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Non-Abelian Anyons and Topological Quantum Computation

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Non-Abelian Anyons and Topological Quantum Computation Non-Abelian Anyons and Topological Quantum Computation Chetan Nayak1,2, Steven H. Simon3, Ady Stern4, Michael Freedman1, Sankar Das Sarma5, 1Microsoft Station Q, University of California, Santa Barbara, CA 93108 2Department of Physics and Astronomy, University of California, Los Angeles, CA 90095-1547 3Alcatel-Lucent, Bell Labs, 600 Mountain Avenue, Murray Hill, New Jersey 07974 4Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel 5Department of Physics, University of Maryland, College Park, MD 20742 Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a fault-tolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasi- particle excitations are neither bosons nor fermions, but are particles known as Non-Abelian anyons, meaning that they obey non-Abelian braiding statistics. Quantum information is stored in states with multiple quasiparticles, which have a topological degeneracy. The unitary gate operations which are necessary for quantum computation are carried out by braiding quasiparticles, and then measuring the multi-quasiparticle states. The fault-tolerance of a topological quantum computer arises from the non-local encoding of the states of the quasiparticles, which makes them immune to errors caused by local perturbations. To date, the only such topological states thought to have been found in nature are fractional quantum Hall states, most prominently the ν = 5/2 state, although several other prospective candidates have been proposed in systems as disparate as ultra-cold atoms in optical lattices and thin film superconductors. In this review article, we describe current research in this field, focusing on the general theoretical concepts of non-Abelian statistics as it relates to topological quantum computation, on understanding non-Abelian quantum Hall states, on proposed experiments to detect non-Abelian anyons, and on proposed architectures for a topological quantum computer. We address both the mathematical underpinnings of topological quantum computation and the physics of the subject using the ν = 5/2 fractional quantum Hall state as the archetype of a non-Abelian topological state enabling fault-tolerant quantum computation. Contents A. ν = 5/2 Qubits and Gates 52 B. Fibonacci Anyons: a Simple Example which is Universal for I. Introduction 1 Quantum Computation 53 C. Universal Topological Quantum Computation 57 II. Basic Concepts 3 D. Errors 59 A. Non-Abelian Anyons 3 1. Non-Abelian Braiding Statistics 3 V. Future Challenges for Theory and Experiment 60 2. Emergent Anyons 6 B. Topological Quantum Computation 7 Acknowledgments 64 1. Basics of Quantum Computation 7 2. Fault-Tolerance from Non-Abelian Anyons 9 A. Conformal Field Theory (CFT) for Pedestrians 64 C. Non-Abelian Quantum Hall States 11 1. Rapid Review of Quantum Hall Physics 11 References 65 2. Possible Non-Abelian States 13 3. Interference Experiments 16 4. A Fractional Quantum Hall Quantum Computer 18 I. INTRODUCTION 5. Physical Systems and Materials Considerations 19 D. Other Proposed Non-Abelian Systems 20 In recent years, physicists’ understanding of the quantum III. Topological Phases of Matter and Non-Abelian Anyons 22 properties of matter has undergone a major revolution pre- A. Topological Phases of Matter 23 cipitated by surprising experimental discoveries and profound 1. Chern-Simons Theory 23 2. TQFTs and Quasiparticle Properties 26 theoretical revelations. Landmarks include the discoveries of B. Superconductors with p + ip pairing symmetry 29 the fractional quantum Hall effect and high-temperature su- 1. Vortices and Fermion Zero Modes 29 perconductivity and the advent of topological quantum field 2. Topological Properties of p + ip Superconductors 31 theories. At the same time, new potential applications for C. Chern-Simons Effective Field Theories, the Jones Polynomial, quantum matter burst on the scene, punctuated by the discov- and Non-Abelian Topological Phases 33 1. Chern-Simons Theory and Link Invariants 33 eries of Shor’s factorization algorithm and quantum error cor- 2. Combinatorial Evaluation of Link Invariants and rection protocols. Remarkably, there has been a convergence Quasiparticle Properties 35 between these developments. Nowhere is this more dramatic D. Chern-Simons Theory, Conformal Field Theory, and Fractional than in topological quantum computation, which seeks to ex- Quantum Hall States 37 1. The Relation between Chern-Simons Theory and Conformal ploit the emergent properties of many-particle systems to en- Field Theory 37 code and manipulate quantum information in a manner which 2. Quantum Hall Wavefunctions from Conformal Field Theory 39 is resistant to error. E. Edge Excitations 43 It is rare for a new scientific paradigm, with its attendant F. Interferometry with Anyons 46 concepts and mathematical formalism, to develop in parallel G. Lattice Models with P, T -Invariant Topological Phases 49 with potential applications, with all of their detailed technical