ИСТИНА

We study[1] the performance of small and 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 an 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 modified BP decoder we study the performance of several known classes of degenerate QLDPC codes including the hypergraph product codes, the hyperbicycle codes, the homological product codes, and the 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 that may be useful for the fault-tolerant syndrome measurement. Besides that we 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 several cases such codes have better performance than the hypergraph product codes. Besides that, we compare (see Fig. 1) the new modification of BP with other known modifications such as the random perturbation, the enhanced feedback, and the matrix augmentation algorithm proposed recently[2]. Moreover, we demonstrate that the performance on the proposed modified BP decoder for some of the new constructed codes ([[1270, 28]] generalized hypergraph code) is better than for the surface [[1201, 1, 25]] code decoded by a near-optimal MPS-based decoder proposed in [3]. [1] P. Panteleev, G. Kalachev, Degenerate Quantum LDPC Codes With Good Finite Length Performance, arXiv, April (2019). [2] J. Rigby, et al., Modified belief propagation decoders for quantum ldpc codes, arXiv, March (2019). [3] S. Bravyi, M. Suchara, and A. Vargo, Phys. Rev. A 90, 032326 (2014)