New methods in quantum error correction and fault-tolerant quantum computing by Christopher Chamberland A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Physics (Quantum Information) Waterloo, Ontario, Canada, 2018 c Christopher Chamberland 2018 Examining Committee Membership The following served on the Examining Committee for this thesis. The decision of the Examining Committee is by majority vote. External Examiner: Stephen Bartlett Professor of Physics, University of Sydney Supervisor: Raymond Laflamme Professor, Dept. of Physics and Astronomy, University of Waterloo Committee Member: Daniel Gottesman Professor, Perimeter Institute for Theoretical Physics Internal Examiner: Richard Cleve Professor, David R. Cheriton School of Computer Science, University of Waterloo Committee Member: Matteo Mariantoni Professor, Dept. of Physics and Astronomy, University of Waterloo ii I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. iii Abstract Quantum computers have the potential to solve several interesting problems in poly- nomial time for which no polynomial time classical algorithms have been found. However, one of the major challenges in building quantum devices is that quantum systems are very sensitive to noise arising from undesired interactions with the environment. Noise can lead to errors which can corrupt the results of the computation. Quantum error correction is one way to mitigate the effects of noise arising in quantum devices. With a plethora of quantum error correcting codes that can be used in various settings, one of the main challenges of quantum error correction is understanding how well various codes perform under more realistic noise models that can be observed in experiments. This thesis proposes a new decoding algorithm which can optimize threshold values of error correcting codes under different noise models. The algorithm can be applied to any Markovian noise model. Further, it is shown that for certain noise models, logical Clifford corrections can further improve a code's threshold value if the code obeys certain symmetries. Since gates and measurements cannot in general be performed with perfect precision, the operations required to perform quantum error correction can introduce more errors into the system thus negating the benefits of error correction. Fault-tolerant quantum computing is a way to perform quantum error correction with imperfect operations while retaining the ability to suppress errors as long as the noise is below a code's threshold. One of the main challenges in performing fault-tolerant error correction is the high resource requirements that are needed to obtain very low logical noise rates. With the use of flag qubits, this thesis develops new fault-tolerant error correction protocols that are applicable to arbitrary distance codes. Various code families are shown to satisfy the requirements of flag fault-tolerant error correction. We also provide circuits using a constant number of qubits for these codes. It is shown that the proposed flag fault-tolerant method uses fewer qubits than previous fault-tolerant error correction protocols. It is often the case that the noise afflicting a quantum device cannot be fully charac- terized. Further, even with some knowledge of the noise, it can be very challenging to use analytic decoding methods to improve the performance of a fault-tolerant scheme. This thesis presents decoding schemes using several state of the art machine learning techniques with a focus on fault-tolerant quantum error correction in regimes that are relevant to near term experiments. It is shown that even in low noise rate regimes and with no knowledge of the noise, noise can be further suppressed for small distance codes. Limitations of ma- chine learning decoders as well as the classical resources required to perform active error correction are discussed. iv In many cases, gate times can be much shorter than typical measurement times of quantum states. Further, classical decoding of the syndrome information used in quantum error correction to compute recovery operators can also be much slower than gate times. For these reasons, schemes where error correction can be implemented in a frame (known as the Pauli frame) have been developed to avoid active error correction. In this thesis, we generalize previous Pauli frame schemes and show how Clifford frame error correction can be implemented with minimal overhead. Clifford frame error correction is necessary if the logical component of recovery operators were chosen from the Clifford group, but could also be used in randomized benchmarking schemes. v Acknowledgements First I would like to acknowledge