Algebraic Methods for Quantum Codes on Lattices

TL;DR

This paper introduces algebraic methods for analyzing quantum error-correcting codes on lattices, focusing on translation-invariant codes, their classification, and an algorithm to transform them into toric codes.

Contribution

It presents a novel algebraic framework for classifying translation-invariant quantum codes and an algorithm to convert certain CSS codes into tensor products of toric codes.

Findings

Classification of translation-invariant quantum codes

Algorithm to transform CSS codes into toric codes

Number of embedded toric codes as a complete invariant

Abstract

This is a note from a series of lectures at Encuentro Colombiano de Computacion Cuantica, Universidad de los Andes, Bogota, 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.