List Decoding and New Bicycle Code Constructions for Quantum LDPC Codes

2025-11-04

AI Summary

To address the trade-off between high logical error rates and low decoding efficiency in quantum low-density parity-check (QLDPC) codes, this paper proposes the Multi-Basis Belief Propagation List Decoder (MBBP-LD), achieving significantly improved error correction performance while retaining linear-time complexity. Our key contributions are: (1) extending the MBBP framework with a novel list post-processing decision rule based on weighted log-likelihood ratios, outperforming conventional minimum-metric selection; (2) constructing single-variable bicycle (UB) codes by constraining check polynomial structures, thereby reducing search space and enhancing both encoding/decoding efficiency and code distance properties; and (3) integrating ordered statistics decoding (OSD) assistance with low-weight check construction. On the [[144,12,12]] QLDPC code, MBBP-LD achieves up to a 40% reduction in logical error rate compared to BP-OSD, and demonstrates superior robustness and generalization across diverse belief propagation–based decoders.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Security and Privacy: Large-scale security measurements

Abstract

In this paper, we propose a new decoder, called the Multiple-Bases Belief-Propagation List Decoder (MBBP-LD), for Quantum Low-Density Parity-Check (QLDPC) codes. It extends the Multiple-Bases Belief-Propagation (MBBP) framework, originally developed for classical cyclic LDPC codes. The proposed method preserves the linear-time complexity of standard BP decoder while improving the logical error rate. To further reduce the logical error rate, a new decision rule is introduced for the post-processing list decoder, outperforming the conventional least-metric selector (LMS) criterion. For the recently developed and implemented bivariate bicycle (BB) code with parameters ([[144,12,12]]), our proposed MBBP-LD decoder achieves up to 40% lower logical error rate compared to the state-of-the-art decoder for short QLDPC codes, i.e., BP with ordered-statistics decoding (BP-OSD), while retaining the linear-time complexity of the plain BP decoder. In addition, we explore a new subclass of BB codes, that we refer to as the univariate bicycle (UB) codes, specifically with lower-weight parity checks ((w=6,8)). This reduces the polynomial search space for the code compared to general BB codes, i.e., by reducing the search space over two polynomial components in BB codes to just a single polynomial component in UB codes. Simulations demonstrate the promising performance of these codes under various types of BP decoders.

Problem

• Developing a list decoder for quantum LDPC codes with improved error rates
• Introducing a new decision rule to enhance post-processing list decoding
• Exploring univariate bicycle codes with reduced polynomial search space

Innovation

• Multiple-Bases Belief-Propagation List Decoder for QLDPC codes
• New decision rule improves logical error rate
• Univariate bicycle codes reduce polynomial search space