Restart Belief: A General Quantum LDPC Decoder

TL;DR

The paper tackles the instability of belief-propagation (BP) decoding for quantum LDPC codes caused by degeneracy, proposing Restart Belief (RB), a general BP-based decoder that leverages branch-and-bound ideas. RB runs an initial root BP, then explores multiple branches by forcing $Z$ errors on qubits with the smallest output LLRs, updating syndromes and re-running BP in parallel until convergence, with early termination rules tied to the code distance $d$ and defect characteristics. Empirical results across diverse QLDPC codes show RB achieving the lowest logical error rates with complexity similar to other BP-based decoders, and RB is demonstrated to be parallelizable and deterministic. The work suggests RB as a practical, hardware-friendly decoder capable of approaching error correction up to the code distance, with potential for scalable deployment on FPGA/ASIC architectures.

Abstract

Hardware-friendly quantum low-density parity-check (QLDPC) decoders are commonly built upon belief propagation (BP) processing. Yet, quantum degeneracy often prevents BP from achieving reliable convergence. To overcome this fundamental limitation, we propose the restart belief (RB) decoder, an iterative BP-based algorithm inspired by branch-and-bound optimization principles. From our analysis we find that the RB decoder represents both the fastest and most accurate decoding algorithm applicable to QLDPC codes to date, conceived with the explicit goal of approaching error correction up to the code distance.

Figures (1)

• Figure 1: Codeword error rate vs. physical error rate varying codes and decoders. QLDPC code classes from left to right: bivariate bicycle, topological, hypergraph product, and generalized bicycle.