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Level Reduction and the Quantum Threshold Theorem Level Reduction and the Quantum Threshold Theorem Thesis by Panos Aliferis In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy arXiv:quant-ph/0703230v2 11 Jul 2011 California Institute of Technology Pasadena, California 2007 (Defended December 11, 2006) ii Copyright notice: Chapters 2 and 3 contain some material from [1] and chapter 4 contains some material from [2] which are both copyrighted by Rinton Press. Chapter 5 contains some material from [3] which is copyrighted by the American Physical Society. The remaining material is c 2007 Panos Aliferis All Rights Reserved iii for Ayis and K., of course iv Foreword Living on the open air of the mountains of research and reaching the point of writing this thesis would not have been possible without the guidance, the encouragement, and the support of many people. To them I write the few thanking words that follow. First, I would like to thank my advisor John Preskill. He provided the vessel in the form of IQI and the guidance for my education at Caltech. From him I learned much. Most importantly, I learned how the successful scientific spirit combines depth with breadth, seriousness with cheerful- ness, abstract proofs with happy dancing. He introduced me to the field of fault-tolerant quantum computation and pointed out interesting problems in this area of research. Many of the results of this thesis came about from my collaboration with him and Daniel Gottesman and would not have been possible without their crucial contributions and insights. Second, I would like to thank my mentor and friend Debbie Leung. I was fortunate to meet her as a starting graduate student at Caltech and collaborate with her on my first research results. From her I also learned much. I remember her emphasizing in several different occasions the value of developing easily composable frameworks for attacking research problems, and it was inspired by these words that I was guided to the idea of level reduction. Her sincere and active interest about both my personal well-being and education were invaluable in keeping me balanced at many difficult times. Third, I would like to thank my collaborators David DiVincenzo and Barbara Terhal for sharing many thoughts and starting new projects with me. My visits to the IBM Quantum Information group in the summers of 2005 and 2006 gave me new ideas and helped me understand in more depth important elements of our common research. Fourth, I would like to thank my collaborators Andrew Cross and Krysta Svore with whom I exchanged many ideas on various topics of this thesis. Andrew carried out the threshold calcula- tions discussed in chapter 5 and welcomed my ideas and suggestions with an enthusiasm that is so characteristic of him. Krysta contributed in the software we used for these calculations and, with her fresh smile and positive attitude, was a source of encouragement and inspiration. Fifth, I would like to thank my fellow graduate students, Kovid Goyal, Hui Khoon Ng, Graeme Smith, and Mike Zwolak, who spent time discussing their research with me. I also owe thanks to v Robin Blume-Kohout, Sergei Bravyi, David Poulin, and Robert Raussendorf whom I met as postdocs at IQI and with whom I have had many insightful discussions. Finally, I do not forget the influence of my three pre-Caltech mentors: my math mentor in Sparta, Stavros Damopoulos, my advisor at NTUA, Heraklis Avramopoulos, and my advisor at the University of Michigan, Duncan Steel. I also appreciate the help of the members of my defense and candidacy committees, Alexei Kitaev, Hideo Mabuchi, Robert McEliece, and Chris Umans. Pasadena, XII 06. vi Abstract Computers have led society to the information age revolutionizing central aspects of our lives from production and communication to education and entertainment. There exist, however, important problems which are intractable with the computers available today and, experience teaches us, will remain so even with the more advanced computers we can envision for tomorrow. Quantum computers promise speedups to some of these important but classically intractable problems. Simulating physical systems, a problem of interest in a diverse range of areas from testing physical theories to understanding chemical reactions, and solving number factoring, a problem at the basis of cryptographic protocols that are used widely today on the internet, are examples of applications for which quantum computers, when built, will offer a great advantage over what is possible with classical computer technology. The construction of a quantum computer of sufficient scale to solve interesting problems is, however, especially challenging. The reason for this is that, by its very nature, operating a quantum computer will require the coherent