Sequential BP-based Decoding of QLDPC Codes
Mohsen Moradi, Salman Habib, Vahid Nourozi, David G. M. Mitchell
TL;DR
This work tackles the challenge of decoding quantum LDPC codes with belief propagation, where conventional BP struggles due to short cycles and degeneracy. It introduces two sequential update schemes—Sequential CN Scheduling ($SCNS$) and Sequential VN Scheduling ($SVNS$)—and integrates them with Belief Propagation Guided Decimation to form SBPGD, achieving faster convergence and lower error rates without changing the code. On standard QLDPC benchmarks under the Pauli-$X$ noise model, the sequential schedules significantly improve the block error rate over flooding BP, with SVNS-BP and SCNS-BP sometimes surpassing post-processing alternatives at similar complexity, and SBPGD reducing the number of decimation rounds. The results demonstrate that updating order alone can markedly enhance both reliability and efficiency of BP-based decoding, offering a practical route toward more scalable quantum fault-tolerance implementations; future work includes schedule optimization and extensions to broader noise models and code families.
Abstract
Quantum low-density parity-check (QLDPC) codes are a leading approach to quantum error correction, yet conventional belief propagation (BP) decoders often perform poorly, primarily due to non-convergence exacerbated by stabilizer constraints, which induce short cycles and degeneracy. We propose two scheduling variants, sequential check node scheduling (SCNS) and sequential variable node scheduling (SVNS), that improve BP's error-correction ability by processing check nodes (CNs) or variable nodes (VNs), respectively, in a fixed order, stabilizing message updates and reducing stalls. We also employ this technique to an improved BP-variant called BP guided decimation (BPGD), where symbols are progressively fixed during decoding iterations. Here, we demonstrate that the sequential BPGD (SBPGD) decoder can further improve the convergence properties and performance of the decoder. On standard QLDPC benchmarks under a Pauli-X noise model, our sequential schedules are shown to lower the block error rate relative to conventional BP, and SBPGD outperforms BPGD while using significantly fewer decimation rounds, translating to lower computational cost. These results demonstrate that changing the update schedule, without altering the code, can improve both the reliability and efficiency of BP-based decoding for QLDPC codes. For the [[1922,50,16]] C2 hypergraph-product code with independent X errors, SVNS-BP surpasses BP-OSD-0 in error correction at roughly the same complexity as standard BP.