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 Gallager’s sum-product algorithm, also known as 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.

由于物理设备和操作不完善，量子信息需要通过量子纠错码进行保护。人们希望对一类量子稳定器码有一种高效和高性能的解码程序。一个潜在的候选者是 Gallager 的 sum-product 算法，也称为 Pearl 的置信传播 (BP)，但它的性能受到量子稳定器代码中固有的许多短周期的影响，尤其是高度退化的代码。普遍的印象是 BP 对拓扑码无效。在本文中，我们提出了一种基于具有附加记忆效应的四元 BP（称为 MBP）的量子码解码算法。这个 MBP 就像一个递归神经网络，在神经元之间（具有负权重的边）有抑制，增强了网络的感知能力。此外，MBP利用了量子码的简并性，因此可以以高概率找到最可能的错误或其简并错误。对于各种量子码，包括量子自行车码、超图积码、曲面码和复曲面码，其解码性能比传统 BP 显着提高。对于表面上的 MBP 和去极化误差上的复曲面码，我们分别观察到 16% 和 17.5% 的误差阈值。