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.
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.
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.
Fault-tolerant quantum computation relies on scaling up quantum error correcting codes in order to suppress the error rate on the encoded quantum states. Topological codes, such as the surface code or color codes, are leading candidates for practical scalable quantum error correction and require efficient and scalable decoders. In this work, we propose and study the efficiency of a decoder for two-dimensional topological color codes on the 4.8.8 lattice (also known as the square-octagon code), by building on the work of Sarvepalli and Raussendorf [Phys. Rev. A 85, 022317 (2012)], for color codes on hexagonal lattices. The decoder is based on a rescaling approach, in which syndrome information on a part of the qubit lattice is processed locally, and then the lattice is rescaled iteratively to smaller sizes. We find a threshold of 6.0% for code capacity noise.
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.
Two-dimensional color codes are a promising candidate for fault-tolerant quantum computing, as they have high encoding rates, transversal implementation of logical Clifford gates, and resource-efficient magic state preparation schemes. However, decoding color codes presents a significant challenge due to their structure, where elementary errors violate three checks instead of just two (a key feature in surface code decoding), and the complexity of extracting syndrome is greater. We introduce an efficient color-code decoder that tackles these issues by combining two matching decoders for each color, generalized to handle circuit-level noise by employing detector error models. We provide comprehensive analyses of the decoder, covering its threshold and sub-threshold scaling both for bit-flip noise with ideal measurements and for circuit-level noise. Our simulations reveal that this decoding strategy nearly reaches the best possible scaling of logical failure ( pfail∼pd/2 ) for both noise models, where p is the noise strength, in the regime of interest for fault-tolerant quantum computing. While its noise thresholds are comparable with other matching-based decoders for color codes ( 8.2% for bit-flip noise and 0.46% for circuit-level noise), the scaling of logical failure rates below threshold significantly outperforms the best matching-based decoders.
We introduce a decoder for the 3D color code with boundaries, which is a variation of the restriction decoder introduced by Kubicka and Delfosse. Specifically, we adapt the lift procedure to efficiently find a correction on qubits adjacent to a boundary. We numerically estimate a threshold of 4%−8% for Pauli X errors and a threshold of 0.7%−0.8% for Pauli Z errors. Our work is a first step towards characterizing the performance of Bombin's recently proposed colorful quantum computing.
We investigate how the lattice geometry influences the error thresholds of two-dimensional toric codes. In the presence of bit- or phase-flip errors, the toric code maps onto the two-dimensional random-bond Ising model (RBIM). We determine the critical behaviour of the RBIM using replica-exchange Monte Carlo simulations. While previous studies have explored thresholds for general lattice geometries under suboptimal decoders such as minimum-weight perfect matching, the rapid development of near-optimal decoders makes resolving the ultimate, maximum-likelihood code capacity relevant. We compute these optimal error thresholds on the Nishimori line for the square, honeycomb, triangular, dice, and kagome lattices. Owing to lattice duality, the phase-flip threshold on a given lattice is equivalent to the bit-flip threshold on its dual. We find that the optimal thresholds of dual-lattice pairs display a characteristic duality-driven splitting around the self-dual square-lattice, mirroring the qualitative behaviour observed for suboptimal decoders. While the average coordination number has the strongest impact on the error threshold, our results demonstrate that the detailed arrangement of vertices and plaquettes also plays a significant role in determining its precise value.