Introduction to Topological Quantum Matter and Quantum Computation

What is'topological'about topological quantum states? How many types of topological quantum phases are there? What is a zero-energy Majorana mode, how can it be realized in a solid-state system, and how can it be used as a platform for topological quantum computation? What is quantum computation and what makes it different from classical computation?Addressing these and other related questions, Introduction to Topological Quantum Matter & Quantum Computation provides an introduction to and a synthesis of a fascinating and rapidly expanding research field emerging at the crossroads of condensed matter physics, mathematics, and computer science. Providing the big picture and emphasizing two major new paradigms in condensed matter physics – quantum topology and quantum information – this book is ideal for graduate students and researchers entering this field, as it allows for the fruitful transfer of ideas amongst different areas, and includes many specific examples to help the reader understand abstract and sometimes challenging concepts. It explores the topological quantum world beyond the well-known topological insulators and superconductors and unveils the deep connections with quantum computation. It addresses key principles behind the classification of topological quantum phases and relevant mathematical concepts and discusses models of interacting and noninteracting topological systems, such as the toric code and the p-wave superconductor. The book also covers the basic properties of anyons, and aspects concerning the realization of topological states in solid state structures and cold atom systems.Topological quantum computation is also presented using a broad perspective, which includes elements of classical and quantum information theory, basic concepts in the theory of computation, such as computational models and computational complexity, examples of quantum algorithms, and key ideas underlying quantum computation with anyons. This new edition has been updated throughout, with exciting new discussions on crystalline topological phases, including higher-order topological insulators; gapless topological phases, including Weyl semimetals; periodically-driven topological insulators; and a discussion of axion electrodynamics in topological materials.Key Features:· Provides an accessible introduction to this exciting, cross-disciplinary area of research.· Fully updated throughout with new content on the latest result from the field.· Authored by an authority on the subject.Tudor Stanescu is a professor of Condensed Matter Theory at West Virginia University, USA. He received a B.S. in Physics from the University of Bucharest, Romania, in 1994 and a Ph.D. in Theoretical Physics from the University of Illinois at Urbana Champaign in 2002. He was a Postdoctoral Fellow at Rutgers University and at the University of Maryland from 2003 to 2009. He joined the Department of Physics and Astronomy at West Virginia University in Fall 2009. Prof. Stanescu's research interests encompass a variety of topics in theoretical condensed matter physics including topological insulators and superconductors, topological quantum computation, ultra-cold atom systems in optical lattices, and strongly correlated materials, such as, for example, cuprate high-temperature superconductors. His