Approximate maximum likelihood decoding with $K$ minimum weight matchings

The minimum weight matching (MWM) and maximum likelihood decoding (MLD) are two widely used and distinct decoding strategies for quantum error correction. For a given syndrome, the MWM decoder finds the most probable physical error corresponding to the MWM of the decoding graph, whereas MLD aims to find the most probable logical error. Although MLD is the optimal error correction strategy, it is typically more computationally expensive compared to the MWM decoder. In this work, we introduce an algorithm that approximates MLD with $K$ MWMs from the decoding graph. Taking the surface code subject to graphlike errors as an example, we show that it is possible to efficiently find the first $K$ MWMs by systematically modifying the original decoding graph followed by finding the MWMs of the modified graphs. For the case where the $X$ and $Z$ errors are correlated, despite the MWM of the decoding hypergraph cannot be found efficiently, we present a heuristic approach to approximate the MLD by finding the $K$ MWMs in the $X$ and $Z$ subgraphs. We benchmark the efficacy of our algorithm for the surface code subject to graphlike errors, the surface-square Gottesman-Kitaev-Preskill (GKP) code and surface-hexagonal GKP code subject to the Gaussian random displacement errors, showing that the fidelity approaches that of the exact MLD (for the first two cases) or the tensor-network decoder (for the last case) as $K$ increases.