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arXiv:2512.13436 · Physical Review A

Decoding 3D color codes with boundaries

Friederike Butt, Lars Esser, Markus Müller

quant-phSubmitted Dec 15, 2025• Announced Dec 16, 2025• 0 citations
v1 · Dec 15, 2025v2 · Dec 22, 2025
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Abstract

Practical large-scale quantum computation requires both efficient error correction and robust implementation of logical operations. Three-dimensional (3D) color codes are a promising candidate for fault-tolerant quantum computation due to their transversal non-Clifford gates, but efficient decoding remains challenging. In this work, we extend previous decoders for two-dimensional color codes [1], which are based on the restriction of the decoding problem to a subset of the qubit lattice, to three dimensions. Including boundaries of 3D color codes, we demonstrate that the 3D restriction decoder achieves optimal scaling of the logical error rate and a threshold value of 1.55(6)% for code-capacity bit- and phase-flip noise, which is almost a factor of two higher than previously reported for this family of codes [2, 3]. We furthermore present qCodePlot3D, a Python package for visualizing 2D and 3D color codes, error configurations, and decoding paths, which supports the development and analysis of such decoders. These advancements contribute to making 3D color codes a more practical option for exploring fault-tolerant quantum computation.

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14 pages, 6 figures

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Practical large-scale quantum computation requires both efficient error correction and robust implementation of logical operations. Three-dimensional (3D) color codes are a promising candidate for fault-tolerant quantum computation due to their transversal non-Clifford gates, but efficient decoding remains challenging. In this work, we extend previous decoders for two-dimensional color codes [1], which are based on the restriction of the decoding problem to a subset of the qubit lattice, to three dimensions. Including boundaries of 3D color codes, we demonstrate that the 3D restriction decoder achieves optimal scaling of the logical error rate and a threshold value of 1.55(6)% for code-capacity bit- and phase-flip noise, which is almost a factor of two higher than previously reported for this family of codes [2, 3]. We furthermore present qCodePlot3D, a Python package for visualizing 2D and 3D color codes, error configurations, and decoding paths, which supports the development and analysis of such decoders. These advancements contribute to making 3D color codes a more practical option for exploring fault-tolerant quantum computation.
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We introduce an efficient decoder of the color code in d ≥ 2 dimensions, the Restriction Decoder, which uses any d -dimensional toric code decoder combined with a local lifting procedure to find a recovery operation. We prove that the Restriction Decoder successfully corrects errors in the color code if and only if the corresponding toric code decoding succeeds. We also numerically estimate the Restriction Decoder threshold for the color code in two and three dimensions against the bit-flip and phase-flip noise with perfect syndrome extraction. We report that the 2D color code threshold p 2D ≈ 10.2 % on the square-octagon lattice is on a par with the toric code threshold on the square lattice.
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.
Color codes are a promising topological code for fault-tolerant quantum computing. Insufficient research on color codes has delayed their practical application. In this work, we address several key issues to facilitate practical fault-tolerant quantum computing based on color codes. First, by introducing decoding graphs with error-rate-related weights, we obtain the threshold of 0.47 % of the 6.6.6 triangular color code under the standard circuit-level noise model, narrowing the gap to that of the surface code. Second, our work first investigates the circuit-level decoding of color code lattice surgery, then gives an efficient decoding algorithm, which is crucial to perform logical operations in a quantum computer with two-dimensional architectures. Last, a state injection protocol of the triangular color code is proposed, reducing the output magic state error rate in one round of 15 to 1 distillation by two orders of magnitude compared to a previous rough protocol. We also prove that our protocol offers the lowest logical error rates for state injection among all possible codes. Published by the American Physical Society 2024
Quantum error correction enables scalable fault-tolerant quantum computation by suppressing logical errors exponentially with code distance once physical error rates fall below a critical threshold. Here we demonstrate below-threshold operation of triangular color codes with distances d=3,5,d=3,5,d=3,5, and 777 on Quantinuum's Helios trapped-ion quantum processor. Leveraging both circuit- and hardware- level leakage reduction schemes, we achieve logical error rates as low as 5−2+4×10−65_{-2}^{+4}\times 10^{-6}5−2+4​×10−6, and demonstrate error suppression by over an order of magnitude, Λ=12−2+3Λ=12_{-2}^{+3}Λ=12−2+3​, with each successive increase in code distance. Moreover, we demonstrate below threshold scaling of error rates for Clifford two-qubit randomized benchmarking circuits when encoded in d=3,5d=3,5d=3,5 color codes, achieving logical error rate of 7−3+3×10−57^{+3}_{-3}\times 10^{-5}7−3+3​×10−5 per logical two-qubit Clifford at distance 555 with a suppression factor of Λ=9−3+8Λ=9_{-3}^{+8}Λ=9−3+8​. We compare logical performance for existing near-optimal algorithmic decoders, and both convolutional neural network and transformer-based neural network decoders capable of real-time decoding an order of magnitude faster than the syndrome extraction rate. These results approach the logical error rates required for megaquop-scale quantum computation and indicate that scaled trapped-ion processors could reach application-relevant performance with substantially smaller code blocks than indicated by previous results in other qubit modalities.