A Construction of Quantum Stabilizer Codes Based on Syndrome Assignment by Classical Parity-Check Matrices

TL;DR

This paper introduces a new method for constructing quantum stabilizer codes using classical parity-check matrices, enhancing error correction capabilities and efficiency, and includes specific code classes like Reed-Muller and quadratic residue codes.

Contribution

The paper presents a simple syndrome assignment-based construction of quantum stabilizer codes from classical parity-check matrices, expanding the set of correctable errors and achieving near-optimal parameters.

Findings

Constructed quantum Reed-Muller codes with improved error correction.

Developed quantum codes inspired by classical quadratic residue codes, some optimal.

Achieved coding efficiency comparable to CSS codes.

Abstract

In quantum coding theory, stabilizer codes are probably the most important class of quantum codes. They are regarded as the quantum analogue of the classical linear codes and the properties of stabilizer codes have been carefully studied in the literature. In this paper, a new but simple construction of stabilizer codes is proposed based on syndrome assignment by classical parity-check matrices. This method reduces the construction of quantum stabilizer codes to the construction of classical parity-check matrices that satisfy a specific commutative condition. The quantum stabilizer codes from this construction have a larger set of correctable error operators than expected. Its (asymptotic) coding efficiency is comparable to that of CSS codes. A class of quantum Reed-Muller codes is constructed, which have a larger set of correctable error operators than that of the quantum Reed-Muller codes developed previously in the literature. Quantum stabilizer codes inspired by classical quadratic residue codes are also constructed and some of which are optimal in terms of their coding parameters.