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.
Highlights
Quantum error-correcting codes are considered as an essential component in the current architectures of quantum computers due to the inherently faulty nature of the quantum hardware
In the first part of the paper, we introduce a new enhancement of the standard belief propagation (BP) decoder for quantum low density parity check (LDPC) (QLDPC) codes using a variant of the well-known decoding algorithm for short classical codes called the ordered statistics decoding (OSD) [13, 14]
We show that with the help of this OSD-like post-processing the performance of the standard BP decoder on many degenerate QLDPC codes can be improved by several orders of magnitude
Summary
Quantum error-correcting codes are considered as an essential component in the current architectures of quantum computers due to the inherently faulty nature of the quantum hardware. There have been proposed a number of interesting families of degenerate QLDPC codes (e.g., hypergraph product codes [7] and homological product codes [8]) with very good asymptotic parameters Their practical error-correcting performance for relatively small code lengths (n < 1000) is largely unexplored, and it is not clear whether their performance is competitive to the best known topological codes. OSD decoder on many known classes of degenerate QLDPC codes, including the already mentioned hypergraph product codes, hyperbicycle codes [9, 10], the homological product codes [8], and Haah’s cubic codes [20] We compare their performance with the performance of the codes, constructed in this work.
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