Quantum Codes for Topological Quantum Computation (SpringerBriefs in Mathematics)

✍ Scribed by Clarice Dias de Albuquerque, Eduardo Brandani da Silva, Waldir Silva Soares Jr.

• Publisher: Springer
• Year: 2022
• Tongue: English
• Leaves: 123
• Edition: 1st ed. 2022
• Category: Library

✦ Synopsis

This book offers a structured algebraic and geometric approach to the classification and construction of quantum codes for topological quantum computation. It combines key concepts in linear algebra, algebraic topology, hyperbolic geometry, group theory, quantum mechanics, and classical and quantum coding theory to help readers understand and develop quantum codes for topological quantum computation.
One possible approach to building a quantum computer is based on surface codes, operated as stabilizer codes. The surface codes evolved from Kitaev's toric codes, as a means to developing models for topological order by using qubits distributed on the surface of a toroid. A significant advantage of surface codes is their relative tolerance to local errors. A second approach is based on color codes, which are topological stabilizer codes defined on a tessellation with geometrically local stabilizer generators. This book provides basic geometric concepts, like surface geometry, hyperbolic geometry and tessellation, as well as basic algebraic concepts, like stabilizer formalism, for the construction of the most promising classes of quantum error-correcting codes such as surfaces codes and color codes.
The book is intended for senior undergraduate and graduate students in Electrical Engineering and Mathematics with an understanding of the basic concepts of linear algebra and quantum mechanics.

✦ Table of Contents

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