Quantum Message Passing for Factor Graphs over Finite Abelian Groups

Quantum Physics [Submitted on 14 Apr 2026] Quantum Message Passing for Factor Graphs over Finite Abelian Groups Avijit Mandal, Henry D. Pfister 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. Subjects: Quantum Physics (quant-ph); Information Theory (cs.IT) Cite as: arXiv:2604.12186 [quant-ph]   (or arXiv:2604.12186v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2604.12186 Focus to learn more Submission history From: Avijit Mandal [view email] [v1] Tue, 14 Apr 2026 01:33:18 UTC (197 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-04 Change to browse by: cs cs.IT math math.IT References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)