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UC Santa Barbara UC Santa Barbara Electronic Theses and Dissertations Title Topological Quantum Computing with Majorana Zero Modes and Beyond Permalink https://escholarship.org/uc/item/04305656 Author Knapp, Christina Publication Date 2019 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY of CALIFORNIA Santa Barbara Topological Quantum Computing with Majorana Zero Modes and Beyond A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics by Christina Paulsen Knapp Committee in charge: Professor Chetan Nayak, Chair Professor Leon Balents Professor Andrea F. Young June 2019 The dissertation of Christina Paulsen Knapp is approved: Professor Leon Balents Professor Andrea F. Young Professor Chetan Nayak, Chair June 2019 Copyright ⃝c 2019 by Christina Paulsen Knapp iii Of course it is happening in your head, Harry, but why on earth should that mean it is not real? -J.K. Rowling, Harry Potter and the Deathly Hallows To my grandparents, John and Betsy Tower, and Harold and Barbara Knapp, for instilling in me the value of learning. iv Acknowledgements I am immensely grateful to many people for making graduate school a successful and en- joyable experience. First, I would like to thank my advisor Chetan Nayak for introducing me to the field of topological phases and for his patience and guidance throughout my Ph.D. Chetan’s balance of providing direction when necessary and encouraging independence when possible has been essential for my growth as a physicist. His good humor and breadth of expertise has fostered a particularly fun and vibrant group that I feel incredibly fortunate to have joined. Next, I would like to thank Parsa Bonderson, who has in many ways acted as my unofficial co-advisor. In addition to leading me deeper into diagrammatic calculus than I ever anticipated I would dive, Parsa’s meticulous revisions of paper drafts and practice talks have held me to a higher standard than I would have otherwise achieved. I am indebted to him for hours of career advice, scientific guidance, and social commentary. I am extremely grateful to my committee members Leon Balents and Andrea Young for their time and advice. Leon’s group meetings have provided me a glimpse into the greater world of topological physics. Andrea graciously welcomed me onto a project that introduced me to the wizardry of graphene experiments and significantly broadened my research directions. I have been extraordinarily fortunate to work with and learn from numerous collaborators. Roman Lutchyn has advised me on several projects, extensively improved my familiarity with field theory, and been a driving force behind my participation in Majorana-based research. Torsten Karzig has been an MVP coauthor, providing lively discussions and reinvigorating my interest in projects at their dullest moments. Dima Pikulin’s optimistic demeanor and grasp of mesoscopic physics, combined with Michael Beverland’s fault tolerance perspective led to a particularly enjoyable collaboration. My foray into fractional Chern insulator physics would not have been possible without Mike Zaletel’s expertise and enthusiasm; his mentorship let to a fascinating project with many interesting future directions. Discussions with Eric Spanton have sharpened my understanding of “experimental accessibility” and improved my intuition for a new system. Meng Cheng and Dong Liu were extremely helpful, and patient, collaborators on my first project. Kaushal Patel endured my frenzied calculations of anyonic entanglement entropies for years and was a calm check on my descent into diagrammatic madness. I continue to learn and benefit from working with Jukka Vayrynen and Dave Aasen. My graduate school experience has been shaped by the past and present Station Q com- munity. My unofficial office in the seminar room has led to many helpful and interesting discussions with Zhenghan Wang, Bela Bauer, Michael Freedman, Chao-Ming Jian, Maissam Barkeshli, David Clarke, Andrey Antipov, and Anna Keselman. I am particularly grateful for Jen Cano’s mentorship, and to all the Q and UCSB graduate students for helpful discussions and a collaborative environment. Every day in Elings has been brightened by Sean Fraer’s friendly greeting and organizational mastery. Finally, I have been supported throughout the last six years and my entire life by an amazing network of friends and family. Their love, encouragement, and much-needed distractions have made me a healthier, happier, and better person. v Curriculum Vitæ Christina Paulsen Knapp Education 2019 Ph.D. in Physics (expected), University of California, Santa Barbara Advisor: Chetan Nayak 2016 M.A. in Physics, University of California, Santa Barbara Advisor: Chetan Nayak 2013 B.A. in Physics (highest