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Fault-Tolerance in Two-Dimensional Topological Systems Jonas Anderson University of New Mexico UNM Digital Repository Physics & Astronomy ETDs Electronic Theses and Dissertations 8-27-2012 Fault-tolerance in two-dimensional topological systems Jonas Anderson Follow this and additional works at: https://digitalrepository.unm.edu/phyc_etds Recommended Citation Anderson, Jonas. "Fault-tolerance in two-dimensional topological systems." (2012). https://digitalrepository.unm.edu/phyc_etds/4 This Dissertation is brought to you for free and open access by the Electronic Theses and Dissertations at UNM Digital Repository. It has been accepted for inclusion in Physics & Astronomy ETDs by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected]. Jonas Tyler Anderson Candidate Physics and Astronomy Department This dissertation is approved, and it is acceptable in quality and form for publication: Approved by the Dissertation Committee: Carlton Caves , Chairperson Andrew Landahl Ivan Deutsch Terry Loring Fault-tolerance in Two-dimensional Topological Systems by Jonas T. Anderson B.S., Physics, West Virginia University, 2006 B.E., Computer Engineering, West Virginia University, 2006 DISSERTATION Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Physics The University of New Mexico Albuquerque, New Mexico July, 2012 c 2012, Jonas T. Anderson iii Acknowledgments First, I would like to thank my parents Dusty and Renée, my grandparents, and my sister Halley for their love and support. Iwouldliketothankmyadvisor,AndrewLandahl,forposingsuchfunandinteresting research problems. I could not have imagined I would be working in such a fascinating area of science. To everyone at CQuIC, thanks for all the great discussions. I hope we stay in touch. Thanks to Alex Tacla, Carl Caves, and Vicky Bird for respectively proposing, buying, and maintaining the Jura espresso machine. Thanks to Ivan Deutsch and Carl Caves for being second advisors to me. Thanks to Chris Cesare and Andrew Landahl for all your help in developing the ideas presented here. Most of my ideas would remain scrawls in notebooks without you guys. Also, thanks to Rolando Somma and Robin Blume-Kohout for many great conversa- tions over coffee. These conversations were always thought-provoking even on the rare occasions when we talked about physics. Last, but certainly not least, thanks to Roya for her love and encouragement, and for too many other things to list here. And to our cats: Makhmal & Mnemosyne for being, well, cats. To the crags, canyons, and mountains of the Southwest and beyond thanks for the humility and wonderment—I will be back. IwassupportedinpartbytheNationalScienceFoundationthroughGrant0829944. IwassupportedinpartbytheLaboratoryDirectedResearchandDevelopmentprogram at Sandia National Laboratories. Sandia National Laboratories is a multi-program lab- oratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. iv Fault-tolerance in Two-dimensional Topological Systems by Jonas T. Anderson B.S., Physics, West Virginia University, 2006 B.E., Computer Engineering, West Virginia University, 2006 Ph.D., Physics, University of New Mexico, 2012 Abstract This thesis is a collection of ideas with the general goal of building, at least in the abstract, alocalfault-tolerantquantumcomputer. The connection between quantum information and topology has proven to be an active area of research in several fields. The introduction of the toric code by Alexei Kitaev demonstrated the usefulness of topology for quantum memory and quantum computation. Many quantum codes used for quantum memory are modeled by spin systems on a lattice, with operators that extract syndrome information placed on vertices or faces of the lattice. It is natural to wonder whether the useful codes in such systems can be classified. This thesis presents work that leverages ideas from topology and graph theory to explore the space of such codes. Homological stabilizer codes are introduced and it is shown that, under a set of reasonable assumptions, any qubit homological stabilizer code is equivalent to either a toric code or a color code. Additionally, the toric code and the color code correspond to distinct classes of graphs. v Many systems have been proposed as candidate quantum computers. It is very desir- able to design quantum computing architectures with two-dimensional layouts and low complexity in parity-checking circuitry. Kitaev’s surface codes provided the first example of codes satisfying this property. They provided a new route to fault tolerance with more modest overheads and thresholds approaching 1%. The recently discovered color codes share many properties with the surface codes, such as the ability to perform