance and advice. I would also like to thank the numerous people with whom I have had interesting discussions and from whom I have received advice. Special mention to David Poulin and Guillume Duclos-Cianci of the University of Sherbrooke. I am thankful to the various students and staff working in the physics department at UQ who made my time during this research project enjoyable and stimulating. My work was funded by the Australian Research Council (ARC) Australian Postgraduate Award scholarship and the ARC Centre for Engineered Quantum Systems (EQuS). Keywords quantum, error correcting, foliated, cluster state, measurement based, computation, fault tolerant, turbo code Australian and New Zealand Standard Research Classifications ANZSRC Code: 020603 Quantum, Information, Computation and Communication, 100%. Fields of Research (FoR) Classification FoR Code: 0206, Quantum Physics, 100%. Contents Contents ix 1 Introduction 1 1.1 Quantum Error Correction . .2 1.2 Cluster States . .6 1.3 Thesis Outline . .7 2 Technical Review 9 2.1 Classical Error Correcting Codes . .9 2.2 Block Codes . 14 2.3 Convolution Codes . 17 2.3.1 Transfer Function Notation . 19 2.3.2 Transfer Function Manipulation . 22 2.3.3 Remarks on Decoding . 25 2.4 Turbo Codes . 26 2.4.1 Serial Turbo Codes . 27 2.4.2 Parallel Turbo Codes . 28 2.4.3 Interleaving . 29 2.5 Decoding Methods . 30 2.5.1 Lookup Table Decoding . 31 ix 2.5.2 Perfect Matching Decoding . 31 2.5.3 Belief Propagation Methods . 33 2.5.4 Tanner Graphs . 40 2.5.5 Trellis Construction . 43 2.5.6 Wolf Trellises . 43 2.5.7 Logical Wolf Trellises . 45 2.5.8 Trellis Decoding . 48 2.6 Quantum Error Correcting Codes . 53 2.6.1 Brief History of Quantum Codes . 53 2.6.2 Principles of Quantum Error Correction . 54 2.7 Stabilizer Codes . 58 2.8 Calderbank-Shor-Steane Codes . 62 2.9 Subsystem Codes . 65 2.10 Topological Codes . 67 2.10.1 Toric Code . 67 2.11 Quantum Convolutional Codes . 70 2.11.1 CSS Convolutional Codes . 70 2.12 Quantum Repeaters . 73 2.13 Summary . 79 3 Foliated Cluster State Codes 81 3.1 Cluster States and Measurements . 82 3.2 Cluster State Codes . 85 3.2.1 Steane Code . 86 3.2.2 Toric Code . 87 3.2.3 Turbo Codes . 88 3.2.4 Cluster State Code Summary . 92 3.3 Foliated Codes . 92 3.3.1 General Construction . 93 3.3.2 Steane Code . 97 3.3.3 Toric Code . 99 3.3.4 Turbo Codes . 101 3.4 Summary . 102 3.A Appendix . 103 4 Foliated Turbo Code Implementation 107 4.1 Quantum Convolutional Codes . 107 4.2 Cluster Convolutional Codes . 113 4.3 Foliated Convolutional Codes . 114 4.4 Decoding Algorithm . 118 4.4.1 Marginal Passing Across Layers . 119 4.4.2 Logical Wolf Trellis Decoder . 122 4.5 Summary . 127 5 Numerical results: Turbo Codes 131 5.1 Sample Turbo Codes . 131 5.1.1 T9 turbo code . 132 5.1.2 T25 turbo code . 134 5.2 Sampling Distribution . 135 5.3 Simulation Results . 136 5.3.1 T9 Turbo Code . 138 5.3.2 T25 Turbo Code . 140 5.4 Analysis . 143 5.4.1 Convolutional code distances . 144 5.4.2 Fault-Tolerance . 149 5.5 Summary . 155 5.A Appendix . 157 6 Numerical Results: Bicycle Codes 161 6.1 LDPC Bicycle Code Construction . 162 6.2 LDPC Foliated Decoder . 164 6.3 Numerical Results . 166 6.4 Analysis . 168 6.5 Summary . 170 6.A Simulation Performance . ..