Quantum CSS LDPC Codes based on Dyadic Matrices for Belief Propagation-based Decoding

TL;DR

This paper introduces a novel algebraic construction of quantum LDPC codes using dyadic matrices, enhancing belief propagation decoding by mitigating short cycles in Tanner graphs for improved quantum error correction.

Contribution

It presents a new dyadic matrix-based method for designing quantum LDPC codes compatible with advanced belief propagation decoders, improving error correction performance.

Findings

Classical quasi-dyadic LDPC codes with girth 6 generated

Quantum CSS LDPC codes constructed with compatible parity-check matrices

Enhanced decoding capability through cycle mitigation in Tanner graphs

Abstract

Quantum low-density parity-check (QLDPC) codes provide a practical balance between error-correction capability and implementation complexity in quantum error correction (QEC). In this paper, we propose an algebraic construction based on dyadic matrices for designing both classical and quantum LDPC codes. The method first generates classical binary quasi-dyadic LDPC codes whose Tanner graphs have girth 6. It is then extended to the Calderbank-Shor-Steane (CSS) framework, where the two component parity-check matrices are built to satisfy the compatibility condition required by the recently introduced CAMEL-ensemble quaternary belief propagation decoder. This compatibility condition ensures that all unavoidable cycles of length 4 are assembled in a single variable node, allowing the mitigation of their detrimental effects by decimating that variable node.