Combining hard and soft decoders for hypergraph product codes

Antoine Grospellier1, Lucien Grouès1, Anirudh Krishna2, and Anthony Leverrier1

1Inria, 2 Rue Simone IFF, CS 42112, 75589 Paris Cedex 12, France
2Université de Sherbrooke, 2500 Boulevard de l'Université, Sherbrooke, QC J1K 2R1, Canada

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Hypergraph product codes are a class of constant-rate quantum low-density parity-check (LDPC) codes equipped with a linear-time decoder called small-set-flip (SSF). This decoder displays sub-optimal performance in practice and requires very large error correcting codes to be effective. In this work, we present new hybrid decoders that combine the belief propagation (BP) algorithm with the SSF decoder. We present the results of numerical simulations when codes are subject to independent bit-flip and phase-flip errors. We provide evidence that the threshold of these codes is roughly 7.5% assuming an ideal syndrome extraction, and remains close to 3% in the presence of syndrome noise. This result subsumes and significantly improves upon an earlier work by Grospellier and Krishna (arXiv:1810.03681). The low-complexity high-performance of these heuristic decoders suggests that decoding should not be a substantial difficulty when moving from zero-rate surface codes to constant-rate LDPC codes and gives a further hint that such codes are well-worth investigating in the context of building large universal quantum computers.

Quantum error correcting codes serve to buffer quantum information against noise. We are on the cusp of demonstrating that error correction is possible in the laboratory. It is clear what sorts of quantum codes will be used in the near term for achieving these milestones. The path further is less clear; quantum codes that succeed in the near term, such as the surface code, seem to require a prohibitive resource overhead as we scale. Can we find quantum codes which circumvent some of the limitations of these approaches?
While out of experimental reach currently, quantum low-density parity-check (LDPC) codes could be the answer in the long term. In theory, they promise a smaller resource cost compared to current techniques. However more research is needed before we can be certain whether this architecture is worth the experimental effort to explore. We take steps towards bridging this gap between theory and practice. To be specific, previous theoretical works only promise that quantum LDPC codes perform well for large quantum circuits. Part of the trouble stemmed from decoding algorithms, i.e. techniques to troubleshoot and correct errors. These algorithms seemed to require far too many qubits before becoming practical.
In this work, we pick a particular class of quantum LDPC codes called hypergraph product codes. We present new decoding algorithms, constructed as hybrids of classical and quantum decoding algorithms. We study different scenarios of increasing complexity, first assuming that the readout (syndrome extraction) can be done perfectly, and then working with models that assume that the readout itself is error prone. Our results show significant improvement over previous work and are supported by different metrics such as the (pseudo-)threshold and the weight of the parity checks. This provides evidence that such codes are well-worth investigating in the context of building large universal quantum computers.

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Cited by

[1] Joschka Roffe, David R. White, Simon Burton, and Earl Campbell, "Decoding across the quantum low-density parity-check code landscape", Physical Review Research 2 4, 043423 (2020).

[2] Ryan Babbush, Jarrod McClean, Michael Newman, Craig Gidney, Sergio Boixo, and Hartmut Neven, "Focus beyond quadratic speedups for error-corrected quantum advantage", arXiv:2011.04149.

[3] Armanda O. Quintavalle, Michael Vasmer, Joschka Roffe, and Earl T. Campbell, "Single-shot error correction of three-dimensional homological product codes", arXiv:2009.11790.

[4] Nicolas Delfosse, Vivien Londe, and Michael Beverland, "Toward a Union-Find decoder for quantum LDPC codes", arXiv:2103.08049.

[5] Nikolas P. Breuckmann and Jens Niklas Eberhardt, "LDPC Quantum Codes", arXiv:2103.06309.

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