Topological Quantum

At the intersection of physics, mathematics, and computer science, an exciting new field of study has formed, known as “Topological Quantum.” This research field examines the deep connections between the theory of knots, special types of subatomic particles known as anyons, certain phases of matter, and quantum computation. This book elucidates this nexus, drawing in topics ranging from quantum gravity to topology to experimental condensed matter physics. Topological quantum has increasingly been a focus point in the fields of condensed matter physics and quantum information over the last few decades, and the forefront of research now builds on the basic ideas presented in this book. The material is presented in a down-to-earth and entertaining way that is far less abstract than most of what is in the literature. While introducing the crucial concepts and placing them in context, the subject is presented without resort to the highly mathematical category theory that underlies the field. Requiring only an elementary background in quantum mechanics, this book is appropriate for all readers, from advanced undergraduates to the professional practitioner. This book will be of interest to mathematicians and computer scientists as well as physicists working on a wide range of topics. Those interested in working in these field will find this book to be an invaluable introduction as well as a crucial reference. Cover Title Copyright Page Preface Contents Introduction: History of Topology, Knots, Peter Tait, and Lord Kelvin Kauffman Bracket Invariant and Relation to Physics The Idea of a Knot Invariant Relation to Physics Twist and Spin-Statistics Blackboard Framing Bras and Kets Quantum Computation with Knots Some Quick Comments about Fractional Quantum Hall Effect Appendix: More Knot Theory Basics Isotopy and Reidemeister Moves Writhe and Linking Chapter Summary Exercises Part I Anyons and Topological Quantum Field Theories Particle Quantum Statistics Single Particle Path Integral Two Identical Particles Many Identical Particles: Preliminaries Paths in (2 + 1)-Dimensions, the Braid Group Paths in (3 + 1)-Dimensions, the Permutation Group Building a Path Integral: Abelian Case (3 + 1)-Dimensions (2 + 1)-Dimensions Nonabelian Case Parastatistics in (3 + 1)-Dimensions Chapter Summary Exercises Aharonov–Bohm Effect and Charge–Flux Composites Review of Aharonov–Bohm Effect Anyons as Charge–Flux Composites Fusion of Anyons Anti-Anyons and the Vacuum Particle Anyon Vacuum on a Torus and Quantum Memory Quantum Memory and Higher Genus Number of Species of Anyons Chapter Summary Exercises Chern–Simons Theory Basics Abelian Chern–Simons Theory Nonabelian Chern–Simons Theory: The Paradigm of TQFT Appendix: Odds and Ends about Chern–Simons Theory Chern–Simons Canonical Quantization for the Abelian Case Multiple Gauge Fields and the Gauge Transformations with Nonabelian Gauge Fields Chern–Simons Action Is Metric Independent Winding Number: The Pontryagin Index Framing of the Manifold—or Doubling the Theory Chern–Simons Theory as the Boundary of a Four-Dimensional Theory Chapter Summary Exercises Short Digression on Quantum Gravity Why This Is Hard Which Approach? Some General Principles? Further Comments on Connections to Quantum Gravity Appendix: No Gravity Waves in (2 + 1)-Dimensions Appendix: Relation of Chern–Simons Theory to (2 + 1)-Dimensional GR Chapter Summary Defining Topological Quantum Field Theory Paraphrasing of Atiyah’s Axioms Adding Particles Particles or No Particles Building Simple 3-Manifolds S3 and the Modular S-matrix S2 × S1 Connected Sums Appendix: Gluing Solid Tori Together Appendix: Cobordisms and Category Theory Chapter Summary Exercises Part II Anyon Basics Fusion and Structure of Hilbert Space Basics of Particles and Fusion—The Abelian Case Multiple Fusion Channels—The Nonabelian Case Example: Fibonacci Anyons Example: Ising Anyons Fusion and the matrices Associativity Application of Fusion: Dimension of Hilbert Space on 2-Manifolds Product Theories Appendix: Tensor Description of Fusion and Splitting Spaces Chapter Summary Exercises Change of Basis and Example: Fibonacci Anyons Example: Ising Anyons Pentagon Gauge Transformations Appendix: Odds and Ends Product Theories Unitarity of with Higher Fusion Multiplicities Chapter Summary Exercises Exchanging Identical Particles Introducing the Locality Some Examples Fibonacci Anyons Ising Anyons Chapter Summary Exercises Computing with Anyons Quantum Computing Universal Quantum Computing in the Quantum Circuit Model Topological Quantum Computing Hilbert Space Measurement (in Brief) and Initialization Universal Braiding Computing with Non-Universal Anyons? Fibonacci Example A Single Fibonacci Qubit Topological Quantum Compiling: Single Qubit Two-Qubit