Efficient near-optimal decoding of the surface code through ensembling

TL;DR

The paper tackles the challenge of fast yet highly accurate quantum error decoding by introducing Harmonization, an ensemble method that perturbs the priors of correlated MWPM decoders and pools their outputs. By generating diverse ensemble members and employing pooling strategies, the approach approaches maximum-likelihood performance on both repetition and surface codes, and supports a layered decoding scheme that preserves accuracy while reducing overhead. Empirical results against tensor-network ML decoders show near-ML performance across circuit-level and phenomenological noise models, with layered decoding achieving most gains at a modest first-pass size. The work provides a practical pathway to real-time, high-accuracy decoding in quantum fault tolerance and suggests broader applicability to other decoders and codes.

Abstract

We introduce harmonization, an ensembling method that combines several "noisy" decoders to generate highly accurate decoding predictions. Harmonized ensembles of MWPM-based decoders achieve lower logical error rates than their individual counterparts on repetition and surface code benchmarks, approaching maximum-likelihood accuracy at large ensemble sizes. We can use the degree of consensus among the ensemble as a confidence measure for a layered decoding scheme, in which a small ensemble flags high-risk cases to be checked by a larger, more accurate ensemble. This layered scheme can realize the accuracy improvements of large ensembles with a relatively small constant factor of computational overhead. We conclude that harmonization provides a viable path towards highly accurate real-time decoding.

Figures (12)

• Figure 1: A toy correlated matching example. The error hypergraph is on the left, with the erring hyperedges and detection events shaded red, and with all edges having equal weight. First, we perform a minimum-weight perfect matching (blue edges) that misidentifies the error, in this case guessing the wrong of two equally valid choices. Second, we use the matching from the first step to infer that the hyperedge error mechanism has likely activated, and shorten the complementary edge. Third, we perform minimum-weight perfect matching on this reweighted graph, correctly identifying the error.
• Figure 2: Different pooling methods at distance-7 for circuit-level noise in the repetition code at $p=0.05$ (left) and the surface code at $p=0.008$ (right). In the repetition code, minimum-weight perfect matching is a most-likely error decoder, which most-likely-error pooling quickly converges to. We observe that vote pooling, which takes into account degeneracy, is the most accurate pooling method. For the surface code, most-likely-error pooling appears to be at least as effective as the other pooling methods once the ensemble is sufficiently large. This suggests that degeneracy plays a comparatively small role in increasing accuracy within the surface code.
• Figure 3: On the left, an error hypergraph with a set of detection events shaded red and the edges chosen by a minimum-weight perfect matching decoder overlaid in blue. On the right, the corresponding edge decomposition graph, with nodes corresponding to chosen edges colored blue. MWPM-based decoding on the edge decomposition graph gives the most likely set of errors, overlaid in green, consistent with the edge set chosen by the matching decoder. In this case, it selects the far left edge-like error mechanism, as well as the only hyperedge-like error mechanism.
• Figure 4: Decoding circuit-level noise in the repetition code at $p=0.05$. At each distance there are three lines: minimum-weight matching (dotted), vote pooled ensembled matching (circles), and TNML decoding (dashed). We observe there is a relatively small difference between the most-likely error decoder and the TNML decoder, and this difference is mostly traversed at larger ensemble sizes. This is explained by the ensembled decoder accounting for degeneracy.
• Figure 5: Decoding phenomenological noise in the surface code at $p=0.04$. At each distance there are three lines: correlated matching (dotted), most-likely-error pooled ensembled matching (circles), and TNML (dashed). We observe that maximum-likelihood pooled MWPM is nearly as accurate as TNML once the ensemble size is large. As the distance increases, the gap between maximum-likelihood pooled MWPM and TNML increases slightly.