For CSS syndrome decoding, the two check matrices impose binary parity-check constraints on the two Pauli error components. The posterior can therefore be written as a binary factor graph with two Tanner graphs coupled by the local joint prior at each qubit. We call the sum-product algorithm on this factorization joint belief propagation (joint BP). Joint BP retains the local channel correlation between the two Pauli components. This note compares joint BP with the four-state Pauli-label factor graph used for four-state BP. The two algorithms are shown to have the same posterior weights, messages, and beliefs after relabeling the four local Pauli states and marginalizing the irrelevant binary component.

翻译：对于CSS症状解码，两个校验矩阵对两个Pauli错误分量施加二元奇偶校验约束。因此，后验概率可表示为二元因子图，其中两个Tanner图通过每个量子比特处的局部联合先验耦合。我们将这种分解上的和积算法称为联合置信传播（joint BP）。联合BP保留了两个Pauli分量之间的局部信道相关性。本文比较了联合BP与用于四态BP的四态Pauli标签因子图。结果表明，在重新标记四个局部Pauli状态并边缘化无关的二元分量后，两种算法具有相同的后验权重、消息和置信度。