The Fourier Transform Approach to Quantum Coding

TL;DR

This paper introduces a Fourier transform-based framework for analyzing and constructing quantum codes, including stabilizer, Clifford, and more general codes, providing new insights into their error detection capabilities.

Contribution

It develops a Fourier inversion approach to quantum coding, enabling the construction and analysis of new code classes from group algebra idempotents.

Findings

Derived conditions for error detection in code sums of Clifford translates

Analyzed codes from non-Abelian error groups

Proposed a unified Fourier-based framework for quantum code analysis

Abstract

Quantum codes are subspaces of the state space of a quantum system that are used to protect quantum information. Some common classes of quantum codes are stabilizer (or additive) codes, non-stabilizer (or non-additive) codes obtained from stabilizer codes, and Clifford codes. We analyze these in the framework of the Fourier inversion formula on a group algebra, the group in question being a subgroup of the error group considered. We study other possible code spaces that may be obtained via such an approach, obtaining codes that are the direct sums of translates of Clifford codes, and more general codes obtained from idempotents in the transform domain. We derive necessary and sufficient conditions for error detection by direct sums of translates of Clifford codes, and provide an example using an error group with non-Abelian index group.