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Fault-Tolerant Quantum Computation: Theory and Practice FAULT-TOLERANT QUANTUM COMPUTATION: THEORY AND PRACTICE FAULT-TOLERANT QUANTUM COMPUTATION: THEORY AND PRACTICE Dissertation for the purpose of obtaining the degree of doctor at Delft University of Technology, by the authority of the Rector Magnificus prof. dr. ir. T.H.J.J. van der Hagen, Chair of the Board of Doctorates, to be defended publicly on Wednesday 15th, January 2020 at 12:30 o’clock by Christophe VUILLOT Master of Science in Computer Science, Université Paris Diderot, Paris, France, born in Clamart, France. This dissertation has been approved by the promotor. promotor: prof. dr. B. M. Terhal Composition of the doctoral committee: Rector Magnificus, chairperson Prof. dr. B. M. Terhal Technische Universiteit Delft, promotor Independent members: Prof. dr. C.W.J. Beenakker Universiteit Leiden Prof. dr. L. DiCarlo Technische Universiteit Delft Prof. dr. R.T. König Technische Universität München Dr. A. Leverrier Inria Paris Prof. dr. ir. L.M.K. Vandersypen Technische Universiteit Delft Prof. dr. R.M. de Wolf Centrum Wiskunde & Informatica Keywords: quantum computing, quantum error correction, fault-tolerance Printed by: Gildeprint - www.gildeprint.nl Front: Kandinsky Vassily (1866-1944), Auf Weiss II, 1923 Photo © Centre Pompidou, MNAM-CCI, Dist. RMN-Grand Palais / image Centre Pompidou, MNAM-CCI Copyright © 2019 by C. Vuillot ISBN 978-94-6384-097-2 An electronic version of this dissertation is available at http://repository.tudelft.nl/. This thesis is dedicated to my daughter Andréa and her mother Lisa. CONTENTS Summary xi Preface xiii 1 Introduction 1 1.1 Introduction to quantum computing . 2 1.1.1 Quantum mechanics. 2 1.1.2 Elementary quantum systems . 5 1.1.3 Quantum computation . 9 1.1.4 Fragility of quantum information . 12 1.2 Quantum error correction. 14 1.2.1 Principle of quantum error correction . 14 1.2.2 Stabilizer formalism . 15 1.2.3 Notable examples . 18 1.3 Fault-tolerance and universality . 25 1.3.1 Fault-tolerance. 25 1.3.2 Magic states and their distillation . 26 1.3.3 Techniques to get to fault-tolerant universality . 28 1.4 Organization of the thesis . 29 References . 29 2 Testing Quantum Fault-Tolerance 35 2.1 Introduction . 36 2.2 Demonstrating fault-tolerance . 36 2.2.1 General approach . 37 2.2.2 The IBM 5Q chip and [[4,2,2]] . 37 2.2.3 Comments on the tested circuits . 40 2.3 Experimental results . 42 2.3.1 Parameters and runs . 42 2.3.2 Performance metric . 42 2.3.3 Comparisons. 42 2.4 Calibration data and additional experiment. 44 2.5 Conclusion . 46 References . 49 3 Quantum error correction with the toric-GKP code 51 3.1 Introduction . 52 3.2 General considerations . 53 3.2.1 Definitions and notations . 53 3.2.2 Maximum-likelihood vs. minimum-energy decoding . 55 vii viii CONTENTS 3.3 Protecting a Single GKP Qubit. 58 3.3.1 Set-up . 58 3.3.2 Decoding Strategies . 60 3.3.3 Numerical Results . 66 3.4 Concatenation: Toric-GKP Code . 67 3.4.1 Setup. 67 3.4.2 Noiseless Measurements & Numerical Results . 68 3.5 Noisy Measurements: 3D Space-Time Decoding . 70 3.5.1 Error model . 70 3.5.2 Equivalent formulation with U(1) symmetry. 73 3.5.3 Decoder and numerical results. 75 3.6 No-Go Result for Linear Oscillator Codes . 81 3.7 Proof of No-Go Theorem . 82 3.7.1 Logical Error Model under Displacement Errors . 83 3.7.2 Eigenvalues of the Covariance Matrix . 85 3.7.3 Existence of a Spread-out & Orthogonal Logical Operator Basis . 87 3.8 Continuous-Variable Toric-Code . 87 3.9 Discussion . 89 References . 89 4 Code Deformation Techniques 93 4.1 Introduction . 94 4.2 Code Deformation and Lattice surgery . 95 4.2.1 Code Deformation . 95 4.2.2 Lattice Surgery . 97 4.3 Gauge Fixing . 99 4.4 Fault-Tolerance Analysis with Gauge Fixing . 100 4.4.1 Fault-Tolerance of Code Deformation . 102 4.4.2 Code Deformation Examples. 106 4.5 Logical operation of a code deformation . 117 4.5.1 Merge operation . 117 4.5.2 Split operation . 120 4.5.3 General code deformation operation . 121 4.6 Discussion . 122 References . 122 5 Quantum Pin Codes 125 5.1 Introduction . 126 5.2 Pin codes . 127 5.2.1 Terminology and formalism . 127 5.2.2 Definition of an (x,z)-pin code. 128 5.2.3 Relation to quantum color codes. 129 5.2.4 Constructing pin codes . 130 5.2.5 Remarks . 134 5.3 Transversal gates and magic state distillation . 136 5.3.1 Weighted polynomials and transversal gates . 137 CONTENTS ix 5.4 Properties of quantum pin codes . 139 5.4.1 Code parameters and basic properties . 139 5.4.2 Colored logicals and unfolding. 141 5.4.3 Gauge pin codes . 142 5.4.4 Transversality . 144 5.4.5 Boundaries and free pins. 144 5.5 Examples and applications . 146 5.5.1 Coxeter groups, hyperbolic color codes . 