DOI: https://doi.org/10.15446/recolma.v50n2.62214

Algebraic Methods for Quantum Codes on Lattices

Jeongwan Haah

Massachusetts Institute of Technology, Cambridge, Massachusetts, USA. jwhaah@microsoft.com

Abstract

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, Clifford group, cleaning lemma, an error correcting criterion, entanglement spectrum, implications of the locality of stabilizer group generators, and the classification 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 Clifford 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 Clifford circuits.

Keywords: quantum stabilizer codes, additive codes, symplectic codes, Laurent polynomial ring, toric code, Clifford circuit.

Mathematics Subject Classification: 81P70, 81R05, 12Y05.

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Recibido: julio de 2016 Aceptado: noviembre de 2016