Quantum Codes for Topological Quantum Computation (SpringerBriefs in Mathematics)

Contents
1 Introduction
	1.1 Historical Summary
	1.2 The Quantum Properties
		1.2.1 Superposition of States
		1.2.2 Entanglement
	1.3 Principle of Quantum Error Correction
		1.3.1 Stabilizer Codes
	1.4 Quantum Bounds
	1.5 Codifying with Topology
2 Review of Mathematical Concepts
	2.1 Classical Error-Correcting Codes
		2.1.1 Basic Definitions
	2.2 Block Codes
		2.2.1 Linear Codes
			Examples
	2.3 Linear Algebra
		2.3.1 Quantum Bit
		2.3.2 Matrices and Operators
	2.4 Quantum Information and Quantum Computation
	2.5 A Glimpse of Quantum Mechanics
		2.5.1 Postulates
		2.5.2 Quantum Gates
	2.6 Introduction to Quantum Error-Correcting Codes
		2.6.1 The 3-Qubit Quantum Code
		2.6.2 Shor Code
	2.7 Quantum Error-Correction Criterion
	2.8 CSS Codes
	2.9 Stabilizer Quantum Codes
		2.9.1 Anti-commutation
		2.9.2 Stabilizer Group
		2.9.3 Stabilizer Code and Examples
	2.10 Hyperbolic Geometry
		2.10.1 Isometries of the Hyperbolic Plane
		2.10.2 Regular Tessellations
3 Topological Quantum Codes
	3.1 Toric Codes
		3.1.1 Toric Codes from the Homology Point of View
		3.1.2 Correction at the Physical Level
	3.2 Projective Plane and Quantum Codes
	3.3 Other Toric Codes
		3.3.1 Polyomino Quantum Codes
	3.4 Hyperbolic Topological Quantum Codes
		3.4.1 Generation of a Surface from a Polygon P
			Classification of Surfaces
			Planar Model of a Surface
			Tessellations
		3.4.2 Constructions of Hyperbolic Topological Quantum Codes
			Classes of Hyperbolic Topological Quantum Codes
4 Color Codes
	4.1 Quantum Color Codes
	4.2 Color Codes on Compact Surfaces
	4.3 Color Codes on Surfaces with Boundary
5 The Interplay Between Color Codes and Toric Codes
	5.1 Introduction
	5.2 Quantum Double Models
		5.2.1 Toric Code
		5.2.2 Color Codes
	5.3 Color Code Equivalence to Two Copies of Toric Codes
Bibliography