IV. Quantum Computing with Anyons 52 issues. However, the physics of topological phases of matter 2 is not only evolving alongside topological quantum computa- ical system ‘condenses’ into a non-Abelian topological phase. tion but is even informed by it. Therefore, this review must In Section III, we describe the universal low-energy, long- necessarily be rather sweeping in scope, simply to introduce distance physics of such phases. We also discuss how they can the concepts of non-Abelian anyons and topological quantum be experimentally detected in the quantum Hall regime, and computation, their inter-connections, and how they may be re- when they might occur in other physical systems. Our focus alized in physical systems, particularly in several fractional is on a sequence of universality classes of non-Abelian topo- quantum Hall states. (For a popular account, see Collins, logical phases, associated with SU(2)k Chern-Simons theory 2006; for a slightly more technical one, see Das Sarma et al., which we describe in section III.A. The first interesting mem- 2006a.) This exposition will take us on a tour extending from ber of this sequence, k = 2, is realized in chiral p-wave knot theory and topological quantum field theory to conformal superconductors and in the leading theoretical model for the field theory and the quantum Hall effect to quantum computa- ν = 5/2 fractional quantum Hall state. Section III.B shows tion and all the way to the physics of gallium arsenide devices. how this universality class can be understood with conven- The body of this paper is composed of three parts, Sections tional BCS theory. In section III.C, we describe how the topo- II, III, and IV. Section II is rather general, avoids technical logical properties of the entire sequence of universality classes details, and aims to introduce concepts at a qualitative level. (of which k = 2 is a special case) can be understood using Section II should be of interest, and should be accessible, to Witten’s celebrated connection between Chern-Simons theory all readers. In Section III we describe the theory of topolog- and the Jones polynomial of knot theory. In section III.D, we ical phases in more detail. In Section IV, we describe how a describe an alternate formalism for understanding the topo- topological phase can be used as a platform for fault-tolerant logical properties of Chern-Simons theory, namely through quantum computation. The second and third parts are proba- conformal field theory. The discussion revolves around the bly of more interest to theorists, experienced researchers, and application of this formalism to fractional quantum Hall states those who hope to conduct research in this field. and explains how non-Abelian quantum Hall wavefunctions Section II.A.1 begins by discussing the concept of braiding can be constructed with conformal field theory. Appendix A statistics in 2+1-dimensions. We define the idea of a non- gives a highly-condensed introduction to conformal field the- Abelian anyon, a particle exhibiting non-Abelian braiding ory. In Section III.E, we discuss the gapless edge excitations statistics. Section II.A.2 discusses how non-Abelian anyons which necessarily accompany chiral (i.e. parity, P and time- can arise in a many-particle system. We then review the ba- reversal T -violating) topological phases. These excitations sic ideas of quantum computation, and the problems of errors are useful for interferometry experiments, as we discuss in and decoherence in section II.B.1. Those familiar with quan- Section III.F. Finally, in Section III.G, we discuss topological tum computation may be able to skip much of this section. We phases which do not violate parity and time-reversal symme- explain in section II.B.2 how non-Abelian statistics naturally tries. These phases emerge in models of electrons, spins, or leads to the idea of topological quantum computation, and ex- bosons on lattices which could describe transition metal ox- plain why it is a good approach to error-free quantum compu- ides, Josephson junction arrays, or ultra-cold atoms in optical tation. In section II.C, we briefly describe the non-Abelian lattices. quantum Hall systems which are the most likely arena for observing non-Abelian anyons (and, hence, for producing a In Section IV, we discuss how quasiparticles in topological topological quantum computer). Section II.C.1 gives a very phases can be used for quantum computation. We first dis- basic review of quantum Hall physics. Experts in quantum cuss the case of SU(2) , which is the leading candidate for Hall physics may be able to skip much of this section. Section 2 the ν = 5/2 fractional quantum Hall state. We show in Sec- II.C.2 introduces non-Abelian quantum Hall states. This sec- tion IV.A how qubits and gates can be manipulated in a gated tion also explains the importance (and summarizes the results) GaAs device supporting this quantum Hall state. We discuss of numericalwork in this field for determiningwhich quantum why quasiparticle braiding alone is not sufficient for universal Hall states are (or might be) non-Abelian.
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