Belief Propagation with Quantum Messages for Symmetric Q-ary Pure-State Channels

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.

💡 Research Summary

This paper extends the framework of belief propagation with quantum messages (BPQM) from binary-input pure‑state channels to the much broader class of symmetric q‑ary pure‑state channels (PSCs) whose output Gram matrices are circulant. The authors observe that for such channels the Gram matrix can be diagonalized by the discrete Fourier transform (DFT) vectors, and therefore the channel is completely characterized by the list of its eigenvalues (the “eigen list”) λ =