Quantum Error Correction

This text presents an algebraic approach to the construction of several important families of quantum codes derived from classical codes by applying the well-known Calderbank-Shor-Steane (CSS) construction, the Hermitian, and the Steane's enlargement construction to certain classes of classical codes. These quantum codes have good parameters and have been introduced recently in the literature. In addition, the book presents families of asymmetric quantum codes with good parameters and provides a detailed description of the procedures adopted to construct families of asymmetric quantum convolutional codes.Featuring accessible language and clear explanations, the book is suitable for use in advanced undergraduate and graduate courses as well as for self-guided study and reference. It provides an expert introduction to algebraic techniques of code construction and, because all of the constructions are performed algebraically, it equips the reader to construct families of codes, rather than only codes with specific parameters. The text offers an abundance of worked examples, exercises, and open-ended problems to motivate the reader to further investigate this rich area of inquiry. End-of-chapter summaries and a glossary of key terms allow for easy review and reference.

Auteur

Giuliano G. La Guardia received a Master's degree in pure mathematics in 1998 and a PhD in electrical engineering in 2008, both from the State University of Campinas (UNICAMP), São Paulo, Brazil. Since 1999 he has been with the Department of Mathematics and Statistics at the State University of Ponta Grossa, where he is an Associate Professor. His research areas include quantum and classical coding theory, matroid theory, category theory and dynamical systems theory.

Contenu

1-Introduction to quantum mechanics

1.1-Vector spaces

1.2- Bases, norms, inner products

1.3- Linear Operators

1.4-Eigenvalues and eigenvectors

1.5- Adjoint, Hermitian and Unitary operators

1.6- Operator functions

1.7- Pauli and generalized Pauli matrices

1.8- Postulates of quantum mechanics

2-Introduction to quantum computation and information

2.1-Single and multiple qubit operations

2.2-Universal quantum gates

2.3- Bit flip and phase shift channels

2.4- Depolarizing channel 2.5-Amplitude damping channel

2.6-Measure of distance of quantum states

2.7-Fidelity

3-Quantum error-correcting codes

3.1-The Shor code

3.2-The Steane code

3.3-Five quibit code

3.4-Quantum Hamming and Singleton bound

3.4-Stabilizer codes

3.5-Calderbank-Shor-Steane code construction

3.6-Hermitian construction

3.7-Ste ane's enlargement construction

3.8-Additive codes

4-Quantum code construction

4.1-Bose-Chaudhuri-Hocquenghem codes 4.2-Reed-Solomon codes

4.3-Reed-Muller codes

4.4-Quadratic residue codes

4.5-Constacyclic codes

4.6- Affine Invariant codes

4.7-Algebraic geometry codes

4.8-Synchronizable codes

5-Asymmetric quantum code construction

5.1- Bose-Chaudhuri-Hocquenghem codes

5.2- Reed-Solomon codes

5.3-Tensor product codes

5.4-Alternalt codes

5.5- Algebraic geometry codes

5.6-New codes from old

6-Quantum convolutional code construction 6.1-Convolutional codes

6.2-Quantum convolutional codes

6.3- Code construction: Bose-Chaudhuri-Hocquenghem,

Reed-Solomon

Reed-Muller 6.4-Construction of asymmetric quantum convolutional codes