honors) & Math, Williams College, Williamstown, MA Advisor: William K. Wootters Awards 2016-2018 NSF Graduate Research Fellow 2016-2019 Outstanding Service to the Department, UCSB Physics 2014 Oustanding Teaching Assistant, UCSB Physics 2013 Ferrando-Fithian Award, UCSB Physics 2013 Phi Beta Kappa and Sigma Xi Honor Societies Publications “Fractional Chern insulator edge states and layer-resolved lattice contacts,” Christina Knapp, Eric M. Spanton, Andrea F. Young, Chetan Nayak, and Michael P. Zaletel. Physical Review B 99, 081114 (2019), arXiv:1810.02325. “Modeling noise and error correction for Majorana-based quantum comput- ing,” Christina Knapp, Michael Beverland, Dmitry I. Pikulin, and Torsten Karzig. Quantum 2, 88 (2018), arXiv:1806.01275. “Dephasing of Majorana-based qubits,” Christina Knapp, Torsten Karzig, Roman Lutchyn, and Chetan Nayak. Physical Review B 97, 125404 (2018), arXiv:1711.03968. vi “Anyonic Entanglement and Topological Entanglement Entropy,” Parsa Bon- derson, Christina Knapp, and Kaushal Patel. Annals of Physics 385 (2017), arXiv:1706.09420. “Scalable Designs for Quasiparticle-Poisoning-Protected Topological Quan- tum Computation with Majorana Zero Modes,” Torsten Karzig, Christina Knapp, Roman Lutchyn, Parsa Bonderson, Matthew Hastings, Chetan Nayak, Jason Alicea, Karsten Flensberg, Stephan Plugge, Yuval Oreg, Charles Mar- cus, and Michael Freedman. Physical Review B 95, 235305 (2017), arXiv:1610.05289. “Nature and Correction of Diabatic Errors in Anyon Braiding,” Christina Knapp, Michael P. Zaletel, Dong E. Liu, Meng Cheng, Parsa Bonderson, and Chetan Nayak. Physical Review X 6, 041003 (2016), arXiv:1601.05790. vii Abstract Topological Quantum Computing with Majorana Zero Modes and Beyond by Christina Paulsen Knapp Topological quantum computing seeks to store and manipulate information in a protected manner using topological phases of matter. Information encoded in the degenerate state space of pairs of non-Abelian anyons or defects is robust to local perturbations, reducing its sus- ceptiblity to environmental errors and potentially providing a scalable approach to quantum computing. However, topological quantum computing faces significant challenges, not least of which is identifying an experimentally accessible platform supporting non-Abelian topologi- cal physics. In this thesis, we critically analyze topological quantum computing with Majorana zero modes, non-Abelian defects of a topological superconductor. We identify intrinsic er- ror sources for Majorana-based systems and propose quantum computing architectures that minimize their effects. Additionally, we consider a new approach for realizing and detecting non-Abelian topological defects in fractional Chern insulators. Topological quantum computing is predicated on the idea that braiding non-Abelian anyons adiabatically can implement quantum gates fault tolerantly. However, any braiding experiment will necessarily depart from the strict adiabatic limit. We begin by analyzing the nature of diabatic errors for anyon braiding, paying particular attention to how such errors scale with viii braiding time. We find that diabatic errors are unfavorably large and worryingly sensitive to details of the time evolution. We present a measurement-based correction protocol for such errors, and illustrate its application in a particular Majorana-based qubit design. We next propose designs for Majorana-based qubits operated entirely by a measurement- based protocol, thereby avoiding the diabatic errors discussed above. Our designs can be scaled into large two dimensional arrays amenable to long-term quantum computing goals, whose core components are testable in near-term devices. These qubits are robust to quasiparticle poison- ing, anticipated to be one of the dominant error sources coupling to Majorana zero modes. We demonstrate that our designs support topologically protected Clifford operations and can be augmented to a universal gate set without requiring additional control parameters. While topological protection greatly suppresses errors, residual coupling to noise limits the lifetimes of our proposed Majorana-based qubits. We analyze the dephasing times for our quasiparticle-poisoning-protected qubits by calculating their charge distribution using a particle number-conserving formalism. We find that fluctuations in the electromagnetic en- vironment couple to an exponentially suppressed topological dipole moment. We estimate dephasing times due to 1=f noise, thermal quasiparticle excitations, and phonons for different qubit sizes. The residual errors discussed above will necessarily require error correction for a suffi- ciently long quantum computation. We develop physically
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