Color code decoder with improved scaling for correcting circuit-level noise
Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, NSW 2006, Australia
| Published: | 2025-01-27, volume 9, page 1609 |
| Editor: | Ujjwal Sen |
| Eprint: | arXiv:2404.07482v2 |
| Doi: | https://doi.org/10.22331/q-2025-01-27-1609 |
| Citation: | Quantum 9, 1609 (2025). |
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Abstract
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 ($p_\mathrm{fail} \sim p^{d/2}$) 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.
Featured image: Concatenated MWPM decoder on the distance-7 triangular color code.
Popular summary
Our work introduces a novel decoding strategy for color codes, designed to perform well even under realistic noise affecting quantum circuits. This approach, called the concatenated MWPM decoder, works by combining multiple steps of matching to identify errors. Through numerical simulations, we showed that, while its thresholds (the maximum noise levels where increasing the code size reduces the logical error rate) are on par with existing methods, its performance below these thresholds significantly surpasses them. This improvement reduces the resource cost required to achieve a desired level of fault tolerance, making large-scale quantum computing more feasible.
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