Abstract: To address the limited search space and local-optimum issue of conventional belief propagationordered statistics decoding (BP-OSD) under a single normalization factor in quantum error-correcting codes, the paper proposes a two-stage multi-factor search BP-OSD algorithm. In Stage I, the normalized min-sum (NMS) algorithm is executed over a set of candidate normalization factors, along with syndrome checking. If a decoding result consistent with the measured syndrome is found, all decoding is terminated and the result is output directly. If no consistent solution is obtained, the algorithm proceeds to the Stage II. In Stage II, according to the most reliable basis-reliability metric, the M' most reliable soft outputs are selected from the M soft outputs associated with the candidate normalization factors to perform OSD, and the final estimate is determined according to the minimum Hamming-weight criterion applied to the error pattern. This design broadens the exploration of the posterior space using only a few BP runs and a few OSD invocations, thereby avoiding the high computational complexity and latency associated with applying OSD to all M soft outputs. Numerical results demonstrate that: 1) for Surface codes, the proposed method consistently outperforms minimum-weight perfect matching and conventional BP decoders across the entire range of physical error rates, significantly reducing the logical error probabilities and increasing the threshold from approximately 15.5% to approximately 16.7%; 2) for quantum low-density parity-check codes, the proposed method achieves notable performance gains over baseline BP and standard BP-OSD, yielding roughly an order-of-magnitude reduction in logical error rate.

Key words: quantum error correction, Surface codes, quantum low-density parity-check codes, belief propagation-ordered statistics decoding algorithm