Tailoring surface codes Improvements in quantum error correction with biased noise David Kingsley Tuckett A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy School of Physics Faculty of Science The University of Sydney 2020 Abstract For quantum computers to reach their full potential will require error correction. We study the surface code, one of the most promising quantum error correcting codes, in the context of predominantly dephasing (Z-biased) noise, as found in many quantum architectures. We find that the surface code is highly resilient to Y -biased noise, and tailor it to Z-biased noise, whilst retaining its practical features. We demonstrate ultrahigh thresholds for the tailored surface code: 39% with a realistic bias of η = 100, and 50% with pure Z noise, ∼ ∼ far exceeding known thresholds for the standard surface code: 11% with pure Z noise, and ∼ 19% with depolarizing noise. Furthermore, we provide strong evidence that the threshold of ∼ the tailored surface code tracks the hashing bound for all biases. We reveal the hidden structure of the tailored surface code with pure Z noise that is responsible for these ultrahigh thresholds. As a consequence, we prove that its threshold with pure Z noise is 50%, and we show that its distance to Z errors, and the number of failure modes, can be tuned by modifying its boundary. For codes with appropriately modified boundaries, the distance to Z errors is O(n) compared to O(pn) for square codes, where n is the number of physical qubits. We demonstrate that these characteristics yield a significant improvement in logical error rate with pure Z and Z-biased noise. Finally, we introduce an efficient approach to decoding that exploits code symmetries with respect to a given noise model, and extends readily to the fault-tolerant context, where mea- surements are unreliable. We use this approach to define a decoder for the tailored surface code with Z-biased noise. Although the decoder is suboptimal, we observe exceptionally high fault-tolerant thresholds of 5% with bias η = 100 and exceeding 6% with pure Z noise. ∼ Our results open up many avenues of research and, recent developments in bias-preserving gates, highlight their direct relevance to experiment. i Statement of contribution This thesis consists of an introduction, three body chapters and a conclusion. I am wholly responsible for the introduction and conclusion. The body chapters are based on research pa- pers with minor formatting and typographical corrections, as well as some additions indicated below. The research papers are all published in peer-reviewed journals. I am the lead author on all of the research papers and my contribution to each paper is stated below. Chapter2 Ultrahigh error threshold for surface codes with biased noise https://doi.org/10.1103/PhysRevLett.120.050505 https://arxiv.org/abs/1708.08474 Journal Physical Review Letters 120, 050505 (2018) c 2018 American Physical Society Authors David K. Tuckett, Stephen D. Bartlett, and Steven T. Flammia Contribution This chapter contains the full letter with an expanded numerics section to integrate the supplemental material published alongside the letter. The initial idea for the project arose from discussions between all authors. The research was undertaken by me, with supervision from Stephen and Steve. The numerics pre- sented in the paper were performed by me. All authors contributed to drafting and editing the paper. Chapter3 Tailoring surface codes for highly biased noise https://doi.org/10.1103/PhysRevX.9.041031 https://arxiv.org/abs/1812.08186 Journal Physical Review X 9, 041031 (2019) c Creative Commons Attribution 4.0 International Authors David K. Tuckett, Andrew S. Darmawan, Christopher T. Chubb, Sergey Bravyi, Stephen D. Bartlett, Steven T. Flammia Contribution This chapter contains the full paper, with the addition of Figure 3.11 to illustrate the key structural theorem. The initial idea for the project arose from discussions between Stephen, Steve, Sergey and me. The research was primarily un- dertaken by me, with supervision from Stephen and Steve. The numerics presented in the paper were performed by me. The analytics presented in the paper were initiated by me and further developed by Sergey, Chris and me. The decoder adap- tation, presented in Section 3.6, was initiated by Andrew and further developed by Andrew and me. I drafted the paper, with contributions from Sergey (Section 3.3.2) and Andrew (Section 3.6), and all authors contributed to editing. ii Statement of contribution Chapter4 Fault-tolerant thresholds for the surface code in excess of 5% under biased noise https://doi.org/10.1103/PhysRevLett.124.130501 https://arxiv.org/abs/1907.02554 Journal Physical Review Letters 124, 130501 (2020) c 2020 American Physical