Gates Controlled Gates Chapter Summary Exercises Anyon Diagrammatics (in Detail) Planar Diagrams Diagrams as Operators Stacking Operators Basis of States One Particle Two Particles Three Particles Again More Particles Causal Isotopy Summary of Planar Diagram Rules in Physics Normalization A Simple Example Appendix: Higher Fusion Multiplicities Chapter Summary Exercises Braiding Diagrams Three-Dimensional Diagrams Braiding Non-Identical Particles Summary of Rules for Evaluating any (2 + 1)-Dimensional Diagram with Physics Normalization The Hexagon Odds and Ends Appendix: Gauge Transformations and Product Theories Appendix: Higher Fusion Multiplicities Chapter Summary Exercises Seeking Isotopy Isotopy Normalization of Diagrams Gauge Choice and Frobenius–Schur Indicator Isotopy-Invariant Unitary Rules Isn’t Chern–Simons Theory Isotopy Invariant and Unitary? What Have We Achieved? Impediments to Isotopy Invariance in Fusion Diagrams Planar Diagrams with Isotopy Invariance Isotopy in Three-Dimensional Diagrams Appendix: Bookkeeping Scheme with Negative Appendix: Is Real Appendix: Spin- Analogy and Why We Have a Frobenius–Schur Sign Appendix: Some Additional Properties of Unitary Fusion Categories Pivotal Property Spherical Property Appendix: Higher Fusion Multiplicities Chapter Summary Exercises Twists Relations between and Appendix: Higher Fusion Multiplicities Chapter Summary Exercises Nice Theories with Planar or Three-Dimensional Isotopy Planar Diagrams Planar Diagrammatic Rules Summary of Diagram Rules Planar Isotopy-Invariant Theories Negative and Unitarity Constraints and Examples Braiding Diagrams Revisited Constraints Gauge Transformations Appendix: Higher Fusion Multiplicities Chapter Summary Exercises Further Structure Quantum Dimension The Unlinking The (Modular) Unitary = Modular Modular Group and Torus Diffeomorphisms Central Charge and Relation to Conformal Field Theory Tables of TQFTs Strand (Kirby Color) Still Further Structure Fermions and Super-Modular Theories Appendix: Perron–Frobenius Theorem Appendix: Algebraic Derivation of the Verlinde Form Appendix: Algebraic Derivation that Quantum Dimensions Form a Representation of the Fusion Algebra Chapter Summary Exercises Some Examples: Planar Diagrams and Anyon Theories Some Simple Examples Fusion Rules Loop Gas Fibonacci Fusion Rules: The Branching Loop Gas Braidings for Fibonacci Anyons The Abelian Theories Ising Fusion Rules Braidings For Ising Fusion Rules More Abelian Theories Fusion Rules General Fusion Rules Anyons with even All Braided Abelian Theories Prime Non-Modular Theories Chapter Summary Exercises Anyons From Discrete Group Elements Group Cohomology Braidings for Abelian Group Simple Examples with Using Non-Commutative Groups? Appendix: Isotopy-Invariant Planar Algebras and Anyon Theories from Cohomology Trivial Cocycle: Anyons Nontrivial Cocycle: Appendix: Cocycles for Chapter Summary Exercises Bosons and Fermions from Group Representations: Rep Some Examples Representations of Quaternion Group in Rep(G) Some Simple Braidings for Rep “Trivial” Braidings: Bosons Fermions and Bosons Other Braidings Parastatistics Revisited Appendix: Further Group Theory Symmetry of Fusion Products in Frobenius–Schur Indicator Clebsch–Gordan Coefficients of Discrete Groups Chapter Summary Exercises Quantum Groups (in Brief) Continuous (Lie) Group Representations? : The Deformation of Representations of Representation Theory of Deformed Representation Theory at Roots of Unity: Other Lie Groups Chapter Summary Exercises Temperly–Lieb Algebra and Jones–Kauffman Anyons Jones–Wenzl Projectors Loop Gas Two Strands in the General Case Three Strands in the General Case Ising Anyons General Values of Unitarization Twisting and Braiding Examples Loop Gases Again Ising Fusion Fusion Chapter Summary Exercises Applications of TQFT Diagrammatics State-Sum TQFTs Simplicial Decomposition and Pachner Moves Two Dimensions Three Dimensions The Turaev–Viro State Sum Proof Turaev–Viro Is a Manifold Invariant Some TQFT Properties Connection to Chern–Simons Theory Connections to Quantum Gravity Revisited Dijkgraaf–Witten Model Other Dimensions Further Comments Chapter Summary Exercises Formal Construction of TQFTs from Diagrams: Surgery and More Complicated 3-Manifolds Surgery Simple Example of Surgery on a 2-Manifold Surgery on 3-Manifolds Representing Manifolds with Knots Lickorish–Wallace Theorem Kirby Calculus Witten-Reshetikhin-Turaev Invariant Some Examples Turaev–Viro Revisited: Chain-Mail and the Turaev–Walker–Roberts Theorem Chapter Summary Exercises Anyon Condensation Condensing Simple Current Bosons Identification Step Orbits of Maximum Size Confinement Step Splitting: Orbits Not of Maximum Size Other Features of Condensation Cosets Dualities and Aliases More General