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

2026-01-29

AI Summary

This work addresses the high complexity of collective measurements in classical-quantum channels, particularly the lack of efficient decoding schemes for non-binary symmetric pure-state channels. It extends Belief Propagation with Quantum Messages (BPQM) to q-ary symmetric pure-state channels whose output Gram matrices are circulant. By leveraging a closed-form recurrence for the eigenvalues of the Gram matrix, the approach efficiently tracks the node-merging process, enabling channel-combination analysis independent of physical implementation. The study establishes, for the first time, a density evolution framework tailored to this class of channels, yielding explicit BPQM unitary operations and analytical bounds on the fidelity of combined channels. These results facilitate the estimation of decoding thresholds for LDPC codes and support the construction of q-ary polar codes.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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.

Problem

• belief propagation
• quantum messages
• q-ary pure-state channels
• symmetric channels
• classical-quantum communication

Innovation

• Belief Propagation with Quantum Messages
• q-ary Pure-State Channels
• Density Evolution
• Gram Matrix Eigenvalues
• Quantum Decoding