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Quantum Bayesian Networks: Compositionality and Typing via Linear Logic

Authors Rémi Di Guardia , Thomas Ehrhard , Claudia Faggian

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Bayesian networks
  • Theory of computation → Quantum computation theory
  • Theory of computation → Linear logic
Keywords
  • Quantum Bayesian Networks
  • Quantum Causal Models
  • Bayesian Networks
  • Proof-Nets
  • Linear Logic

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Abstract

Quantum Bayesian networks [Henson et al., 2014] 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 [Pearl, 2009] - 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, and Claudia Faggian. Quantum Bayesian Networks: Compositionality and Typing via Linear Logic. In 11th International Conference on Formal Structures for Computation and Deduction (FSCD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 378, pp. 16:1-16:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.FSCD.2026.16

Author Details

Rémi Di Guardia
  • Université Paris Cité, CNRS, Inria, IRIF, F-75013, Paris, France
Thomas Ehrhard
  • Université Paris Cité, CNRS, IRIF, F-75013, Paris, France
Claudia Faggian
  • Université Paris Cité, CNRS, IRIF, F-75013, Paris, France

Funding

by the Plan France 2030 through the PEPR project EPiQ (ANR-22-PETQ0007) and by the European Union through the MSCA SE project QCOMICAL (Grant Agreement ID: 101182520).

Acknowledgements

We are grateful to Benoît Valiron for insightful discussions and to the anonymous referees for their valuable comments and the suggested references.

References

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