syndrome extraction locally in two dimensions. Some families of color codes admit a transversal implementation of the entire Clifford group. This work investigates color codes on the 4.8.8 lattice known as triangular codes. I develop a fault-tolerant error-correction strategy for these codes in which repeated syndrome measurements on this lattice generate a three-dimensional space-time combi- natorial structure. I then develop an integer program that analyzes this structure and determines the most likely set of errors consistent with the observed syndrome values. Iimplementthisintegerprogramtofindthethresholdfordepolarizingnoiseonsmall versions of these triangular codes. Because the threshold for magic-state distillation is likely to be higher than this value and because logical CNOT gates can be performed by code deformation in a single block instead of between pairs of blocks, the threshold for fault-tolerant quantum memory for these codes is also the threshold for fault-tolerant quantum computation with them. Since the advent of a threshold theorem for quantum computers much has been im- proved upon. Thresholds have increased, architectures have become more local, and gate sets have been simplified. The overhead for magic-state distillation has been studied, but not nearly to the extent of the aforementioned topics. A method for greatly reducing this overhead, known as reusable magic states, is studied here. While examples of reusable magic states exist for Clifford gates, I give strong reasons to believe they do not exist for non-Clifford gates. vi Contents List of Figures xiv 1Introduction 1 1.1 A brief history of error correction ...................... 1 1.2 Overview ................................... 5 2IntroductiontoQuantumErrorCorrection 7 2.1 Quantum Channels .............................. 8 2.2 Quantum Channels and Error Correction .................. 11 2.3 Commonly Studied Error Channels ..................... 13 2.4 The Pauli Group ............................... 15 2.5 The Clifford Group .............................. 16 2.6 Stabilizer Formalism ............................. 20 2.6.1 Stabilizer Update Rules ....................... 23 2.7 Stabilizer Codes ................................ 26 vii Contents 2.7.1 Pauli Group Alchemy ......................... 28 2.7.2 More on Logical Operators ...................... 30 2.7.3 CSS codes ............................... 31 2.7.4 Stabilizer Subsystem Codes ..................... 34 2.8 Fault-tolerance ................................ 35 2.8.1 The quest for an easy-to-implement fault-tolerant gate set. .... 38 2.8.2 Transversal gates and CSS codes .................. 40 2.8.3 The Magic Protocol .......................... 44 2.9 Universality .................................. 49 3IntroductiontoTopologicalCodes 53 3.1 Creating and Moving Defects ........................ 55 3.2 Braiding Defects ............................... 59 4HomologicalStabilizerCodes 65 4.1 Introduction .................................. 66 4.2 Graph Theory ................................. 68 4.3 Non-planar surfaces .............................. 74 4.4 Kitaev’s toric code .............................. 78 4.5 Color Codes .................................. 81 4.6 Homological Stabilizer Codes ........................ 82 viii Contents 4.7 Label Set Equivalence ............................ 88 4.8 4-colorable graphs ............................... 91 4.9 Optimal HSCs ................................. 94 4.10 Punctures ................................... 96 4.11 Boundaries .................................. 98 4.12 Conclusion ................................... 100 5 FTQC with the Topological Color Codes 102 5.1 Introduction .................................. 104 5.2 Noise and control model ........................... 108 5.3 Fault-tolerant error correction of color codes ................ 112 5.3.1 Code family .............................. 112 5.3.2 Syndrome extraction ......................... 115 5.3.3 Decoding algorithm .......................... 120 5.4 Numerical estimate of the accuracy threshold for fault-tolerant quantum error correction ................................ 128 5.4.1 Code capacity noise model ...................... 128 5.4.2 Phenomenological noise model .................... 133 5.4.3 Circuit-level noise model ....................... 138 5.5 Analytic bound on the accuracy threshold for fault-tolerant quantum error correction ................... 142 ix Contents 5.5.1 Code capacity noise model ...................... 143 5.5.2 Phenomenological noise model .................... 144 5.6 Fault-tolerant computation with color codes ................ 146 5.6.1 Fault-tolerance by transversal gates ................
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