Algebraic Methods for Quantum Codes on Lattices

This is a note from a series of lectures at Encuentro Colombiano de Computación Cuántica, Universidad de los Andes, Bogotá, Colombia, 2015. The purpose is to introduce additive quantum error correcting codes, with emphasis on the use of binary representation of Pauli matrices and modules over a tran...

Full description

Autores:
Haah, Jeongwan
Tipo de recurso:
Article of journal
Fecha de publicación:
2016
Institución:
Universidad Nacional de Colombia
Repositorio:
Universidad Nacional de Colombia
Idioma:
spa
OAI Identifier:
oai:repositorio.unal.edu.co:unal/66450
Acceso en línea:
https://repositorio.unal.edu.co/handle/unal/66450
http://bdigital.unal.edu.co/67478/
Palabra clave:
51 Matemáticas / Mathematics
quantum stabilizer codes
additive codes
symplectic codes
Laurent polynomial ring
toric code
Cliord circuit
Rights
openAccess
License
Atribución-NoComercial 4.0 Internacional
Description
Summary:This is a note from a series of lectures at Encuentro Colombiano de Computación Cuántica, Universidad de los Andes, Bogotá, Colombia, 2015. The purpose is to introduce additive quantum error correcting codes, with emphasis on the use of binary representation of Pauli matrices and modules over a translation group algebra. The topics include symplectic vector spaces, Cliord group, cleaning lemma, an error correcting criterion, entanglement spectrum, implications of the locality of stabilizer group generators, and the classication of translation-invariant one-dimensional additive codes and two-dimensional CSS codes with large code distances. In particular, we describe an algorithm to find a Cliord quantum circuit (CNOTs) to transform any two-dimensional translation-invariant CSS code on qudits of a prime dimension with code distance being the linear system size, into a tensor product of finitely many copies of the qudit toric code and a product state. Thus, the number of embedded toric codes is the complete invariant of these CSS codes under local Cliord circuits.