The correlated matching decoder for the 4.8.8 color code

TL;DR

The paper introduces a correlated MWPM decoder for the 4.8.8 color code by leveraging correlations between restricted lattices and a surface-code mapping. This approach yields higher decoding thresholds than existing restricted decoders and closely matches the unified decoder at very low physical error rates, under both code-capacity and phenomenological noise models ($p_{th}^{cc,code}=10.38\%$, $p_{th}^{cc,phen}=3.13\%$; surface-code analogues: $p_{th}^{surface,code}=16.62\%$, $p_{th}^{surface,phen}=3.52\%$). The method preserves the low overhead of restricted decoders while improving decoding performance via a two-stage, correlated matching procedure and boundary-weight adjustments to mitigate edge-case failures. The work clarifies why weight-consistency matters in MWPM for color codes and demonstrates a practical, scalable decoder with potential applicability to other code families and noise models.

Abstract

Color codes present distinct advantages for fault-tolerant quantum computing, such as high encoding rates and the transversal implementation of Clifford gates. However, existing matching-based decoders for the color codes such as the restricted decoder (Kubica and Delfosse, 2023), suffer from limited decoding performance. Inspired by the global decoding insight of the unified decoder (Benhemou et al., 2023), this paper introduces a correlated decoder for the 4.8.8 color code, which improves upon the conventional restricted decoder by leveraging correlations between restricted lattices, and is derived by mapping the correlated matching decoder for the surface code onto the color code lattice. Analytical and numerical results show that the correlated decoder achieves higher thresholds than the restricted and unified decoders, while matching the performance of the unified decoder at very low physical error rates. Under the code capacity and phenomenological noise models, the estimated thresholds for the color code against bit-flip error are 10.38% and 3.13%, respectively. Furthermore, by applying the surface-color code mapping, the thresholds of 16.62% and 3.52% are obtained for the surface code against depolarizing noise.

Figures (11)

• Figure 1: The surface-color code mapping. (a) The color code can be obtained by encoding physical qubits of a pair of surface codes (blue and green) using $\llbracket 4,2,2 \rrbracket$ codes. (b) Depolarizing errors on the blue and green surface codes are mapped to the two-qubit bit-flip errors on the color code (i.e., logical $X$ errors on the $\llbracket 4,2,2 \rrbracket$ codes). We emphasize that $\overline{X}_1\equiv \overline{Z}_2$ and $\overline{Z}_1\equiv \overline{X}_2$ with a Hadamard rotation under the surface code duplication. (c) The single-qubit depolarizing errors with their syndromes on the surface codes. (d) Stabilizers and logical operators of the $\llbracket 4,2,2 \rrbracket$ code defined on a square with qubits on its vertices. (e,f) For blue surface code (e), each physical qubit is encoded into the first logical qubit of a $\llbracket 4,2,2 \rrbracket$ code. The $X$ ($Z$) checks of the blue surface code are mapped to blue (green) octagonal $X$ ($Z$) checks of the color code. A similar mapping holds for the green surface code (f) after a transversal Hadamard rotation.
• Figure 2: The restricted and correlated decoders for the color code can be derived from their surface code counterparts via the color-surface mapping. (a) The independent $X/Z$ decoder (i) and the restricted decoder (ii) exhibit an analogous limitation: while decoding the degenerate weight-$O(d/2)$ error patterns (containing $Y$ or diagonal errors), they overestimate the matching weight of the correct path, results in a success probability of only 1/2. (b) This issue is resolved by the correlated decoders for both codes (i, ii) by setting the weight of edges identified in the first matching to zero during the second matching. This ensures that the predicted error weight is consistent with the total matching weight, thus achieving more accurate decoding.
• Figure 3: Matching graph of the correlated decoder. Dangling edges are connected to a virtual boundary check, defined as the product of all the checks. (a) In $\mathcal{R}_\textbf{b}$ lattice, edges adjacent to the top and bottom red checks are assigned a weight of 0.999 to handle boundary errors, while others have weight 1. (b) In $\mathcal{R}_\textbf{g}$ lattice, edge weights are set to 0 or 1 depends on the decoding results from the $\mathcal{R}_\textbf{g}$ lattice.
• Figure 4: The space-time matching graph for $r=3$ rounds under the phenomenological noise model. Time progresses upward from the bottom, with each horizontal plane corresponding to a round of stabilizer measurements. The vertical dashed edges represent space-time locations where measurement errors may occur.
• Figure 5: Comparative difficulty of correcting errors on the boundary versus the interior of the lattice. Errors in the bulk (middle) are correctable by the correlated decoder, whereas identical error configurations on the boundary have a probability 1/2 of logical failure.