Quantum Computation and Information Storage in Quantum Double Models Thesis by Anna Kómár In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2018 Defended May 3, 2018 ii © 2018 Anna Kómár ORCID: 0000-0002-4701-4931 All rights reserved iii ‘Would you tell me, please, which way I ought to go from here?’ ‘That depends a good deal on where you want to get to,’ said the Cat. ‘I don’t much care where–’ said Alice. ‘Then it doesn’t matter which way you go,’ said the Cat. ‘–so long as I get somewhere,’ Alice added as an explanation. ‘Oh, you’re sure to do that,’ said the Cat, ‘if only you walk long enough.’ Alice in Wonderland [1,2] Lewis Carroll iv ACKNOWLEDGEMENTS I would like to thank John Preskill, for taking the risk of becoming my advisor, and for guiding my academic growth throughout the years. I could always count on his physical intuition and patience during our discussions. I thank Olivier Landon- Cardinal for being my mentor, for supporting me, both academically and personally, throughout the years; and for putting up with my trivial questions about physics. I thank my thesis committee: Xie Chen, Manuel Endres, Alexei Kitaev, and John Preskill for their valuable comments about the first version of this thesis, especially those regarding Chapter4. I would like to thank my collaborators and colleagues: Kristan Temme, for helping me take my first steps as a researcher in the field; Elizabeth Crosson, for helping me appreciate Markov chains more; Sujeet Shukla, Burak Sahinoglu and Dave Aasen, for their time and patience while discussing quantum doubles and ribbon operators with me; Aleksander Kubica, for always being there when I needed advice. I thank Alexei Kitaev for his time he spent discussing quantum doubles with me; and David Poulin and Anirudh Krishna for making me feel welcome in Sherbrooke during my visit. Spending five years at Caltech made me appreciate the importance of personal relationships. I would like to thank all my Caltech friends for being there during these five years, for throwing board game nights, coming to dinners, and playing Dungeons & Dragons together. You’re all awesome. Special thanks is due to Ioana Craiciu, my mentee then friend, with whom I could discuss topics encompassing quantum physics, philosophy, travel, arts and crafts, and so much more. To my other best friend, Lena Murchikova: you’re the strongest person I know, and I feel honored to be your friend. I wouldn’t be here without the two people who helped me start my career in science, my former advisors: Gergő Pokol and Tünde Fülöp. Thank you for helping me realize that doing research is fun, and for supporting me even after I left the field. Last, but not least, I thank my family for their continuing support, in particular my parents, Antal and Erzsébet, for raising me to be a curious, skeptical, hard-working adult, for teaching me to never give up. I thank you both for encouraging and consoling me throughout all these years, and for accepting that I decided to move thousands of miles away from home. I thank my brother, Péter, for being a friend, a role model, someone who’s always there to help. Thank you for being a caring big brother. Finally, I would like to thank my husband, Dávid, for being there with me along the road. I thank you for listening to my ramblings about quantum physics, for celebrating milestones with me, and for picking up my pieces when needed. You have been my anchor throughout these years. v ABSTRACT The results of this thesis concern the real-world realization of quantum computers, specifically how to build their “hard drives” or quantum memories. These are many- body quantum systems, and their building blocks are qubits, the same way bits are the building blocks of classical computers. Quantum memories need to be robust against thermal noise, noise that would other- wise destroy the encoded information, similar to how strong magnetic field corrupts data classically stored in magnetic many-body systems (e.g., in hard drives). In this work I focus on a subset of many-body models, called quantum doubles, which, in addition to storing the information, could be used to perform the steps of the quantum computation, i.e., work as a “quantum processor”. In the first part of my thesis, I investigate how long a subset of quantum doubles (qudit surface codes) can retain the quantum information stored in them, referred to as their memory time. I prove an upper bound for this memory time, restricting the maximum possible performance of qudit surface codes. Then, I analyze the structure of quantum doubles, and find two interesting properties. First, that the high-level description of doubles, utilizing only their quasi-particles to describe their states, disregards key components of their microscopic proper- ties. In short, quasi-particles (anyons) of quantum doubles are not in a one-to-one correspondence with the energy eigenstates of their Hamiltonian. Second, by in- vestigating phase transitions of a simple quantum double, D¹S3º, I map its phase diagram, and interpret the physical processes