Quantum Message Passing for Factor Graphs over Finite Abelian Groups

TL;DR

This paper introduces a quantum message-passing framework for factor graphs over finite abelian groups, enabling analysis of quantum channels and codes with potential applications in quantum error correction.

Contribution

It extends the BPQM framework to non-cyclic groups and general factor-graph constraints, providing explicit update rules for quantum message passing.

Findings

Diagonalization of the Gram matrix by the character basis

Explicit update rules for quantum messages in factor graphs

Applicability to standard code families like polar and LDPC codes

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.