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Topological Quantum Compiling Layla Hormozi Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2007 Topological Quantum Compiling Layla Hormozi Follow this and additional works at the FSU Digital Library. For more information, please contact [email protected] THE FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES TOPOLOGICAL QUANTUM COMPILING By LAYLA HORMOZI A Dissertation submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy Degree Awarded: Fall Semester, 2007 The members of the Committee approve the Dissertation of Layla Hormozi defended on September 20, 2007. Nicholas E. Bonesteel Professor Directing Dissertation Philip L. Bowers Outside Committee Member Jorge Piekarewicz Committee Member Peng Xiong Committee Member Kun Yang Committee Member Approved: Mark A. Riley , Chair Department of Physics Joseph Travis , Dean, College of Arts and Sciences The Office of Graduate Studies has verified and approved the above named committee members. ii ACKNOWLEDGEMENTS To my advisor, Nick Bonesteel, I am indebted at many levels. I should first thank him for introducing me to the idea of topological quantum computing, for providing me with the opportunity to work on the problems that are addressed in this thesis, and for spending an infinite amount of time helping me toddle along, every step of the process, from the very beginning up until the completion of this thesis. I should also thank him for his uniquely caring attitude, for his generous support throughout the years, and for his patience and understanding for an often-recalcitrant graduate student. Thank you Nick — I truly appreciate all that you have done for me. Next, I should thank Dimitrije Stepanenko for all his help, for spending many valuable hours answering my questions about quantum computing, and for making me feel at home when I first joined the group. Many thanks to Steve Simon for generously sharing his advice, his ideas, and his codes, and for our close collaboration during the past three years. I owe much of my understanding of the non-Abelian aspects of the fractional quantum Hall states to Steve, and for that, I am most grateful. I should also thank Georgios Zikos for actually finding the many braid sequences that are presented in this thesis. Thanks to my professors, Kun Yang, Pedro Schlottmann, Laura Reina, Jorge Piekarewicz, Vladimir Dobrosavljevic, and Elbio Dagotto for all that they have taught me. Thanks also to Professors Peng Xiong, Oskar Vafek, Mark Riley, Per Rikvold, Dragana Popovic, Steve von Molnar, Jan Jaroszynski, Lloyd Engle, James Brooks, and Luis Balicas for their supportive attitude and their willingness to share their good advice. Thanks to the folks at the computer support group, especially Tom Combs and Jim Berhalter, for their much-needed help over the years. Thanks also to Alice Hobbs, Andrea Durham, and Arshad Javed at the Magnet Lab, and Sherry Tointigh at the Physics Department for being so helpful, and for wonderfully taking care of all my paperwork during the past several years. iii Many thanks to my friends at the Magnet Lab, in particular, Gonzalo Alvarez, Yong Chen, Qinghong Cui, Maxim Dzero, John Janik, Mohammad Moraghebi, Ivana Raicevic, and especially Matthew Case. I should also thank all the students and postdocs in Prof. Brooks’ group for bearing with the impurity sitting in their area of the lab for the past couple of years. Thanks also to Jamaa Bouhattate, Fernando Cordero, Amin Dezfuli, Parisa Mahjour, Callie Maidhof, Asal Mohammadi, Mahtab Munshi, Afi Sachi-Kocher, Jelena Trbovic, and many others, for their friendship during my time in grad school. I should also thank Fatemeh Khalili-Araghi and Guiti Zolfagharkhani, with whom I started this journey. Their friendship and support, through our frequent phone calls and visits over the years, has been a constant motivating force. Thanks also to Nazly Emadi, Zahra Fakhraai, Akbar Jaefari, Thalat Monajemi, Tahereh Mokhtari, Saman Rahimian, Azadeh Tadjpour, and Shadi Tamadon for their long-time friendship and their positive influence. This list would definitely be incomplete without thanking my wonderful roommates, Rhia Obrecht and Mahsa Saedirad, for being so much fun, and for their supportive attitude, especially at times that I needed them the most. I should especially thank Rhia: without her friendship, I wouldn’t have survived the first years in Tallahassee. Finally, thanks to my parents and my sister, for their endless love and support, for their wonderful sense of humor, and for their constant attention and their efforts to cheer me up whenever things were gloomy. I love you so much! The research presented in this thesis has been supported by US Department of Energy through Grant No. DE-FG02-97ER45639. iv TABLE OF CONTENTS List of Tables ...................................... vi List of Figures ..................................... vii Abstract ........................................ xvii 1. Introduction .................................... 1 1.1 Quantum Computing Basics ......................... 2 1.2 Non-Abelian Phases of Matter ........................ 10 1.3 The Quantum Hall Effect .......................... 17 1.4 Quantum Computing with FQH States ................... 37 1.5 Outline of The Thesis ............................ 38 2. Compiling Braids for Fibonacci Anyons ..................... 40 2.1 SU(2)k Particles: Fusion Rules and Hilbert Space ............. 41 2.2 SU(2)3 and Fibonacci Anyons ....................... 43 2.3 Fibonacci Anyon Basics ........................... 45 2.4 Qubit Encoding and General Computation Scheme ............ 48 2.5 Compiling Three-Braids and Single-Qubit Gates .............. 52 2.6 Two-Qubit Gates ............................... 62 2.7 What’s Special about k = 3? ......................... 82 2.8 Summary ................................... 87 3. Compiling Braids for SU(2)k Anyons ....................... 89 3.1 SU(2)k Revisited ............................... 90 3.2 Encoding Qubits and Single-Qubit Gates .................. 94 3.3 Two-Qubit Gates ............................... 98 3.4 Summary ................................... 114 APPENDIX ...................................... 115 A. The Pentagon Equation .............................. 115 REFERENCES ..................................... 119 BIOGRAPHICAL SKETCH ............................. 124 v LIST OF TABLES 1.1 Ground state wavefunctions, the corresponding potential, the thin cylinder limit, ground state degeneracy on a cylinder and the corresponding filling frac- tion. Superscripts B and F correspond to bosons and fermions respectively. L, MR and RRk correspond to Laughlin, Moore-Read and Read-Rezayi at level k, respectively. ................................ 36 1.2 Wavefunctions in the presence of quasiholes, the thin cylinder limit and the charge of the corresponding elementary excitations. Superscripts B and F correspond to bosons and fermions respectively. L, MR and RRk correspond to Laughlin, Moore-Read and Read-Rezayi at level k, respectively. ...... 36 2.1 Values of b′ for different values of a and b after applying the F weave as shown in Fig. 2.23, and the phase applied to the resulting state by a phase weave with zero winding. The value of b′ is determined by the fact that b′ = 1 when a = 0 and b′ = b when a = 1, as shown in the text. ............... 79 3.1 Values of d′ as a function of a and b. ...................... 101 vi LIST OF FIGURES 1.1 Representation of a qubit in the Bloch sphere. 0 and 1 (with unit length) which correspond to classical states of a qubit, provide| i a| basisi for an arbitrary state of a qubit, ψ to be expressed in terms of the two angles, θ and φ. .. 5 | i 1.2 Single-qubit gates as rotations. Single-qubit gates, U~α, can be represented by a vector, ~α, in a solid sphere of radius 2π. the direction of ~α represents the axis of rotation, and its magnitude determines the rotation angle. ...... 6 1.3 A universal set of quantum gates. Any quantum computation can be carried out by applying gates from a universal set, consisting of single-qubit gates, U, and a controlled-NOT gate. ........................... 7 1.4 The Loss-DiVincenzo model of a quantum computer. It consists of an array of quantum dots (dashed ovals) each with one electron trapped in it. Spin of each trapped electron plays the role of a qubit. ................. 9 1.5 Statistics of identical particles in two versus three dimensional space. (a) Taking one particle around another, in two dimensions, cannot be smoothly deformed into the identity, therefore the corresponding unitary operation is non-trivial. (b)-(d) Taking one particle around another, in three dimensions can be smoothly deformed into the identity, shown in (d) without one particle having to cut through the trajectory of the other particle. Therefore the corresponding unitary operation is trivial. ................... 12 1.6 Exchanging particles in 2+1 dimensions. (a) Exchanging particles in a two-dimensional space corresponds to braiding their world-lines in three- dimensional space-time. (b) Exchanging particles in a clockwise manner leads to a braid which is topologically distinct from the braid resulting from exchanging particles in a counterclockwise manner. .............. 14 vii 1.7 The Braid group. The braid group, Bn, can be generated using a set of braid generators σ1,σ2, ... σn−1, acting on n strands (a). (b) σi corresponds to exchanging strands i and i+1 in a clockwise manner while a counterclockwise −1 exchange corresponds to σi . (c) An example of a group element.
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