Quantum information needs to be protected by quantum error-correcting codes due to imperfect physical devices and operations. One would like to have an efficient and high-performance decoding procedure for the class of quantum stabilizer codes. A potential candidate is Pearl's belief propagation (BP), but its performance suffers from the many short cycles inherent in a quantum stabilizer code, especially highly-degenerate codes. A general impression exists that BP is not effective for topological codes. In this paper, we propose a decoding algorithm for quantum codes based on quaternary BP with additional memory effects (called MBP). This MBP is like a recursive neural network with inhibitions between neurons (edges with negative weights), which enhance the perception capability of a network. Moreover, MBP exploits the degeneracy of a quantum code so that the most probable error or its degenerate errors can be found with high probability. The decoding performance is significantly improved over the conventional BP for various quantum codes, including quantum bicycle, hypergraph-product, surface and toric codes. For MBP on the surface and toric codes over depolarizing errors, we observe error thresholds of 16% and 17.5%, respectively.

翻译：由于物理设备和操作的不完美，量子信息需要借助量子纠错码来保护。我们希望针对量子稳定子码这一类编码拥有高效且高性能的译码程序。一个潜在的候选方案是Pearl的置信传播（BP），但其性能受到量子稳定子码中固有的许多短环影响，尤其是高度退化的编码。普遍存在一种印象，即BP对于拓扑码并不有效。在本文中，我们提出了一种基于四元BP并附加记忆效应的量子码译码算法（称为MBP）。此MBP类似于一种带有神经元间抑制（具有负权重的边）的递归神经网络，这增强了网络的感知能力。此外，MBP利用了量子码的退化性，使得最可能错误或其退化错误能够以高概率被找到。与传统的BP相比，该译码性能在多种量子码上均有显著提升，包括量子自行车码、超图积码、表面码和环面码。针对去极化错误下的表面码和环面码上的MBP，我们观察到错误阈值分别为16%和17.5%。