Topological quantum / Steven H. Simon.

Format:

Book

Author/Creator:

Simon, Steven H., author.

Series:

Oxford scholarship online.

Oxford scholarship online

Language:

English

Subjects (All):

Physics.

Mathematics.

Computer science.

Physical Description:

1 online resource (641 pages)

Place of Publication:

Oxford University Press 2023

Summary:

This publication discusses topological quantum, drawing in topics ranging from quantum gravity to topology to experimental condensed matter physics.

Contents:

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

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

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

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

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

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

Change of Basis and

Pentagon

Gauge Transformations

Appendix:

Odds and Ends

Unitarity of

with Higher Fusion Multiplicities

Exchanging Identical Particles

Introducing the

Locality

Some Examples

Fibonacci Anyons

Ising Anyons

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

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

Braiding Diagrams

Three-Dimensional Diagrams

Braiding Non-Identical Particles

Summary of Rules for Evaluating any (2 + 1)-Dimensional Diagram with Physics Normalization

The Hexagon

Appendix: Gauge Transformations and

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

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

Twists

Relations between

and

Nice Theories with Planar or Three-Dimensional Isotopy

Planar Diagrammatic Rules

Summary of Diagram Rules Planar Isotopy-Invariant Theories

Negative

and Unitarity

Constraints and Examples

Braiding Diagrams Revisited

Constraints

Appendix: Higher Fusion Multiplicities.

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

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

General

Anyons

with

even

All Braided Abelian Theories

Prime Non-Modular Theories

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:

Nontrivial Cocycle:

Appendix: Cocycles for

Bosons and Fermions from Group Representations: Rep

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

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

Temperly-Lieb Algebra and Jones-Kauffman Anyons

Jones-Wenzl Projectors

Two Strands in the General Case

Three Strands in the General Case

General Values of

Unitarization

Twisting and Braiding

Examples

Loop Gases Again

Ising Fusion

Fusion

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

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

Turaev-Viro Revisited: Chain-Mail and the Turaev-Walker-Roberts Theorem

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

Toric Code Basics

Introducing Quantum Error Correction

Classical Versus Quantum Information

Memories

Errors

Classical Error Correction

Quantum No-Cloning Theorem

Quantum Error Correction.

Notes:

Also issued in print: 2023.

Includes bibliographical references and index.

Description based on online resource and publisher information; title from PDF title page (viewed on September 22, 2023).

Other Format:

Print version: Simon, Steven H. Topological Quantum

ISBN:

0-19-199442-1

0-19-888677-2

OCLC:

1398633334