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 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.

• 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