Prof. Raymond Laflamme for his help, support and guidance throughout my PhD. Ray has provided a very vibrant research environment and allowed me to explore many different research avenues. Without this freedom I would never have been able to learn so many different aspects of quantum error correction and fault-tolerance. I would like to thank Tomas Jochym-O'Connor for his mentorship at the beginning of my PhD. I learned so many new ideas, especially at an early stage in my career, which were crucial for many of the projects I worked on during my PhD. I would like to thank David Poulin and Barbara Terhal for inviting me to visit Sher- brooke and Delft for several months to work on research ideas in fault-tolerance. These visits have been very rewarding and the exchanges of ideas pushed me in new directions expanding my understanding of fault-tolerance and error correction. I would like to thank the QuArC group at Microsoft for giving me the opportunity to do an internship there during the summer of 2017. In particular, I would like to thank Michael Beverland for his support and mentorship during my internship. Michael was incredibly patient and would always take time to discuss new ideas which lead to a paper of which I am very proud of. Throughout my PhD I was very fortunate to have several scientific and engaging discus- sions with very talent scientists. In particular, I would like to thank Joel Wallman, Debbie Leung, Nicolas Delfosse, Pavithran Iyer, Pooya Ronagh, Guillaume Verdon-Akzam, Ben Criger, Annie Park, Nikolas P. Breuckmann, Jonathan Conrad, Theerapat Tansuwannont, Thomas O'Brien, Christophe Vuillot, Daniel Weigand, Francesco Battisel, Xiatong Ni, Joel Klassen, Michael Chen, Hemant Katiyar, JunAn Lin, Brandon Buonacorsi, Jason Pye, Theodore Yoder and Ryuji Takagi. Lastly, I would like to thank my parents for their constant support and encouragement throughout my PhD and for giving me the opportunity to receive the level of education that allowed me to be where I am today. vi Dedication To everyone who is passionate about building a quantum computer. vii Table of Contents List of Tables xii List of Figures xvi 1 Brief review of error correction and fault-tolerance methods1 1.1 Review of the stabilizer formalism.......................2 1.1.1 The [[7; 1; 3]] Steane code, CSS codes and the Eastin-Knill theorem.4 1.2 Fault-tolerant quantum error correction....................7 1.2.1 Overview of fault-tolerant error correction using Shor-EC......7 2 Hard decoding algorithm for optimizing thresholds under general Marko- vian noise 14 2.1 Introduction and motivation.......................... 14 2.2 Stabilizer codes in the process matrix formalism............... 17 2.2.1 Process matrix formalism for noise at the physical level....... 18 2.2.2 Stabilizer codes............................. 19 2.2.3 Effective process matrix at the logical level.............. 20 2.2.4 Process matrix for two-qubit correlated noise............. 24 2.2.5 Effective noise channels for concatenated codes............ 24 2.2.6 Thresholds for noise models...................... 26 2.2.7 Specific decoders............................ 26 viii 2.3 Hard decoding algorithm for optimizing error-correcting codes....... 27 2.3.1 Infidelity-optimized decoding..................... 30 2.3.2 Resolving ties.............................. 31 2.4 Numerically calculating threshold hypersurfaces............... 32 2.4.1 Symmetric threshold hypersurfaces.................. 32 2.4.2 Threshold hypersurfaces for our infidelity-optimized decoder.... 32 2.5 Thresholds and infidelities for amplitude-phase damping.......... 32 2.6 Thresholds for coherent errors......................... 37 2.7 Correlated noise channel............................ 41 2.8 The effect of Pauli twirling on thresholds and the benefits of using transversal operations.................................... 42 2.9 Sensitivity and robustness of our hard decoding optimization algorithm to perturbations of the noise model........................ 44 2.10 Summary and outlook............................. 46 3 Flag fault-tolerant error correction with arbitrary distance codes 48 3.1 Introduction and formalism.......................... 48 3.1.1 Definitions, noise model and pseudo-threshold calculations..... 50 3.2 Flag error correction for small distance codes................. 52 3.2.1 Definitions and Flag 1-FTEC with distance-3 codes......... 52 3.2.2 Flag 2-FTEC with distance-5 codes.................. 57 3.3 Examples of flag 2-FTEC applied to d = 5 codes............... 61 3.4 Flag error correction protocol for arbitrary distance codes.......... 64 3.4.1 Conditions and protocol.......................