control of the quantum state of a very large number of particles. Fortunately, the theory of quantum error correction and fault-tolerant quantum computation gives us confidence that such quantum states can be created, can be stored in memory and can also be manipulated provided the quantum computer can be isolated to a sufficient degree from sources of noise. One of the central results in the theory of fault-tolerant quantum computation, the quantum threshold theorem shows that a noisy quantum computer can accurately and efficiently simulate any ideal quantum computation provided that noise is weakly correlated and its strength is below a critical value known as the quantum accuracy threshold. This thesis provides a simpler and more transparent non-inductive proof of this theorem based on the concept of level reduction. This concept is also used in proving the quantum threshold theorem for coherent and leakage noise and for quantum computation by measurements. In addition, the proof provides a methodology which allows us to establish improved rigorous lower bounds on the value of the quantum accuracy threshold. vii Contents Foreword iv Abstract vi 1 Prologue 1 1.1 TheQuantumComputer............................... ... 1 1.2 IntroductiontoQuantumComputation . ...... 2 1.2.1 QuantumMechanicsandQubits . 3 1.2.2 Open Systems and the Theory of Physical Operations . ...... 6 1.2.3 TheQuantumCircuitModel ........................... 8 1.2.4 Accuracy ...................................... 10 1.2.5 Quantum Universality . 14 1.3 QuantumCoding..................................... 15 1.3.1 Quantum Error-Correction Criteria . .... 17 1.3.2 Stabilizer Codes . 19 1.3.3 CSSConstruction ................................. 21 1.3.4 Examples ...................................... 22 2 The Syntax of Fault-Tolerant Quantum Computation 24 2.1 Introduction...................................... ... 24 2.2 LocalNoise ........................................ 25 2.3 Fault-Tolerant Quantum Circuit Simulations . ..... 28 2.4 Accuracy ......................................... 34 2.5 HistoryandAcknowledgements . ..... 39 3 RecursiveSimulationsandtheQuantumThresholdTheorem 40 3.1 Introduction...................................... ... 40 3.2 Recursive Fault-Tolerant Simulations . ...... 41 3.3 TheLevel-ReductionLemma . .. .. .. .. .. .. .. .. .. .. .. ... 43 viii 3.3.1 Independent Bad Rectangles as Simulated Faults . ..... 45 3.3.2 The Combinatorics for Overlapping Rectangles . .... 48 3.4 TheQuantumThresholdTheoremforLocalNoise . ....... 48 3.4.1 Improving the Threshold Estimate . .. 50 3.5 HistoryandAcknowledgements . ..... 52 4 Further Applications of Level Reduction 54 4.1 Introduction...................................... ... 54 4.2 LeakageNoise...................................... .. 55 4.2.1 TheLocalLeakageNoiseModel . 56 4.2.2 TheLeakage-ReductionLemma. .. 57 4.2.2.1 Bad Level-0 Rectangles As Simulated Faults . 60 4.2.3 The Quantum Threshold Theorem for Local Leakage Noise . ...... 61 4.2.4 Contracting Level-0 Rectangles . ... 62 4.3 Measurement-BasedQuantumComputation . ........ 63 4.3.1 Gate Teleportation and Quantum Computation by Measurements ...... 64 4.3.2 TheModel-ConversionLemma . 66 4.3.3 The Threshold Theorem for Quantum Computation by Measurements .... 68 4.4 HistoryandAcknowledgements . ..... 69 5 Lower Bounds on the Quantum Accuracy Threshold 70 5.1 Introduction...................................... ... 70 5.2 GadgetConstructions ............................... .... 71 5.2.1 Fault-Tolerant Quantum Error Correction . ..... 71 5.2.1.1 SyndromeMeasurementwithCatStates . 71 5.2.1.2 Syndrome Measurement with Encoded Pauli Eigenstates . .. 74 5.2.1.3 Syndrome Measurement via Logical Teleportation . 75 5.2.1.4 Satisfying Property 1(a) . 76 5.2.2 Encoded Quantum Universality . 78 5.2.2.1 Encoded Clifford-Group Operations . 78 5.2.2.2 Quantum Software and Encoded Non-Clifford Gates . 79 5.2.3 Fault-Tolerant Preparation and Measurement . ...... 80 5.3 Quantum Fault Tolerance with the Bacon-ShorCode . ........ 80 5.3.1 Syndrome Measurement using the Gauge Qubits . .... 82 5.3.2 Standard Syndrome Measurement Procedures . ....... 84 5.3.3 AccuracyThresholdLowerBounds . ... 85 5.3.3.1 The9-QubitBacon-ShorCode . 86 ix 5.3.3.2 The25-QubitBacon-ShorCode . 89 5.3.3.3 SummaryofResults........................... 91 5.3.4 UniversalQuantumComputation. .. 92 5.3.4.1 TheLogicalPhaseGate......................... 92 5.3.4.2 TheLogicalToffoliGate ........................ 95 5.3.4.3 Boundsonthe AccuracyofQuantumSoftware . 100 5.4 HistoryandAcknowledgements . ..... 103 6 Epilogue 104 A Deriving the Bacon-Shor code 105 A.1 StartingfromShor’sCode. .. .. .. .. .. .. .. .. .. .. .. .. .... 105 A.2 StartingfromBravyiandKitaev’sSurfaceCode . ........ 105 B Teleporting into Code Blocks
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