Approximate maximum likelihood decoding with $K$ minimum weight matchings

TL;DR

The paper introduces the $K$-MWM decoder, a deterministic method to approximate maximum-likelihood decoding by enumerating the first $K$ minimum-weight matchings on decoding graphs. It develops a tree-based framework to efficiently generate successive MWMs, and extends the approach to correlated errors and GKP-based codes by leveraging lattice coset structures. Numerical results across surface codes and GKP codes show fidelity improvements toward optimal MLD with increasing $K$, sometimes matching tensor-network decoders for certain codes. The work highlights a scalable, tunable trade-off between decoding accuracy and compute, with broad applicability to graphlike, correlated, and continuous-variable quantum codes.

Abstract

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.

Figures (10)

• Figure 1: The MWM decoder for the surface code subject to graphlike errors. (a) The $d=5$ surface code where the black dots represent the data qubits and the $X$-type and $Z$-type stabilizers are shown in orange and green respectively. (b) The model graph for the $Z$-type errors where the white vertices and edges correspond to the $X$-type stabilizers and data qubits of the code respectively. The green circles represent virtual vertices that are all connected via edges with weight zero which are not shown for clarity reason. (c) The decoding graph where the red vertices represent nontrivial stabilizer measurement outcomes, and the weights of the edges are given in Eq. \ref{['eq:weight_e_k']}. (d) An example of MWM which matches the highlighted vertices pairwise with lowest possible weight.
• Figure 2: Illustrations of reduced decoding graphs, and their matchings. (a) Reproduction of the decoding graph $G$ in Fig. \ref{['fig:MWM']}(d) with an ordering of the edges in $\mathcal{M}_1(G)$ included. The set of green edges represents a cycle $\mathcal{C}$, and $\mathcal{C}\cup \mathcal{M}_1(G)$ is one candidate for $\mathcal{M}_2(G)$. (b) An illustration of the reduced decoding graph $G^{(1)}$ defined in Eq. \ref{['eq:def_G_j_1']}-\ref{['eq:def_G_j_2']}, obtained by removing $e_1$ from $G$ while retaining the same set of highlighted vertices. The set of blue edges represents $\mathcal{M}_1(G^{(1)})$, the MWM of the reduced graph $G^{(1)}$, which is another candidate for $\mathcal{M}_2(G)$. (c) An illustration of the reduced decoding graph $G^{(2)}$, which is obtained by removing $e_1$ and $e_2$ from $G$ and flipping the states of vertices in $e_1$. The set of blue edges represents $\mathcal{M}_1(G^{(2)})$, the MWM of the reduced graph $G^{(2)}$. (d) Another candidate for $\mathcal{M}_2(G)$ is obtained by taking the union of $e_1$ and $\mathcal{M}_1(G^{(2)})$, which matches the highlighted vertices in $G$ pairwise.
• Figure 3: An example decoding tree for a decoding graph $G$. The root of the decoding tree is $\mathcal{M}_1(G)$, the MWM of the graph, and all of its descendants are subsequent matchings of the graph. As defined in Eq. \ref{['eq:general_M']}, each node has three pieces of information: the corresponding matching $\mathcal{M}$, the corresponding reduced graph $G'$ and the edges $\mathcal{E}"$ needed for matching completion. To identify the children of a node, which are defined in Eq. \ref{['eq:def_children_M']}, we follow the protocol in Sec. \ref{['sec:Finding the second matching for the syndrome graph']} to find the candidates of $\mathcal{M}_2(G')$ followed by using $\mathcal{E}"$ for matching completion. The number of children of a node is given by the number of candidates of $\mathcal{M}_2(G')$. Each level of the tree consists only of the child nodes from a parent node and the $k$-th level of the tree has at most $|\mathcal{M}_{k-1}(G)|+1$ nodes. The leaf nodes up to the $k$-th level (nodes with no descendants if we cut the tree between the $k$-th and $(k+1)$-th levels) form the set of all the candidates for $\mathcal{M}_k(G)$.
• Figure 4: Results of $K$-MWM decoder for surface-square GKP code. (a) The fidelity for $d=13$ as a function of noise variance $\sigma$. The MLD is approximated with $K=400$ MWMs. The fidelity achieved for various values of $K$, ranging from $K=1$ to $K=400$ are shown for $\sigma=0.596, 0.597, \cdots, 0.607$. Each data point is obtained from $10^6$ Monte-Carlo samples. When finding the first $400$ MWMs, we explored much more candidate MWMs. The fidelity for including all explored MWMs is also shown, which is slightly higher than that without the additional MWMs. The black squares are the optimal fidelity achieved via the BSV decoder with $10^7$ samples. The inset shows the weights of the first 400 MWMs for $d=13$, averaged over 480 Monte-Carlo samples. The weight differences between the matchings become smaller as we increase $K$, which is consistent with the observation from the main plot that the fidelity improvement also becomes smaller as $K$ increases. (b) The minimum value of $K$ needed to achieve decoding inaccuracy $0.7\%$ as a function of distance at $\sigma=0.607$ (in logarithmic scale), which increases slightly faster than exponentially with the distance, agreeing with the fact that exact MLD is in general a NP-hard problem. The inset shows the decoding inaccuracy for various values of $K$ for $d=15$ with the horizontal solid line indicates $0.7\%$ inaccuracy. (c) The runtime needed to achieve decoding inaccuracy $0.7\%$ as a function of distance (in logarithmic scale), which also scales exponentially with $d$ consistent with that found in (b). The inset shows that the runtime scales linearly as a function of the number of MWMs for a given distance, as expected. The stars label the values of $K$ plotted in (b). (d). The accuracy improvement as a function of distance for various values of $K$.
• Figure 5: Threshold behavior of the $K$-MWM decoder. (a) The fidelity for $d=11$ and $d=13$ as a function of noise variance and the number of MWMs used. Error bars represent two standard errors, approximately $0.001$ with $10^6$ samples. The inset shows results obtained with the optimal BSV decoder, where the error bars are approximately $0.0003$ based on $10^7$ samples. (b) Logical fidelities for $d = 13$ and $15$ under the $K$-MWM and BSV decoders.