On the Fourier Transform Approach to Quantum Error Control

TL;DR

This paper explores a Fourier transform framework for analyzing various quantum error-correcting codes, including new classes beyond Clifford codes, characterizing their error detection capabilities and providing examples and computational results.

Contribution

It introduces a Fourier transform approach to analyze and construct quantum codes, extending beyond traditional Clifford codes to more general classes.

Findings

Characterization of error detection in new code classes

Construction of example codes using the framework

Computer search results for code properties

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. These are analyzed in a framework using the Fourier transform on finite groups, the finite group in question being a subgroup of the quantum error group considered. All the classes of codes that can be obtained in this framework are explored, including codes more general than Clifford codes. The error detection properties of one of these more general classes ("direct sums of translates of Clifford codes") are characterized. Examples codes are constructed, and computer code search results presented and analysed.