Florida State University Libraries 2015 Non-Abelian Quantum Error Correction Weibo Feng Follow this and additional works at the FSU Digital Library. For more information, please contact [email protected] FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES NON-ABELIAN QUANTUM ERROR CORRECTION By WEIBO FENG A Dissertation submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2015 Copyright c 2015 Weibo Feng. All Rights Reserved. Weibo Feng defended this dissertation on August 31, 2015. The members of the supervisory committee were: Nicholas E Bonesteel Professor Directing Dissertation Philip L Bowers University Representative Jorge Piekarewicz Committee Member Kun Yang Committee Member Peng Xiong Committee Member The Graduate School has verified and approved the above-named committee members, and certifies that the dissertation has been approved in accordance with university requirements. ii ACKNOWLEDGMENTS Here I present my deepest gratitude to my advisor, Nick Bonesteel, without whom the work can never easily be done. Prof. Bonesteel has been constantly supportive of my study here at the Maglab from the first day I joint it. I thank him not only for introducing me to the field of topological quantum computation, but also for his patient guidance and motivation through my research in this field. I shall be forever benefited from what I have learnt from him during this precious period of time. Also to the committee members of my defense, Prof. Philip Bowers, Prof. Takemichi Okui, Prof. Jorge Piekarewicz, Prof. Oskar Vafek, Prof. Pen Xiong and Prof. Kun Yang, I thank them for their challenging questions and helpful comments which have inspired me greatly both in my Prospectus and final Dissertation. I would like to thank my colleagues here in the Maglab, Daniel Zeuch and Julia Wildeboer for their continuous support and feedback throughout the writing and defending of my Dissertation. I thank Julia for the encouragement she gave to me academic-wise and nonacademic-wise. The cakes and cookies she brought to me when I was striving hard to finish my Dissertation were the best! My sincere thanks also go to my good friends in Tally: the list of which can be a really long one and I want to thank them for all the fun we have experienced together here in this lovely town. Our friendship has always been the greatest comfort to me when I was in my hardest time. Finally, I want to thank my parents and my sister. In this past several years their love and support to me are the anchor of my life here in a foreign land. They are the reasons why I never give up on this long life voyage. May my work here and what I shall achieve in the future repay them well. iii TABLE OF CONTENTS ListofFigures ......................................... ... vi Abstract............................................. xiii 1 Introduction 1 1.1 Classical and Quantum Computation . 1 1.1.1 FromBitstoQubits ................................ 2 1.1.2 QuantumGates................................... 4 1.1.3 Decoherence and Quantum Error Correction . 7 1.2 Topological Quantum Computation . 9 1.3 Non-Abelian Quantum Error Correction . 11 1.4 BriefOverviewofDissertation. 12 2 Fibonacci Anyons And Levin-Wen Models 13 2.1 Fibonacci Anyons . 13 2.1.1 Basic Algebraic Theory of Fibonacci Anyons . 16 2.1.2 Pentagon and Hexagon Equations . 21 2.1.3 Quantum Computing by Braiding Fibonacci Anyons . 29 2.2 FibonacciLevin-WenModels . 33 2.3 Summary .......................................... 44 3 Non-Abelian Quantum Error Correction 45 3.1 Previous Work on Measuring Qv and Bp ......................... 45 3.1.1 VertexErrors.................................... 47 3.1.2 Plaquette Errors Measured by Plaquette Reduction . 49 3.2 New Method for Measuring and Correcting Vertex and Plaquette Errors . 60 3.2.1 VertexErrorsonStringEnds . 61 3.2.2 Plaquette Swapping . 65 3.2.3 Plaquette Swapping: Abelian Case . 73 3.3 MovingErrors ...................................... 75 3.3.1 MovingVertexErrors ............................... 76 3.3.2 PlaquetteErrorsRevisited . 79 3.3.3 MovingPlaquetteErrors. .. 81 3.4 FusingErrors ...................................... 85 3.5 SummaryofResults ................................... 87 4 Summary of Dissertation and Future Work 89 iv Appendices A Quantum Circuits to Verify the Pentagon Equation 92 B Quantum Circuits to Verify the Hexagon Equation 94 Bibliography .......................................... 96 BiographicalSketch ..................................... 