Quantum Message Passing for Factor Graphs over Finite Abelian Groups

2026-04-13

AI Summary

This work addresses the efficient implementation of quantum message passing on factor graphs over finite Abelian groups to solve quantum state discrimination under group-covariant pure-state channels. By diagonalizing the Gram matrix using the character basis of the dual group, the channel is represented as an eigenvalue list indexed by characters, and message update rules for check, equality, and homomorphism factors are derived, ensuring that messages remain closed within the mixture class of group-covariant pure-state channels. The study extends quantum belief propagation for the first time to non-cyclic alphabets and general Abelian group homomorphism constraints, unifying diverse coding structures—including polar codes, LDPC codes, convolutional codes, and turbo codes—into a single theoretical framework for quantum decoding on tree-structured factor graphs.

Technology Category

• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

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.

Problem

• quantum message passing
• factor graphs
• finite abelian groups
• group-covariant pure-state channels
• quantum state discrimination

Innovation

• quantum message passing
• finite abelian groups
• group-covariant pure-state channel
• factor graphs
• character basis diagonalization