Is A Quantum Stabilizer Code Degenerate or Nondegenerate for Pauli Channel?

TL;DR

This paper investigates whether quantum stabilizer codes are degenerate or nondegenerate for Pauli channels, introducing new criteria and methods to analyze their error syndromes and minimum distance, and demonstrating performance advantages of degenerate codes.

Contribution

It provides necessary and sufficient conditions for degeneracy of stabilizer codes over GF(2) and proposes a new approach to determine their minimum distance, highlighting the superior performance of degenerate codes.

Findings

Degenerate quantum codes outperform or match nondegenerate codes for Pauli channels.

Error syndrome matrices are determined by check matrices, similar to classical codes.

A new method to find the minimum distance of quantum stabilizer codes was developed.

Abstract

Mapping an error syndrome to the error operator is the core of quantum decoding network and is also the key step of recovery. The definitions of the bit-flip error syndrome matrix and the phase-flip error syndrome matrix were presented, and then the error syndromes of quantum errors were expressed in terms of the columns of the bit-flip error syndrome matrix and the phase-flip error syndrome matrix. It also showed that the error syndrome matrices of a stabilizer code are determined by its check matrix, which is similar to the classical case. So, the error-detection and recovery techniques of classical linear codes can be applied to quantum stabilizer codes after some modifications. Some necessary and/or sufficient conditions for the stabilizer code over GF(2) is degenerate or nondegenerate for Pauli channel based on the relationship between the error syndrome matrices and the check matrix was presented. A new way to find the minimum distance of the quantum stabilizer codes based on their check matrices was presented, and followed from which we proved that the performance of degenerate quantum code outperform (at least have the same performance) nondegenerate quantum code for Pauli channel.