Introduction to Topological Quantum Matter & Quantum Computation

✍ Scribed by Тudоr D. Stаnеsсu

• Publisher: CRC Press
• Year: 2024
• Tongue: English
• Leaves: 449
• Edition: 2
• Category: Library

✦ Synopsis

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.

The fundamental tasks of quantum computation can be summarized as follows:

I. Develop quantum computational models. The fundamental model of quantum computation is the quantum circuit model, which defines a computation as a sequence of quantum gates – the quantum analog of classical logic gates. It can be demonstrated that there is a small set of so-called universal gates with the property that any quantum computation can be expressed in terms of these gates.

II. Design quantum algorithms. The fundamental underlying question concerns the ultimate limits of quantum computation. What class of computations can be performed using a quantum computer? Are there computational tasks that a quantum computer can perform better than a classical computer? So far, there are only a few examples of quantum algorithms that provide an advantage over known classical algorithms. Examples include the Deutsch–Jozsa and Shor algorithms, quantum search algorithms (e.g., the Grover algorithm), and quantum simulations (simulating a physical quantum mechanical system).

III. Develop quantum error-correction codes. The underlying fundamental question is: how can one reliably perform quantum information processing in the presence of noise? The concept of fault-tolerant quantum computation reveals that the encoding and decoding of quantum states can tolerate errors and, in addition, one can perform logical operations on encoded quantum states using imperfect underlying gate operations. An important result is the so-called threshold theorem: it is possible to efficiently perform an arbitrarily large quantum computation provided the noise in individual quantum gates is below a certain constant threshold.

IV. Identify specific physical systems and quantum properties to be used for the realization of quantum computers. The basic requirements for quantum computation concern the abilities to a) robustly represent quantum information, b) prepare the initial states, c) perform universal unitary transformations, and d) measure the output.

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.

✦ Table of Contents

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