Quantum Bayesian Networks: Compositionality and Typing via Linear Logic
arXiv:2604.26059 · doi:10.4230/LIPIcs.FSCD.2026.16
The paper introduces a compositional and typed framework for quantum Bayesian networks using linear logic proof‑nets, showing that it matches classical Bayesian semantics when causes are classical and reduces to tensor networks in the fully quantum case.
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.
23 pages, preprint of a FSCD paper