# Degenerate Quantum LDPC Codes With Good Finite Length Performance

Faculty of Mechanics and Mathematics, Moscow State University, GSP-1, Leninskie Gory, Moscow, 119991, Russian Federation

### Abstract

We study the performance of medium-length quantum LDPC (QLDPC) codes in the depolarizing channel. Only degenerate codes with the maximal stabilizer weight much smaller than their minimum distance are considered. It is shown that with the help of OSD-like post-processing the performance of the standard belief propagation (BP) decoder on many QLDPC codes can be improved by several orders of magnitude. Using this new BP-OSD decoder we study the performance of several known classes of degenerate QLDPC codes including hypergraph product codes, hyperbicycle codes, homological product codes, and Haah's cubic codes. We also construct several interesting examples of short generalized bicycle codes. Some of them have an additional property that their syndromes are protected by small BCH codes, which may be useful for the fault-tolerant syndrome measurement. We also propose a new large family of QLDPC codes that contains the class of hypergraph product codes, where one of the used parity-check matrices is square. It is shown that in some cases such codes have better performance than hypergraph product codes. Finally, we demonstrate that the performance of the proposed BP-OSD decoder for some of the constructed codes is better than for a relatively large surface code decoded by a near-optimal decoder.

The conference talk at the 5th International Conference on Quantum Error Correction (QEC’19) – held from 29th July to 2nd August 2019 at Senate House in London.

Surface codes, as well as other quantum codes with geometrically local stabilizers, are usually considered as leading candidates for fault-tolerant architectures of quantum computers. However, the overhead of such architectures grows significantly with the code distance. On the other hand, quantum LDPC codes, where long-range interaction between qubits is allowed, can potentially be used to provide fault-tolerant quantum computations with constant overhead. Unfortunately, the finite length performance of the known classes of QLDPC codes is far from optimal under the state-of-the-art decoders. In the current work, we propose a new decoding algorithm called BP-OSD, which combines the standard BP decoder with a post-processing algorithm called Ordered Statistics Decoding (OSD), an idea borrowed from classical error-correcting codes. We demonstrate on many examples that this combined decoding strategy improves the error-correcting performance of the BP decoder by several orders of magnitude. We also propose a number of new QLDPC codes and show that for the standard depolarizing noise model the error-correcting performance of such codes under the BP-OSD decoder can be better than for surface codes even if the latter ones are decoded using a near-optimal decoder.

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### Cited by

[1] Joschka Roffe, "Towards practical quantum LDPC codes", Quantum Views 5, 63 (2021).

[2] Leonid P. Pryadko, Vadim A. Shabashov, and Valerii K. Kozin, "QDistRnd: A GAP package for computing the distance of quantum error-correcting codes", Journal of Open Source Software 7 71, 4120 (2022).

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