146 5.5.2 Pin codes from chain complexes . 147 5.5.3 Puncturing triply-even spaces . 149 5.5.4 Logical circuits of CCZ s . 151 5.6 Discussion . 151 References . 152 6 Conclusion 157 Acknowledgements 159 Curriculum Vitæ 161 List of Publications 163 SUMMARY Quantum computation is the modern version of Schrödinger’scat experiment. It is backed up in principle by the theory and thinking about it can make people equally uncomfort- able and excited. Besides, its practical realization seems so extremely challenging that some people even doubt it is possible. On the other hand, we are nowadays much closer to realizing quantum computation and in addition, it has much more implications than Schrödinger’s original cat experiment. One of the major difficulties in realizing quantum computation is the inevitable pres- ence of noise in realistic quantum devices which makes the direct realization of quan- tum computers impossible. In order to protect quantum information and quantum pro- cesses against noise, quantum error correction and fault-tolerance have been devised. Although the gap between experiments and the requirements of fault-tolerance is still daunting, the field of quantum error correction and fault-tolerance extends and influ- ences architectural decisions from the hardware to the ideal quantum programs that we want to run. That is why it has the potential to make or break the practicality of quantum computation and a lot of research effort goes into this field. In this thesis we investigate and improve several aspects of fault-tolerant schemes and quantum error correction. We implement an experiment which validates on a small device the usefulness of fault-tolerance for quantum computation. We investigate the advantages of harnessing quantum continuous degrees of freedom present in the lab to protect discrete quantum information in a scalable way. We establish a framework to analyze the fault-tolerant properties of code deformation techniques which are versatile techniques to process quantum information protected by an error correcting code. We also present some novel code deformation techniques with the potential to increase re- liability. Finally we define a new class of quantum error correcting codes, quantum pin codes, with built in capabilities for fault-tolerant quantum gates. We give some practical constructions and show some protocols with interesting parameters. The roads towards universal and fault-tolerant quantum computation are still steep but research efforts are pushing in the right directions. xi PREFACE HYSHOULD we build a quantum computer? With this foreword I would like to try W to present a case for investing in building a quantum computer. This case mainly consists in what makes me excited about working in the field of quantum computing. Besides, this is the humble opinion of a young and inexperienced researcher probably lacking a more broad view of science, technology and academia. One of the fascinating aspects of science is that advancing our understanding actu- ally allows us to modify the world around us. Not necessarily in a useful way: making fireworks, drawing beautiful fractals, going to the moon or trapping and monitoring the presence of a single photon for milliseconds are not really directly useful. When we un- cover such ways of conjuring something new, and when someone takes on themselves to figure out how to use it, it often has dramatic consequences. However, the marvelous part to me is more in the former than the latter. The field of computing is a prime example which obviously dramatically changed the world. It is also constructed from many such marvelous understandings that were neces- sary to produce the final object, a computer. The original question that was raised was: is it possible to compute using physical mechanized processes, or really any kind of phys- ical process? The first surprising results about this question was that there are problems that cannot be solved by any machine we could come up with. Problems that would re- quire such a machine computing forever before giving an answer. The second surprising answer is that there is a simple machine that can perform all the feasible computations. Sure enough this machine, the computer, got built at some point. On the physics side, an ever growing understanding of condensed matter and its electrical properties brought us to a place where we can build these computers so efficiently and so small that they can do billions of operations per seconds and we can store trillions of bits in our pockets. Computers are truly fascinating objects even before considering all the work they do for us. The field of quantum physics is also a major example filled with surprising under- standings.
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