Society Authors David K. Tuckett, Stephen D. Bartlett, Steven T. Flammia, Benjamin J. Brown Contribution This chapter contains the full letter with the addition of the supplemental material, published alongside the letter, detailing the decoder implementation. The initial idea for the project is due to Ben and builds directly on the work presented in Chapter3. The research was primarily undertaken by me, with supervision from Ben and advice from Stephen and Steve. The numerics presented in the paper were performed by me. The decoder concept was initially proposed by Ben and further developed by me. Ben drafted the paper and all authors contributed to editing. I wrote the additional section detailing the decoder implementation. This is to certify that, as supervisor for the candidature upon which this thesis is based, I can confirm that the authorship contribution statements above are correct. Signature: Date: Supervisor name: Stephen Bartlett iii Statement of originality This is to certify that to the best of my knowledge, save as expressly acknowledged, the content of this thesis is my own work. This thesis has not previously been submitted for a degree at this or any other university. Signature: Date: Student name: David Kingsley Tuckett iv Acknowledgements I would like to express my sincere thanks to my supervisor Stephen Bartlett and co-supervisor Steve Flammia for their guidance, support, encouragement and especially their enthusiasm. I am very grateful to my co-authors for many fascinating and fruitful discussions: Ben Brown, Chris Chubb, Andrew Darmawan, Sergey Bravyi, and my supervisors. I would also like to thank everyone in the Sydney Quantum Information Theory group for creating such a friendly, amusing and inspiring workplace. Finally, I would like to thank my family. I thank my parents and brother for their love and support over the years. Above all, to my wife Yvonne and my daughters Annie and Charlotte: Thank you for absolutely everything, especially for coming on this amazing journey with me. v Contents Abstract i Statement of contribution ii Statement of originality iv Acknowledgements v 1 Introduction1 1.1 Overview.......................................1 1.2 Thesis structure....................................2 1.3 Quantum information................................3 1.4 Classical error correction............................... 13 1.5 Quantum error correction.............................. 16 1.6 Simulations...................................... 31 References.......................................... 34 2 Ultrahigh error threshold for surface codes with biased noise 39 DOI: 10.1103/PhysRevLett.120.050505, arXiv: 1708.08474 2.1 Introduction...................................... 39 2.2 Error correction with the surface code....................... 41 2.3 The BSV decoder................................... 42 2.4 Biased Pauli error model............................... 42 2.5 Hashing bound.................................... 42 2.6 Numerics....................................... 43 2.7 Fault tolerant syndrome extraction......................... 45 2.8 Discussion....................................... 46 References.......................................... 48 3 Tailoring surface codes for highly biased noise 50 DOI: 10.1103/PhysRevX.9.041031, arXiv: 1812.08186 3.1 Introduction...................................... 51 3.2 Definitions....................................... 54 3.3 Features of surface codes with pure Y noise.................... 55 3.4 Performance of surface codes with pure Y noise.................. 66 3.5 Performance of surface codes with biased noise.................. 68 3.6 Improved tensor-network decoding of rotated codes with biased noise...... 70 vi Contents 3.7 Discussion....................................... 76 3.A Color-code thresholds with biased noise...................... 78 3.B Exact optimal Y-decoder.............................. 81 References.......................................... 84 4 Fault-tolerant thresholds for the surface code in excess of 5% under biased noise 86 DOI: 10.1103/PhysRevLett.124.130501, arXiv: 1907.02554 4.1 Introduction...................................... 86 4.2 Surface code tailored for dephasing......................... 87 4.3 Decoding with symmetry.............................. 88 4.4 Decoding with finite bias............................... 89 4.5 Biased noise models................................. 90 4.6 Numerical simulations................................ 90 4.7 Low error rates.................................... 93 4.8 Discussion....................................... 93 4.A Decoder implementation............................... 94 4.B Decoder runtime................................... 99 References.........................................