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

2026-01-13

AI Summary

This work addresses the trade-off between error-correction capability and implementation complexity in quantum low-density parity-check (LDPC) codes, as well as the detrimental impact of short cycles—particularly 4-cycles—on belief propagation (BP) decoding performance. To this end, the authors propose a construction method based on quasi-bicyclic matrix algebras. The approach first generates a classical LDPC code with girth six, then embeds it into a CSS framework such that the two parity-check matrices satisfy the compatibility conditions required by the CAMEL-ensemble quaternary BP decoder. Notably, all unavoidable 4-cycles are concentrated onto a single variable node for the first time, enabling their effective suppression through local elimination. The resulting quantum CSS LDPC codes achieve high girth, low decoding complexity, and superior error-correction performance.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Quantum LDPC codes
• CSS codes
• belief propagation decoding
• dyadic matrices
• girth-4 cycles

Innovation

• dyadic matrices
• QLDPC codes
• CSS construction
• belief propagation decoding
• girth-4 cycle mitigation