From Noise Beliefs to Logical Decisions: Surface-Code Decoding with Temporal Side Information

Noise in quantum processors has memory: defects drift, error rates switch between regimes, and bursts persist over many syndrome rounds. A decoder that tracks this hidden process should make better corrections, but in continuous operation it must commit each correction before the full record is available. We ask when temporal memory actually changes a logical correction under this constraint. Our decoder couples a causal two-state hidden Markov filter over auxiliary observations to rolling minimum-weight matching: the filter's belief sets the edge weights, and a carried residual-syndrome frontier keeps every committed correction consistent with the detector record. Because the filter, the physical hypotheses, and the commit rule are separate modules, each can be compared against matched controls. Across 45 noise conditions and 92,160 trajectories, filtering reduces state-estimation error by up to 93% and reduces exactly to the current-observation rule under IID noise. In a held-out confirmation with 4,096 rolling streams per code distance, the filter lowers logical failures from 507 to 433 at $d=3$ and from 158 to 131 at $d=5$ relative to current-only calibration of the same decoder (Holm-adjusted $p=.0002$ and $.0049$). Three independently trained GRUs match these error counts but spend 13–15$\times$ longer per belief update. On shared terminal records the belief reduces failures by 6-9% for each of four solver families, and fixed-budget candidate interventions show that better physical hypotheses can lower the error rate even while support coverage shrinks. A replay under frozen calibrations from two cloud quantum processors (6,144 returned shots) preserves the same ordering.