research uses a combination of analytical and numerical tools and focuses on understanding the emergence of exotic states of matter in solid state and cold atom structures, for example, topological superconducting phases that host Majorana zero modes, and on investigating the possibilities of exploiting these states as physical platforms for quantum computation. Cover Half Title Title Page Copyright Page Dedication Contents Preface to the second edition Preface to the first edition SECTION I: Topological Quantum Phases: Basic Theory, Classification, and Modeling CHAPTER 1: Topology and Quantum Theory 1.1. QUANTUM AMPLITUDES AND KNOT INVARIANTS 1.2. TOPOLOGY AND DIFFERENTIAL GEOMETRY: MATHEMATICAL HIGHLIGHTS 1.3. GEOMETRIC PHASES: EXAMPLES AND OVERVIEW 1.3.1. Classical and quantum holonomies 1.3.2. Historical overview and conceptual distinctions 1.4. PHASE CHANGES DURING CYCLIC QUANTUM EVOLUTIONS 1.4.1. The Berry phase 1.4.2. The non-Abelian adiabatic phase 1.4.3. The Aharonov–Anandan phase 1.5. THE MATHEMATICAL STRUCTURE OF GEOMETRIC PHASES 1.5.1. Elementary introduction to fiber bundles 1.5.2. Holonomy interpretations of geometric phases CHAPTER 2: Symmetry and Topology in Condensed Matter Physics 2.1. THEMES IN MANY-BODY PHYSICS 2.2. LANDAU THEORY OF SYMMETRY BREAKING 2.2.1. Construction of the Landau functional 2.2.2. Phases and phase transitions 2.3. TOPOLOGICAL ORDER, SYMMETRY, AND QUANTUM ENTANGLEMENT 2.4. TOPOLOGY AND QUANTUM COMPUTATION 2.5. TOPOLOGY AND EMERGENT PHYSICS CHAPTER 3: Topological Insulators and Superconductors 3.1. INTRODUCTION 3.2. SYMMETRY CLASSIFICATION OF GENERIC NONINTERACTING HAMILTONIANS 3.2.1. Time-reversal symmetry 3.2.2. Particle-hole and chiral symmetries 3.2.3. Classification of random Hamiltonians 3.3. TOPOLOGICAL CLASSIFICATION OF BAND INSULATORS AND SUPERCONDUCTORS 3.3.1. The origin of topology in gapped noninteracting systems 3.3.2. Classification of topological insulators and superconductors 3.4. TOPOLOGICAL INVARIANTS: CHERN NUMBERS, WINDING NUMBERS, AND Z2 INVARIANTS 3.4.1. Hall conductance and the Chern number 3.4.2. Chern numbers and winding numbers 3.4.3. The Z2 topological invariant CHAPTER 4: Extensions of the Noninteracting Topological Classification 4.1. TOPOLOGICAL CRYSTALLINE INSULATORS AND SUPERCONDUCTORS 4.1.1. Weak topological phases and fragile topology 4.1.2. Crystalline topological phases 4.1.3. Higher order topological phases 4.2. GAPLESS TOPOLOGICAL PHASES 4.2.1. Weyl semimetals in three-dimensional solids 4.2.2. Topological semimetals and nodal superconductors 4.3. FLOQUET TOPOLOGICAL INSULATORS CHAPTER 5: Interacting Topological Phases 5.1. TOPOLOGICAL PHASES: ORGANIZING PRINCIPLES 5.1.1. Systems with no symmetry constraints 5.1.2. Systems with symmetry constraints 5.2. QUANTUM PHASES WITH TOPOLOGICAL ORDER 5.2.1. Effective theory of Abelian fractional quantum Hall liquids 5.2.2. The toric code 5.3. SYMMETRY PROTECTED TOPOLOGICAL QUANTUM SATES 5.3.1. SPT phases in one dimension 5.3.2. SPT phases in two and three dimensions CHAPTER 6: Theories of Topological Quantum Matter 6.1. TOPOLOGICAL BAND THEORY: CONTINUUM DIRAC MODELS 6.1.1. Graphene and Dirac fermions 6.1.2. Quantum spin Hall state: The Kane–Mele model 6.1.3. Three-dimensional four-component Dirac Hamiltonian 6.2. TOPOLOGICAL BAND THEORY: TIGHT-BINDING MODELS 6.2.1. Haldane model 6.2.2. Mercury telluride quantum wells: