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arXiv:1308.6207 · Physical Review A

Decoding color codes by projection onto surface codes

Nicolas Delfosse

Published Jan 16, 2014• 90 citations
arXivHTMLPDF

Abstract

We propose a general strategy to decode color codes, which is based on the projection of the error onto three surface codes. This provides a method to transform every decoding algorithm of surface codes into a decoding algorithm of color codes. Applying this idea to a family of hexagonal color codes, with the perfect matching decoding algorithm for the three corresponding surface codes, we find a phase error threshold of approximately $8.7%$. Finally, our approach enables us to establish a general lower bound on the error threshold of a family of color codes depending on the threshold of the three corresponding surface codes. These results are based on a chain complex interpretation of surface codes and color codes.

DOI
10.1103/physreva.89.012317

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We propose a general strategy to decode color codes, which is based on the projection of the error onto three surface codes. This provides a method to transform every decoding algorithm of surface codes into a decoding algorithm of color codes. Applying this idea to a family of hexagonal color codes, with the perfect matching decoding algorithm for the three corresponding surface codes, we find a phase error threshold of approximately 8.78.7%8.7. Finally, our approach enables us to establish a general lower bound on the error threshold of a family of color codes depending on the threshold of the three corresponding surface codes. These results are based on a chain complex interpretation of surface codes and color codes.
The quantum erasure channel models phenomena such as loss or leakage of qubits. Using quantum codes, we can recover from such errors. In this paper, we are interested in studying the performance of two-dimensional color codes over the quantum erasure channel. Our approach makes use of the local equivalence between color codes and surface codes. We propose a variety of decoding algorithms for color codes over the erasure channel. First, we propose algorithms that decode by projecting the erasures on the color to surface codes. Then, instead of directly decoding on the color code or on the equivalent copies of surface codes, we decode jointly on the color code and the equivalent surface codes. We observe a threshold of 44.3% for the color code on the square octagonal lattice.
The development of practical, high-performance decoding algorithms reduces the resource cost of fault-tolerant quantum computing. Here we propose a decoder for the surface code that finds low-weight correction operators for errors produced by the depolarising noise model. The decoder is obtained by mapping the syndrome of the surface code onto that of the color code, thereby allowing us to adopt more sophisticated color-code decoding algorithms. Analytical arguments and exhaustive testing show that the resulting decoder can find a least-weight correction for all weight d/2 depolarising errors for even code distance d . This improves the logical error rate by an exponential factor O(2d/2) compared with decoders that treat bit-flip and dephasing errors separately. We demonstrate this improvement with analytical arguments and supporting numerical simulations at low error rates. Of independent interest, we also demonstrate an exponential improvement in logical error rate for our decoder used to correct independent and identically distributed bit-flip errors affecting the color code compared with more conventional color-code decoding algorithms.
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.
Topological color codes defined by the 4.8.8 semiregular lattice feature geometrically local check operators and admit transversal implementation of the entire Clifford group, making them promising candidates for fault-tolerant quantum computation. Recently, several efficient algorithms for decoding the syndrome of color codes were proposed. Here, we modify one of these algorithms to account for errors affecting the syndrome, applying it to the family of triangular 4.8.8 color codes encoding one logical qubit. For a three-dimensional bit-flip channel, we report a threshold error rate of 0.0208(1), compared with 0.0305(4) previously reported for an integer-program-based decoding algorithm. When we account for circuit details, this threshold is reduced to 0.00143(1) per gate, compared with 0.00672(1) per gate for the surface code under an identical noise model.
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