99 v LIST OF FIGURES 1.1 A classical and quantum bit. (Right) A classical bit can be represented by an arrow which points up if it is in the state 0 or down if it is the state 1. (Left) A qubit can be represented by an arrow which points to a point on the Bloch sphere. Like a classical bit, when the arrow is pointing up or down along the z axis the qubit is in the state 0 or 1 , respectively. However, unlike the classical bit, the qubit can point in an | i | i arbitrary direction, corresponding to a quantum superposition of the states 0 and 1 . 3 | i | i 1.2 One and two qubit quantum gates shown in quantum circuit notation. (a) A single- qubit gate. This gate which applies the unitary operation U to a qubit in the state ψ . The horizontal line represents the qubit with time flowing from left to right. (b) | i A controlled-U two-qubit quantum gate. This gate applies a unitary operation to the bottom qubit in the state ψ if the control qubit is in the state 1 , but otherwise | i | i doesnothing.......................................... 5 1.3 Universal gate set and example of a quantum circuit. (a) The set of all single qubit operations together with the ability to carry out controlled-NOT gates is a universal set of gates, meaning that any unitary operation acting on many qubits can be carried out using them. (b) Example of a quantum circuit acting on 7 qubits. 6 1.4 Toffoli-class quantum gates. (a) A controlled-controlled-U three-qubit gate. This gate applies the operation U to the bottom qubit in the state ψ if and only if the top two | i qubits are both in the state 1 , otherwise it does nothing. (b) A controlled-controlled- | i NOT gate which is also known as a Toffoli gate. (c) A controlled-controlled-controlled- U gate. Such multiqubit controlled gates play an important role in the quantum circuits developed in Chapter 3 of this Dissertation. 8 2.1 A recursive calculation shows that the total quantum dimension of n Fibonacci anyons fusing together follows the Fibonacci series. The first term on the right hand side describes the case when the first two particles fused into a total charge 1, which turns out to be Nn−1. The second term shows when the fused charge of the first two is 0, in which case the number fusion channels is Nn−2 .................... 15 2.2 Two different ways of combining a group of 3 anyons b, a and d are connected by a unitary operation, i.e. the F -tensor. The oval diagram on the left shows how we choose to combine different pairs of anyons( b,a or a,d ). The world line diagram { } { } on the right tells the same story with time flows from top to bottom. 17 2.3 The R-move switches two anyons before combining into one. The difference between ab the original and the rotated wave functions is the so called R-matrix Rc . 18 2.4 The tube version of the R-move. The space-time line of each anyon is represented by a elastic tube that can track how exactly the line is twisted. The anyon pair a and b then undergoes a counter-clockwise exchange. After that we proceed to ”tighten” the vi diagram by pulling the tubes away from each other, which results in a clockwise twist for both of the anyon a and b. Note that the anyon c which represents their total charge is still not rotated yet. To finish the exchange we have to continue to untwist the tube c, giving us a counter-clockwise π rotation. 19 2.5 The topological spin of a Fibonacci anyon a, which is unique to the 2+1 dimensional phases of matter. Rotating the quasi-particle in 2π is equivalent to a full twist on its tube version, which results in an overall phase θa. .................... 21 2.6 The pentagon equation which tells the story of two equivalent path of basis changing. Starting from the basis to the very left, one can end at the basis to the very right either through two F -moves(the upper path), of three F -moves(the lower path), putting together to be a pentagon diagram. The diagram is also isotropic: meaning that every basis can be the starting point or the end point. There are always two equivalent path connecting in between — one only needs to change the direction of the F -moves accordingly........................................... 22 2.7 The world line version of the pentagon equation. Time flows from top to bottom. Each line segment represents an anyon before it merges with another one to create a new anyon that has their total charge. 23 2.8 A deformed version of the F -move equation, which better demonstrates the symme- tries underneath its structure. The dashed red lines mark the symmetric lines of the equation. ........................................... 24 2.9 The F -move for Fibonacci anyons in its every component. For each fusion pattern(tree diagram) the black thick line represents a particle of charge 1, while the light thin line marks a charge 0 anyon. (a)&(b) are the cases where the target particle e is solely | i determined by the fusion rule Eq. 2.1. (c) shows the only two non-trivial F -moves here, which needs to be solved in further discussions.