Bias-Aware BP Decoding of Quantum Codes via Directional Degeneracy

TL;DR

This paper introduces a bias-aware belief propagation decoding method for quantum CSS codes that leverages directional degeneracy to improve error correction performance, demonstrating significant error-rate reductions in simulations.

Contribution

It formalizes directional degeneracy in quantum codes, integrating anisotropic priors into BP decoding without altering code design, and provides theoretical bounds and practical improvements.

Findings

Significant logical error-rate reductions observed in simulations.

Directional degeneracy bounds relate to code parameters.

Method enhances hardware-aware decoding performance.

Abstract

We study directionally informed belief propagation (BP) decoding for quantum CSS codes, where anisotropic Tanner-graph structure and biased noise concentrate degeneracy along preferred directions. We formalize this by placing orientation weights on Tanner-graph edges, aggregating them into per-qubit directional weights, and defining a \emph{directional degeneracy enumerator} that summarizes how degeneracy concentrates along those directions. A single bias parameter~$\beta$ maps these weights into site-dependent log-likelihood ratios (LLRs), yielding anisotropic priors that plug directly into standard BP$\rightarrow$OSD decoders without changing the code construction. We derive bounds relating directional and Hamming distances, upper bound the number of degenerate error classes per syndrome as a function of distance, rate, and directional bias, and give a MacWilliams-type expression for the directional enumerator. Finite-length simulations under code-capacity noise show significant logical error-rate reductions -- often an order of magnitude at moderate physical error rates -- confirming that modest anisotropy is a simple and effective route to hardware-aware decoding gains.