Generalized Belief Propagation Algorithms for Decoding of Surface Codes

Josias Old1,2 and Manuel Rispler1,2,3

1Institute for Quantum Information, RWTH Aachen University, Aachen, Germany
2Institute for Theoretical Nanoelectronics (PGI-2), Forschungszentrum Jülich, Jülich, Germany
3QuTech, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands

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Abstract

Belief propagation (BP) is well-known as a low complexity decoding algorithm with a strong performance for important classes of quantum error correcting codes, e.g. notably for the quantum low-density parity check (LDPC) code class of random expander codes. However, it is also well-known that the performance of BP breaks down when facing topological codes such as the surface code, where naive BP fails entirely to reach a below-threshold regime, i.e. the regime where error correction becomes useful. Previous works have shown, that this can be remedied by resorting to post-processing decoders outside the framework of BP. In this work, we present a generalized belief propagation method with an outer re-initialization loop that successfully decodes surface codes, i.e. opposed to naive BP it recovers the sub-threshold regime known from decoders tailored to the surface code and from statistical-mechanical mappings. We report a threshold of $\textit{17%}$ under independent bit-and phase-flip data noise (to be compared to the ideal threshold of $\textit{20.6%}$) and a threshold value of $\textit{14%}$ under depolarizing data noise (compared to the ideal threshold of $\textit{18.9%}$), which are on par with thresholds achieved by non-BP post-processing methods.

Quantum computing devices suffer from operational errors and decoherence. Methods to keep errors in check and advance towards fault-tolerant quantum computing involve $\textit{quantum error correcting codes}$ and fast and accurate decoding algorithms. Belief propagation (BP) is well-known as a low complexity decoding algorithm with a strong performance for important classes of quantum error correcting codes, e.g. notably for the quantum low-density parity check (LDPC) code class of random expander codes. However, it is also well-known that the performance of BP breaks down when facing topological codes such as the surface code, where naive BP fails entirely to reach a below-threshold regime, i.e. the regime where error correction becomes useful. Previous works have shown, that this can be remedied by resorting to post-processing decoders outside the framework of BP. In this work, we present a generalized belief propagation method with an outer re-initialization loop that successfully decodes surface codes within the framework of BP, achieving a below-threshold regime.

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Cited by

[1] Joschka Roffe, Lawrence Z. Cohen, Armanda O. Quintavalle, Daryus Chandra, and Earl T. Campbell, "Bias-tailored quantum LDPC codes", Quantum 7, 1005 (2023).

[2] Christopher A. Pattison, Anirudh Krishna, and John Preskill, "Hierarchical memories: Simulating quantum LDPC codes with local gates", arXiv:2303.04798, (2023).

[3] Laura Caune, Brendan Reid, Joan Camps, and Earl Campbell, "Belief propagation as a partial decoder", arXiv:2306.17142, (2023).

The above citations are from SAO/NASA ADS (last updated successfully 2024-02-26 10:36:17). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2024-02-26 10:36:16).