Quantum Codes for Topological Quantum Computation

ISBN-13: 9783031068324 / Angielski / Miękka / 2022

Clarice Dias de Albuquerque;Eduardo Brandani da Silva;Waldir Silva Soares Jr.

Opis

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.

Spis treści

[preliminary]

1 An Overview on Quantum Codes

1.1 Previous Results

1.2 Goals

1.3 Some Classes of Quantum Error-Correcting Codes

1.4 Quantum Error-Correcting Codes

1.4.1 Formalism of Stabilizer Codes

1.5 Topological Quantum Codes

1.5.1 Topological Stabilizer Codes

1.6 CSS Codes

1.7 Surface Codes

1.8 Toric Quantum Code, g = 1

1.9 Hyperbolic Surface Codes, g ≥ 2

1.10 Color Codes

2 Preliminaries

2.1 Upper Half-Plane Model

2.2 Unit Open Disc Model

2.3 Geometrical Properties in H2 and [Delta]

2.4 Tessellations in Euclidean and Hyperbolic Planes

3 Surface Codes 29

3.1 Toric Codes, g = 1

3.2 Projective Plane Codes, g = 0

3.3 Homological Quantum Codes, g = 1

3.4 g-Toric Codes, g ≥ 2

4 Color Codes

4.1 Quantum Color Codes

4.2 Hyperbolic Color Codes

4.3 Polygonal Color Codes

O autorze

Clarice Dias de Albuquerque is an adjoint professor at the Federal University of Cariri, Brazil. She holds Bachelor's and Master's degrees from the Federal University of Ceará, Brazil, and a PhD in Electrical Engineering from the State University of Campinas, Brazil.

Eduardo Brandani da Silva is an Associate Professor at the State University of Maringá, Brazil. He holds Bachelor's (1988) and Master's degrees (1992) in Mathematics from the State University of Campinas, Brazil, and a PhD in Electrical Engineering (2000) from the same university. 

Waldir Silva Soares Júnior is a Professor at the Federal Technological University of Paraná, Brazil. He holds Bachelor's (2004) and Master's degrees (2008) in Mathematics from the State University of Maringá, and a PhD in Mathematics (2017) from the same university.