An unreviewed preprint posted to arXiv proposes approximate maximum-likelihood decoding (AMLD), a framework designed to close the performance gap between the tractable but suboptimal minimum-weight decoder (MWD) and the theoretically optimal but #P-hard maximum-likelihood decoder (MLD) for stabilizer codes under Pauli noise. The central idea is to repurpose the candidate recovery operators that stochastic inner decoders already generate — and ordinarily discard — to construct per-logical-class truncated free-energy estimates. The authors prove these estimates are bounded below by the exact free energy and above by the empirical minimum weight, giving the method a well-defined relationship to both extremes.
In benchmarks using a simulated-annealing Ising decoder on toric and color codes under bit-flip and depolarizing noise, AMLD reportedly closes up to 83% of the threshold gap between MWD and MLD. The largest single improvement the paper claims is on the 6.6.6 color code under depolarizing noise, where the threshold shifts from 17.28% to 18.62%. The authors also apply AMLD to the [[144,12,12]] bivariate-bicycle code — a hypergraph-structure problem — using a BP-OSD candidate pool, reporting a 13% reduction in logical error rate at physical error rate p = 0.05 relative to MWD on the same pool, without requiring matching-based enumeration or code-specific tensor-network contraction.
Because AMLD wraps around existing stochastic solvers as a black-box post-processing step with linear classical overhead, the authors present it as broadly applicable across decoder architectures. The results are promising, but as with all arXiv preprints this work has not yet undergone peer review, and independent replication of the threshold and error-rate claims remains to be established.