Improved Single-Shot Decoding of Higher-Dimensional Hypergraph-Product Codes

In this work, we study the single-shot performance of higher-dimensional hypergraph product codes decoded using belief propagation and ordered-statistics decoding P. Panteleev and G. Kalachev, Quantum 5, 585 (2021). We find that decoding data-qubit and syndrome-measurement errors together in a single stage leads to single-shot thresholds that greatly exceed all previously observed single-shot thresholds for these codes. For the three-dimensional toric code and a phenomenological noise model, our results are consistent with a sustainable threshold of 7.1% for Z errors, compared to the threshold of 2.90% previously found using a two-stage decoder Quintavalle et al., 2021. For the four-dimensional (4D) toric code, for which both X and Z error correction is single shot, our results are consistent with a sustainable single-shot threshold of 4.3%, which is even higher than the threshold of 2.93% for the two-dimensional toric code for the same noise model but using L rounds of stabilizer measurement. We also explore the performance of balanced-product and 4D hypergraph-product codes, which we show lead to a reduction in qubit overhead compared the surface code for phenomenological error rates as high as 1%.