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

TL;DR

This paper extends belief propagation with quantum messages to symmetric q-ary pure-state channels, providing a framework for analyzing and designing quantum-assisted error correction codes.

Contribution

It generalizes BPQM to q-ary channels, deriving closed-form recursions for Gram-matrix eigenvalues and enabling threshold estimation and code construction.

Findings

Derived explicit BPQM unitaries for q-ary channels

Established analytic bounds on channel fidelities

Provided a density-evolution framework for code design

Abstract

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.