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A decoder for the 3d color code with boundaries

Skylar Turner

Published Aug 16, 2021• 0 citations
OpenAlex

Abstract

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.

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Decoding 3D color codes with boundaries

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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 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 Bombín's recently proposed "colorful quantum computing."

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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.

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Efficient color code decoders in d ≥ 2 dimensions from toric code decoders

Aleksander Kubica, Nicolas Delfosse

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.

Quantum•Feb 21, 2023•29 citations
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%4\% - 8\%4%−8% for Pauli XXX errors and a threshold of 0.7%−0.8%0.7\% - 0.8\%0.7%−0.8% for Pauli ZZZ errors. Our work is a first step towards characterizing the performance of Bombin's recently proposed colorful quantum computing.
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.
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%4\% - 8\%4%−8% for Pauli XXX errors and a threshold of 0.7%−0.8%0.7\% - 0.8\%0.7%−0.8% for Pauli ZZZ errors. Our work is a first step towards characterizing the performance of Bombín's recently proposed "colorful quantum computing."
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.
In quantum computers, each logical qubit must be encoded in several physical qubits to protect against noise in physical qubits. The tetrahedral color code is such an encoding. In the primal lattice, the three-dimensional tetrahedral color code is a bitruncated octahedral lattice, and in the dual lattice, the tetrahedral color code is a four-colorable body-centered cubic lattice. This color code can be utilized for measurement-based quantum computing, for which all entanglement is present in a cluster state at the beginning of the computation and gate operations are performed by local measurements and classical operations. Moreover, this color code admits a gate set that is both transversal, i.e., realized by qubit-wise operations, and universal when supplemented by measurement and classical computing. During cluster state preparation and computation, errors may occur on the physical qubits. Therefore, it is necessary to have a decoder that, using the syndrome of these errors, finds a set of qubits to correct and performs the operations needed to correct those qubits. The decoder is considered successful if and only if the correction does not cause a logical error on the logical qubit. In this thesis, I present a decoder for Pauli Z, or phase-flip, errors on the three-dimensional tetrahedral color code with nonperiodic boundaries. This decoder for Z-errors includes as a subroutine a decoder for Pauli X errors. The decoder uses a restriction procedure to map the tetrahedral color code to a toric code. The toric code is another quantum error correcting code. Then, an existing toric code decoder interprets the error syndrome. This research on the bounded color code builds upon previous work on the unbounded color code and the two-dimensional color code. Under independently identically distributed noise, evidence indicates an error probability threshold for Z-errors between 0.01% and 0.02% and for X-errors between 2.5% and 3.3%. I also present example errors and illustrate how the decoder attempts to correct those errors.
Efficient color code decoders in d≥2d\geq 2d≥2 dimensions from toric code decoders
We introduce an efficient decoder of the color code in d≥2d\geq 2d≥2 dimensions, the Restriction Decoder, which uses any ddd-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 p2D≈10.2%p_{\textrm{2D}} \approx 10.2\%p2D​≈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.