The BHZ model 6.2.3. p-Wave superconductors in one and two dimensions 6.3. TOPOLOGICAL FIELD THEORY CHAPTER 7: Axion Electrodynamics in Topological Quantum Matter 7.1. QUANTIZED MAGNETO-ELECTRIC EFFECT IN TOPOLOGICAL INSULATORS AND AXION INSULATORS 7.2. DYNAMICAL AXION FIELDS IN TOPOLOGICAL MAGNETIC INSULATORS 7.3. TOPOLOGICAL ELECTROMAGNETIC RESPONSE OF WEYL SEMIMETALS 7.4. AXION “GRAVITOELECTROMAGNETISM” IN TOPOLOGICAL SUPERCONDUCTORS CHAPTER 8: Majorana Zero Modes in Solid-State Heterostructures 8.1. THEORETICAL BACKGROUND 8.1.1. Majorana zero modes 8.1.2. “Synthetic” topological superconductors 8.2. REALIZATION OF MAJORANA ZERO MODES: PRACTICAL SCHEMES 8.2.1. Semiconductor-superconductor hybrid structures 8.2.2. Shiba chains 8.3. EXPERIMENTAL DETECTION OF MAJORANA ZERO MODES 8.3.1. Tunneling spectroscopy 8.3.2. Fractional Josephson effect 8.3.3. Nonlocal transport 8.4. EFFECTS OF DISORDER IN HYBRID MAJORANA NANOWIRES CHAPTER 9: Topological Phases in Cold Atom Systems 9.1. BRIEF HISTORICAL PERSPECTIVE 9.2. MANY-BODY PHYSICS WITH ULTRACOLD GASES: BASIC TOOLS 9.2.1. Cooling and trapping of neutral atoms 9.2.2. Optical lattices 9.2.3. Feshbach resonances 9.3. LIGHT-INDUCED ARTIFICIAL GAUGE FIELDS 9.3.1. Geometric gauge potentials 9.3.2. Abelian gauge potentials: The  scheme 9.3.3. Non-Abelian gauge potentials: The tripod scheme and spin-orbit coupling 9.4. TOPOLOGICAL STATES IN COLD ATOM SYSTEMS 9.4.1. Realization of the Haldane model with ultracold atoms 9.4.2. Majorana fermions in optical lattices SECTION II: Quantum Information and Quantum Computation: Introductory Concepts CHAPTER 10: Elements of Quantum Information Theory 10.1. INTRODUCTION 10.2. CLASSICAL INFORMATION THEORY 10.3. OPERATIONAL QUANTUM MECHANICS 10.3.1. Noiseless quantum theory 10.3.2. Noisy quantum theory 10.4. QUANTUM INFORMATION THEORY: BASIC CONCEPTS 10.4.1. Quantum bits 10.4.2. Quantum operations 10.4.3. No cloning 10.5. ENTROPY AND INFORMATION 10.6. DATA COMPRESSION 10.6.1. Schumacher’s noiseless quantum coding theorem 10.7. ACCESSIBLE INFORMATION 10.7.1. The Holevo bound 10.8. ENTANGLEMENT-ASSISTED COMMUNICATION 10.8.1. Superdense coding 10.8.2. Quantum teleportation 10.9. QUANTUM CRYPTOGRAPHY 10.9.1. Quantum key distribution CHAPTER 11: Introduction to Quantum Computation 11.1. INTRODUCTION 11.2. CLASSICAL THEORY OF COMPUTATION 11.2.1. Computational models: The Turing machine 11.2.2. Computational complexity 11.2.3. Energy and computation 11.3. QUANTUM CIRCUITS 11.4. QUANTUM ALGORITHMS 11.4.1. Deutsch’s algorithm 11.4.2. Quantum search: Grover’s algorithm 11.4.3. Quantum Fourier transform: Shor’s algorithm 11.4.4. Simulation of quantum systems 11.5. QUANTUM ERROR CORRECTION CHAPTER 12: Anyons and Topological Quantum Computation 12.1. QUANTUM COMPUTATION WITH ANYONS 12.1.1. Abelian and non-Abelian anyons 12.1.2. Braiding 12.1.3. Particle types, fusion rules, and exchange properties 12.1.4. Fault-tolerance from non-Abelian anyons 12.1.5. Ising anyons 12.1.6. Fibonacci anyons 12.2. ANYONS AND TOPOLOGICAL QUANTUM PHASES 12.2.1. Abelian Chern–Simons field theories 12.2.2. Non-Abelian Chern–Simons field theories 12.3. TOPOLOGICAL QUANTUM COMPUTATION WITH MAJORANA ZERO MODES 12.3.1. Non-Abelian statistics 12.3.2. Fusion of Majorana zero modes 12.3.3. Quantum information processing 12.4. OUTLOOK: QUANTUM COMPUTATION AND TOPOLOGICAL QUANTUM MATTER Bibliography Index