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Graphical Symplectic Algebra

Authors Robert I. Booth , Titouan Carette , Cole Comfort

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LIPIcs.FSCD.2026.7.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
  • Theory of computation → Rewrite systems
Keywords
  • graphical algebra
  • symplectic geometry
  • string diagrams
  • category theory
  • classical mechanics
  • quantum mechanics
  • graph theory

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Abstract

We introduce a family of diagrammatical equational theories unifying two research programs: categorical quantum mechanics and graphical linear algebra. We prove their completeness with respect to denotational semantics described in terms of relations between vector spaces equipped with symplectic structure. This provides versatile graphical languages encompassing both affinely constrained classical mechanical systems, as well as odd-prime-dimensional stabiliser and Gaussian quantum circuits. Terms are described by labelled graphs with input and output interfaces, and the languages are equipped with equational theories amenable to standard graph rewriting techniques.
In order to reason about large composite systems, we introduce a compact scalable notation where the vertices are themselves labelled by graphs. This notation allows us to state new and powerful rewrite rules which operate on diagrams at a large scale. We also show how this notation neatly captures some important constructions, such as graph states of quantum computing and the impedance and admittance matrices of electrical networks.

Cite As Get BibTex

Robert I. Booth, Titouan Carette, and Cole Comfort. Graphical Symplectic Algebra. In 11th International Conference on Formal Structures for Computation and Deduction (FSCD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 378, pp. 7:1-7:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.FSCD.2026.7

Author Details

Robert I. Booth
  • University of Oxford, United Kingdom
Titouan Carette
  • LIX, CNRS, École polytechnique, Institut Polytechnique de Paris, 91120, Palaiseau, France
Cole Comfort
  • Université Paris-Saclay, CNRS, ENS Paris-Saclay, Inria, CentraleSupélec, Laboratoire Méthodes Formelles, France

Funding

  • Booth, Robert I.: Engineering and Physical Sciences Research Council grant reference EP/Z002230/1: (De)constructing quantum software (DeQS).
  • Comfort, Cole: This work was partially supported by the project NEASQC funded from the European Union’s Horizon 2020 research and innovation programme (grant agreement No 951821). This work has been partially funded by the European Union through the MSCA SE project QCOMICAL and within the framework of "Plan France 2030", under the research projects EPIQ ANR-22-PETQ-0007 and HQI-R&D ANR-22-PNCQ-0002.

Acknowledgements

We thank our many reviewers, who have helped to significantly clarify the exposition and correct a few minor mistakes.

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