Introduction to Topological Quantum Computation

✍ Scribed by Jiannis K. Pachos

• Publisher: Cambridge University Press
• Year: 2012
• Tongue: English
• Leaves: 225
• Category: Library

✦ Table of Contents

Front Cover
Back Cover
Front Matter
Contents
Acknowledgements
Part I. Preliminaries
Chapter 1: Introduction
1.1 Particle exchange and quantum physics
1.2 Anyons and topological systems
1.3 Quantum computation with anyons
1.4 Abelian and non-Abelian anyonic statistics
1.5 What are anyonic systems?
1.5.1 Two-dimensional wave functions and quasiparticles
1.5.2 Symmetry, degeneracy and quantum correlations
Summary
Exercises
Chapter 2: Geometric and topological phases
2.1 Quantum phases and gauge fields
2.1.1 Charged particle in a magnetic field
2.1.2 The Aharonov-Bohm effect
2.1.3 Anyons and Aharonov-Bohm effect
2.2 Geometric phases and holonomies
2.2.1 Spin-1/2 particle in a magnetic field
2.2.2 Non-Abelian geometric phases
2.2.3 Properties of geometric evolutions
2.2.4 Anyons and geometric phases
2.3 Example I: Integer quantum Hall effect
2.3.1 Wave function of a charged particle in a magnetic field
2.3.2 Current behavior and Hall conductivity
2.3.3 Laughlin's thought experiment and geometric phases
Summary
Exercises
Chapter 3: Quantum computation
3.1 Qubits and their manipulations
3.1.1 Quantum bits
3.1.2 Decoherence and mixed states
3.1.3 Quantum gates and projectors
3.2 Quantum circuit model
3.2.1 Quantum algorithm and universality
3.2.2 Computational complexity
3.3 Other computational models
3.3.1 One-way quantum computation
3.3.2 Adiabatic quantum computation
3.3.3 Holonomic quantum computation
Summary
Exercises
Chapter 4: Computational power of anyons
4.1 Anyons and their properties
4.1.1 Particle types
4.1.2 Fusion rules of anyons
4.1.3 Anyonic Hilbert space
4.1.4 Exchange properties of anyons
4.1.5 Pentagon and hexagon identities
4.1.6 Spin and statistics
4.2 Anyonic quantum computation
4.2.1 Anyonic setting
4.2.2 Stability of anyonic computation
4.3 Example I: Ising anyons
4.3.1 The model and its properties
4.3.2 F and R matrices
4.4 Example II: Fibonacci anyons
Summary
Exercises
Part II. Topological Models
Chapter 5: Quantum double models
5.1 Error correction
5.1.1 Quantum error correcting codes
5.1.2 Stabiliser codes
5.2 Quantum double models
5.2.1 The toric code
5.2.2 General D(G) quantum double models
5.3 Example I: Abelian quantum double models
5.4 Example II: The non-Abelian D(S3) model
5.5 Quantum doubles as quantum memories
5.5.1 Non-Abelian information encoding and manipulation
Summary
Exercises
Chapter 6: Kitaev's honeycomb lattice model
6.1 Introducing the honeycomb lattice model
6.1.1 The spin lattice Hamiltonian
6.1.2 Majorana fermionisation
6.1.3 Emerging lattice gauge theory
6.2 Solving the honeycomb lattice model
6.2.1 The no-vortex sector
6.2.2 Vortex sectors
6.3 Ising anyons as Majorana fermions
Summary
Exercises
Chapter 7: Chern-Simons quantum field theories
7.1 Abelian Chern-Simons theories
7.1.1 Four-dimensional electromagnetism
7.1.2 Three-dimensional electromagnetism
7.1.3 Abelian anyons and topological invariants
7.2 Non-Abelian Chern-Simons theories
7.2.1 Non-Abelian gauge theories
7.2.2 Wilson loops and anyonic worldlines
7.2.3 The braiding evolution
7.3 Example I: Braiding for the SU(2) Chern-Simons theory
7.4 Example II: From bulk to boundary
7.4.1 Abelian case
7.4.2 Non-Abelian case
7.5 Example III: Non-Abelian anyones and their fusion rules
7.5.1 Number of anyonic species
7.5.2 Fusion rules
Summary
Exercises
Part III. Quantum Information Perspectives
Chapter 8: The Jones polynomial algorithm
8.1 From link invariance to Jones polynomials
8.1.1 Reidemeister moves
8.1.2 Skein relations and Kauffman brackets
8.1.3 Jones polynomial
8.2 From the braid group to Jones polynomials
8.2.1 The braid group
8.2.2 The Temperley-Lieb algebra
8.2.3 Markov traces and Jones polynomials
8.3 Analogue quantum computation of Jones polynomials
8.4 Example I: Kauffman brack of simple links
8.5 Example II: Jones polynomials from Chern-Simons theories
Summary
Exercises
Chapter 9: Topological entanglement entropy
9.1 Entanglement entropy and topological order
9.2 Topological entropy and its properties
9.2.1 Definition of topological entropy
9.2.2 Properties of topological entropy
9.2.3 Topological entropy and Wilson loops
9.3 Example I: Quantum double models
9.3.1 Hamiltonian and its ground state
9.3.2 Topological entropy
Summary
Exercises
Chapter 10: Outlook
References
Index