Improved single-shot decoding of higher dimensional homological product codes

In this work we study the single-shot performance of higher dimensional homological product codes decoded using belief-propagation and ordered-statistics decoding. 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 thresholds for these codes. For the 3D toric code and a phenomenological noise model, we find a sustainable threshold of 7.08%, compared to the threshold of 2.90% previously found using a two-stage decoder~[Quintavalle et al., 2021]. For the 4D toric code, we observe a sustainable single-shot threshold of 4.29%. We also explore the performance of other product codes that generalise beyond the toric code, including some 4D homological product codes which we show lead to a significant reduction in qubit overhead compared the surface code for phenomenological error rates as high as 1%.