We develop a quantum message-passing framework for factor graphs over finite abelian groups. Our starting point is the task of discriminating between a collection of quantum states indexed by the elements of a finite abelian group $\mathcal{G}$ whose overlaps respect the structure of a group-covariant pure-state channel (PSC). For such channels, we show that the Gram matrix constructed from the output states is diagonalized by the character basis of the dual group $\widehat{\mathcal{G}}$. Hence, the channel is characterized, up to isometric equivalence, by its character-indexed eigen list. Based on this representation, we analyze the induced classical-quantum channels associated with check, equality, homomorphism, marginalization, and automorphism factors. For each factor, we derive explicit update rules showing that if the incoming messages are heralded mixtures of group-covariant PSCs, then the outgoing message remains in the same class. This provides a closed quantum message-passing framework for tree-structured factor graphs assembled from these primitives. The framework applies directly to several standard code families over finite abelian groups, including polar codes, LDPC codes, and convolutional and turbo codes. It recovers the previously studied $q$-ary formulation as the special case $(\mathcal{G}=\mathbb{Z}_q)$, while extending the belief propagation with quantum messages (BPQM) framework introduced by Renes to non-cyclic alphabets and more general factor-graph constraints described by homomorphisms between products of abelian groups.

翻译：我们为有限阿贝尔群上的因子图发展了一个量子消息传递框架。我们的出发点是对由有限阿贝尔群 $\mathcal{G}$ 的元素索引的量子态集合进行区分，这些量子态的重叠服从群协变纯态信道（PSC）的结构。对于此类信道，我们证明由输出态构建的Gram矩阵通过其对偶群 $\widehat{\mathcal{G}}$ 的特征基对角化。因此，该信道在等距等价意义下由其以特征指数为索引的特征值列表刻画。基于这一表示，我们分析了与校验因子、等式因子、同态因子、边缘化因子和自同构因子相关的诱导经典-量子信道。对于每个因子，我们推导出显式更新规则，表明如果输入消息是群协变PSC的预告混合态，那么输出消息仍保持同一类别。这为从这些基本组件组装而成的树形结构因子图提供了一个封闭的量子消息传递框架。该框架直接适用于有限阿贝尔群上的多种标准码族，包括极化码、LDPC码、卷积码和Turbo码。它将先前研究的 $q$ 元公式作为特例 $(\mathcal{G}=\mathbb{Z}_q)$ 重新得到，同时将Renes提出的带量子消息的信度传播（BPQM）框架扩展至非循环字母表以及由阿贝尔群乘积之间的同态描述的更一般因子图约束。