We study directionally informed belief propagation (BP) decoding for quantum CSS codes, where anisotropic Tanner-graph structure and biased noise concentrate degeneracy along preferred directions. We formalize this by placing orientation weights on Tanner-graph edges, aggregating them into per-qubit directional weights, and defining a \emph{directional degeneracy enumerator} that summarizes how degeneracy concentrates along those directions. A single bias parameter~$β$ maps these weights into site-dependent log-likelihood ratios (LLRs), yielding anisotropic priors that plug directly into standard BP$\rightarrow$OSD decoders without changing the code construction. We derive bounds relating directional and Hamming distances, upper bound the number of degenerate error classes per syndrome as a function of distance, rate, and directional bias, and give a MacWilliams-type expression for the directional enumerator. Finite-length simulations under code-capacity noise show significant logical error-rate reductions -- often an order of magnitude at moderate physical error rates -- confirming that modest anisotropy is a simple and effective route to hardware-aware decoding gains.

翻译：我们研究了量子CSS码的方向性信息置信传播（BP）解码，其中各向异性的Tanner图结构和偏置噪声将简并性集中在优选方向上。我们通过为Tanner图边分配方向权重、将其聚合为每个量子位的方向权重，并定义一个**方向简并枚举器**来总结简并性如何沿这些方向集中，从而形式化这一过程。单个偏置参数~$β$将这些权重映射为与位置相关的对数似然比（LLR），产生各向异性的先验信息，可直接插入标准BP$\rightarrow$OSD解码器而无需改变码构造。我们推导了方向距离与汉明距离之间的关系界，给出了每个校验子对应的简并错误类数量作为距离、码率和方向偏置函数的上界，并给出了方向枚举器的MacWilliams型表达式。在码容量噪声下的有限长度仿真显示逻辑错误率显著降低——在中等物理错误率下通常可达一个数量级——证实适度的各向异性是实现硬件感知解码增益的简单有效途径。