Belief propagation with quantum messages (BPQM) provides a low-complexity alternative to collective measurements for communication over classical--quantum channels. Prior BPQM constructions and density-evolution (DE) analyses have focused on binary alphabets. Here, we generalize BPQM to symmetric q-ary pure-state channels (PSCs) whose output Gram matrix is circulant. For this class, we show that bit-node and check-node combining can be tracked efficiently via closed-form recursions on the Gram-matrix eigenvalues, independent of the particular physical realization of the output states. These recursions yield explicit BPQM unitaries and analytic bounds on the fidelities of the combined channels in terms of the input-channel fidelities. This provides a DE framework for symmetric q-ary PSCs that allows one to estimate BPQM decoding thresholds for LDPC codes and to construct polar codes on these channels.

翻译：基于量子消息的置信传播（BPQM）为经典-量子信道通信提供了一种低复杂度的集体测量替代方案。先前的BPQM构造与密度演化（DE）分析主要集中于二进制字母表。本文中，我们将BPQM推广至输出格拉姆矩阵为循环矩阵的对称q元纯态信道（PSCs）。针对此类信道，我们证明比特节点与校验节点的合并可通过格拉姆矩阵特征值的闭式递推进行高效追踪，且与输出态的具体物理实现无关。这些递推关系给出了显式的BPQM酉变换，并基于输入信道保真度推导出合并信道保真度的解析边界。这为对称q元纯态信道建立了一个密度演化框架，可用于估计LDPC码的BPQM译码阈值，并在此类信道上构造极化码。