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arXiv · 2604.26059

Quantum Bayesian Networks: Compositionality and Typing via Linear Logic

Rémi Di Guardia·Thomas Ehrhard·Claudia Faggian
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Abstract

Quantum Bayesian networks provide a mathematical formalism to describe causal relations, to analyse correlations, and to predict the probabilities of measurement outcomes, in systems involving both classical and quantum data. They generalize Pearl's Bayesian networks -- prominent graphical models for classical probabilistic reasoning and inference. The goal of this paper is to bring compositional principles and a typing discipline into this setting. A key feature of our compositional semantics is that when all causes are classical, it coincides with the standard factor-based semantics of Bayesian networks, while in the purely quantum case it reduces to tensor networks. We then propose a typed formalism based on linear logic proof-nets, where types ensure well-behaved composition of systems, and which we prove sound and complete with respect to quantum Bayesian networks.

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Rémi Di Guardia, Thomas Ehrhard, Claudia Faggian. 2026-04-28. Quantum Bayesian Networks: Compositionality and Typing via Linear Logic. https://doi.org/10.4230/lipics.fscd.2026.16

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