List Decoding and New Bicycle Code Constructions for Quantum LDPC Codes

TL;DR

This paper introduces a new decoding algorithm, MBBP-LD, for quantum LDPC codes that improves error rates while maintaining linear complexity, and explores new bicycle code subclasses with promising performance.

Contribution

It proposes the MBBP-LD decoder for QLDPC codes with improved error rates and introduces univariate bicycle codes with reduced complexity.

Findings

MBBP-LD achieves up to 40% lower logical error rate for 0144,12,12 codes.

MBBP-LD maintains linear-time complexity of standard BP decoding.

Univariate bicycle codes have lower polynomial search space and promising simulation results.

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.