the theory undergoes through terms borrowed from the Landau theory of phase transitions. vi PUBLISHED CONTENT AND CONTRIBUTIONS Anna Kómár, Olivier Landon-Cardinal, and Kristan Temme. Necessity of an energy barrier for self-correction of Abelian quantum doubles. Physical Review A, 93 (5):052337, 2016. doi: 10.1103/PhysRevA.93.052337. The project idea was conceived during discussions between AK, OLC and KT, as a non-trivial extension of the pioneering work done by KT [3]. AK derived the main result (upper bound on memory time), and proved that it is a constant. The value of the prefactor of the bound was derived and proven collaboratively by AK and OLC. The writing of the manuscript was led by AK, and collaboratively done by all authors. Anna Kómár, and Olivier Landon-Cardinal. Anyons are not energy eigenspaces of quantum double Hamiltonians. Physical Review B, 96(19):195150, 2017. doi: 10.1103/PhysRevB.96.195150. The project idea was conceived during discussions between AK and OLC. AK derived the main results (construction of projectors, including their commuta- tive and orthogonality properties). The proofs and interpretations of the results were finalized during discussions between AK and OLC. The writing of the manuscript was led by AK, and collaboratively done by the authors. Anna Kómár, and Olivier Landon-Cardinal. Topological phase diagram of D¹S3º induced by forbidding charges and fluxes. arXiv preprint: 1805.00032. The project idea was conceived during discussions between AK and OLC. AK derived the main results (what new phases emerge in each case of phase transition), and categorized the intermediate processes during these phase transitions. The writing of the manuscript was led by AK, and collaboratively done by the authors. vii TABLE OF CONTENTS Acknowledgements............................... iv Abstract..................................... v Published Content and Contributions...................... vi Table of Contents................................ vii Nomenclature.................................. x Chapter I: Overview............................... 1 1.1 Topological Quantum Computation.................. 3 1.1.1 Topological order....................... 4 1.1.2 Anyons ............................ 4 1.1.3 Error Correction ....................... 6 1.1.4 Toric code........................... 8 1.2 Summary of results .......................... 11 1.2.1 Memory time of Abelian quantum doubles.......... 12 1.2.2 Computation with non-Abelian quantum doubles . 13 Chapter II: Self correction requires an energy barrier for Abelian quantum doubles ................................... 17 2.1 Introduction .............................. 17 2.2 Framework............................... 20 2.2.1 Abelian quantum doubles................... 20 2.2.2 Thermal noise model..................... 24 2.3 Generalized energy barrier....................... 26 2.3.1 Definition of the generalized energy barrier ......... 26 2.3.2 Arrhenius upper bound on the mixing time.......... 28 2.3.3 Generalized energy barrier is constant for Abelian doubles . 29 2.3.4 Length of the local errors path................ 33 2.3.5 The effect of defect lines ................... 36 2.4 Discussion............................... 39 2.4.1 Possible improvements.................... 39 2.4.2 Implication for entropy protection .............. 39 2.5 Details of the derivation........................ 41 2.5.1 Diagonalizing the Hamiltonian, Jump operators . 42 2.5.2 Construction of the Dirichlet matrix and the variance matrix 45 2.5.3 Bounds on the gap (comparison theorem and canonical paths) 48 2.5.4 Evaluation of the bound and the generalized energy barrier . 50 2.6 Conclusions .............................. 54 2.A Maximum cardinality and sum of multisets without a zero-sum subset 54 Chapter III: Anyons are not energy eigenspaces of quantum double Hamiltonians 58 3.1 Introduction .............................. 58 3.2 The Drinfeld double construction and the quantum double models . 60 viii 3.2.1 Non-Abelian Aharonov-Bohm effect............. 61 3.2.2 Kitaev’s quantum double on a lattice............. 66 3.3 Refined quantum double Hamiltonian for arbitrary group . 71 3.3.1 Refined quantum double construction ............ 71 3.3.2 Example of G = S3 ...................... 77 3.4 Hilbert space splitting......................... 78 3.4.1 Two distinct yet consistent ways to split the Hilbert space . 78 3.4.2 Dimension of the proper Hilbert space of a site . 80 3.4.3 Diagrammatic representation and energy sectors . 81 3.5 Local degrees of freedom ....................... 85 3.5.1 Disagreement between anyons and energy sectors . 85 3.5.2 The role of finite lattice spacing ............... 86 3.5.3 Local vs. global degrees of freedom............. 88 3.6 Discussion............................... 89 3.6.1 Consequences for quantum computation........... 89 3.6.2 Consequences for quantum memories ............ 90 3.6.3 Holography between local, topological, and fusion degrees of freedom?.......................... 90 3.A Mathematical proofs.........................