Condensations Condensation and Boundary Modes Chapter Summary Exercises Toric Code Basics Introducing Quantum Error Correction Classical Versus Quantum Information Memories Errors Classical Error Correction Quantum No-Cloning Theorem Quantum Error Correction Qubit-Flip Correcting Code Nine-Qubit Shor Code Chapter Summary Exercises Introducing the Toric Code Toric Code Hilbert Space Vertex and Plaquette Operators Operators Commute Is This a Complete Set of Operators? (Not Quite!) Building the Code Space Errors and Error Correction Errors Combinations of and Other Errors The Toric Code on Different Lattices and Different Topologies Toric Code (Briefly) Code Space Errors Chapter Summary Exercises The Toric Code as a Phase of Matter and a TQFT Excitations Statistical Properties of Excitations Braiding Vertex Defects with Plaquette Defects Properties of the Fermion and Charge–Flux Model Toric Code (Briefly) Chapter Summary Exercises Robustness of Topologically Ordered Matter Perturbed Hamiltonian Robustness of Ground-State Degeneracy Quasiparticles Topologically Ordered Matter Importance of Rigidity The Notion of Topological Order Defining a Topological Phase of Matter Appendix: Brillouin-Wigner Perturbation Theory Chapter Summary Exercises Abstracting the Toric Code: Introducing the Tube Algebra Toric Code as a Loop Gas Preview of Coming Attractions: Generalizations of the Toric Code The Tube Algebra and Quasiparticle Excitations States on the Annulus Composition of States Quasiparticle Idempotents Twists and Direct Calculation of Braiding and Fusion Generalization to Toric Code (Briefly) Chapter Summary Exercises More General Loop-Gas and String-Net Models Kitaev Quantum Double Model Defining the Model Vertex and Plaquette Operators Code Space Hamiltonian How the Kitaev Model Generalizes the Toric Code Kitaev Ground State Is Topological Continuum Model Brief Comments on Twisted Kitaev Theory Ground-State Degeneracy on a Torus Quasiparticles Quasiparticle Basis on Torus Quasiparticle Ribbon Operators on a Lattice Relation to Gauge Theory Gauge Theory in (3 + 1)-Dimensions Appendix: Two Expressions for the Ground-State Degeneracy on a Torus Chapter Summary Exercises Doubled-Semion Model A Microscopic Model Magnetic String Operator Attempted Graphical Analysis Gauge Choice and Unitary Diagram Algebra Doubled Semions Comments Chapter Summary Exercises Levin–Wen String-Net Graphical Example: Doubled-Fibonacci Quasiparticle Types Torus States A Nicer Torus Basis More General Diagram Algebras Modular Case Microscopic Hamiltonian Fat-Lattice Construction Vertex and Plaquette Operators Levin–Wen Hamiltonian Ground State Is Topological Quasiparticle String Operators Toric Code: A Non-Modular Example Doubled Semions: A Modular Example More Generally Appendix: Explicit Form of Certain Operators Plaquette Operator String Operator Example Chapter Summary Exercises Entanglement and Symmetries Topological Entanglement Entanglement in the Toric Code Generalizing to Arbitrary TQFTs Topological Entanglement Entropy Is Robust Appendix: Three Derivations of Topological Entanglement Entropy Kitaev-Preskill Argument Entanglement Hamiltonian Argument Surgery Argument Chapter Summary Exercises Symmetry-Protected Topological Phases of Matter Symmetry Protection Some Short Comments on SPT History On-Site Symmetries Example: Ising ( ) Symmetry Relation to Loop Models Twisted Boundaries and Symmetry Defects Edge Modes Relation to Twisted Kitaev Model Relation to Dijkgraaf–Witten Model Appendix: One-Dimensional SPTs Chapter Summary Exercises Anyon-Permuting Symmetry Symmetry Defects of Anyon-Permuting Symmetry Symmetric Form of the Toric Code Symmetry Defects in the Toric Code More General Symmetry Defects Symmetry-Enriched Topological Phases Chapter Summary Exercises Further Thoughts Experiments (In Brief!) Fractional Quantum Hall Effects Abelian Fractional Quantum Hall States Nonabelian Fractional Quantum Hall States Fractional Chern Insulators Bosonic Fractional Quantum Hall Effects Gapped Quantum Spin Liquids Kitaev Honeycomb Model Frustrated Antiferromagnets Conventional Superconductors Majorana Materials Quantum Simulation Chapter Summary Final Comments What Is Not in This Book Theory of Fractional Quantum Hall Effect Connections to Conformal Field Theory Tensor Networks Topological Insulators Final Words Appendix: Kac and Other Resources for TQFTs Kac Other Resources Appendix: Some Mathematical Basics Manifolds Some Examples: Euclidean Spaces and Spheres Unions of Manifolds Products of Manifolds: Manifolds with Boundary Boundaries of Manifolds: Removing Bits Groups Some Examples of Groups More Features of Groups Lie Groups and Lie Algebras Representations of Groups Fundamental Group Examples of Fundamental Groups Chapter Summary References Index