Local Probabilistic Decoding of a Quantum Code

T. R. Scruby and K. Nemoto

Okinawa Institute of Science and Technology, Okinawa, 904-0495, Japan

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Abstract

$\tt{flip}$ is an extremely simple and maximally local classical decoder which has been used to great effect in certain classes of classical codes. When applied to quantum codes there exist constant-weight errors (such as half of a stabiliser) which are uncorrectable for this decoder, so previous studies have considered modified versions of $\tt{flip}$, sometimes in conjunction with other decoders. We argue that this may not always be necessary, and present numerical evidence for the existence of a threshold for $\tt{flip}$ when applied to the looplike syndromes of a three-dimensional toric code on a cubic lattice. This result can be attributed to the fact that the lowest-weight uncorrectable errors for this decoder are closer (in terms of Hamming distance) to correctable errors than to other uncorrectable errors, and so they are likely to become correctable in future code cycles after transformation by additional noise. Introducing randomness into the decoder can allow it to correct these "uncorrectable" errors with finite probability, and for a decoding strategy that uses a combination of belief propagation and probabilistic $\tt{flip}$ we observe a threshold of $\sim5.5\%$ under phenomenological noise. This is comparable to the best known threshold for this code ($\sim7.1\%$) which was achieved using belief propagation and ordered statistics decoding [Higgott and Breuckmann, 2022], a strategy with a runtime of $O(n^3)$ as opposed to the $O(n)$ ($O(1)$ when parallelised) runtime of our local decoder. We expect that this strategy could be generalised to work well in other low-density parity check codes, and hope that these results will prompt investigation of other previously overlooked decoders.

Most quantum error correcting codes detect the presence of errors in a quantum system via the measurement of operators called stabilisers. The outcomes of these measurements are then passed to a classical algorithm called a decoder which uses this measurement data to calculate a correction that counteracts the effects of the errors. A common requirement for decoding algorithms is that they should be able to reliably calculate corrections for all errors under a certain size. In this work we examine the necessity of this requirement and study the performance of certain "bad" decoding algorithms which fail to correct some very small errors. We present numerical results which demonstrate that in some cases these decoders can perform very well despite their failings, and identify the reasons behind this. The algorithms we study in this work are very simple and efficient but nonetheless achieve performance competitive with the best known decoder for the same code, and so we argue that being less strict about what properties we require for decoders can lead to the discovery of more practically useful algorithms.

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Cited by

[1] Lucas Berent, Timo Hillmann, Jens Eisert, Robert Wille, and Joschka Roffe, "Analog Information Decoding of Bosonic Quantum Low-Density Parity-Check Codes", PRX Quantum 5 2, 020349 (2024).

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

[3] Nithin Raveendran, Emmanuel Boutillon, and Bane Vasić, 2023 12th International Symposium on Topics in Coding (ISTC) 1 (2023) ISBN:979-8-3503-2611-6.

The above citations are from Crossref's cited-by service (last updated successfully 2024-06-22 09:24:16) and SAO/NASA ADS (last updated successfully 2024-06-22 09:24:17). The list may be incomplete as not all publishers provide suitable and complete citation data.