Abstract:
This article proposes a novel low-complexity syndrome-based linear programming (SB-LP) decoding algorithm for decoding quantum low-density parity-check codes. Under the code-capacity model, the SB-LP decoder can be used as a standalone decoder; however, it is particularly powerful when used as a postprocessing step following SB min-sum (SB-MS) decoding. In the latter case, the proposed decoder is shown to be capable of significantly reducing the error floor of the SB-MS decoder for both flooded and layered SB-MS scheduling. Also, an early stopping criterion is introduced to decide when to activate the SB-LP algorithm, avoiding executing a predefined maximum number of iterations for the SB-MS decoder. Simulation results show, for some example hypergraph and generalized bicycle (GB) codes, that the proposed decoder can lower the error floor by one to three orders of magnitude compared to SB-MS for the same total number of decoding iterations. Furthermore, for the class of GB codes, it is shown that as the minimum distance of the code increases, the logical error rate provided by the proposed decoder also improves, indicating that the solution is scalable. Under the circuit-level noise model, it is shown that while the SB-LP decoder does not fully replace the need for ordered statistics decoding (OSD) when flooded SB-MS is used as a preliminary step, it reduces the number of calls to the OSD postprocessor, which directly impacts the overall latency. In addition, the method offers a syndrome-matching decoder and improves the accuracy of the logical error rate for bivariate bicycle codes of distances 6 to 18, particularly at low error rates, when compared to the belief propagation+OSD benchmark.