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Minimality in Finite-Dimensional ZW-Calculi

Authors Marc de Visme , Renaud Vilmart

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ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Theory of computation → Equational logic and rewriting
  • Theory of computation → Semantics and reasoning
Keywords
  • Quantum Computing
  • Categorical Quantum Mechanics
  • ZW-calculus
  • Qudits
  • Finite Dimensional Hilbert Spaces
  • Completeness
  • Minimality

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Abstract

The ZW-calculus is a graphical language capable of representing 2-dimensional quantum systems (qubit) through its diagrams, and manipulating them through its equational theory. We extend the formalism to accommodate finite dimensional Hilbert spaces beyond qubit systems. 
First we define a qudit version of the language, where all systems have the same arbitrary finite dimension d, and show that the provided equational theory is both complete - i.e. semantical equivalence is entirely captured by the equations - and minimal - i.e. none of the equations are consequences of the others. We then extend the graphical language further to allow for mixed-dimensional systems. We again show the completeness and minimality of the provided equational theory.

Cite As Get BibTex

Marc de Visme and Renaud Vilmart. Minimality in Finite-Dimensional ZW-Calculi. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 49:1-49:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CSL.2025.49

Author Details

Marc de Visme
  • Université Paris-Saclay, Inria, CNRS, ENS Paris-Saclay, Laboratoire Méthodes Formelles, 91190, Gif-sur-Yvette, France
Renaud Vilmart
  • Université Paris-Saclay, Inria, CNRS, ENS Paris-Saclay, Laboratoire Méthodes Formelles, 91190, Gif-sur-Yvette, France

Funding

This work is supported by the PEPR integrated project EPiQ ANR-22-PETQ-0007 part of Plan France 2030, by the projects ANR-22-PNCQ-0002 and ANR-22-CE47-0012.

Acknowledgements

The authors would like to thank Antoine Guilmin-Crépon for discussions about the minimality of the present equational theories. The diagrams of the present paper were drawn using the TikZit tool [Aleks Kissinger, 2019].

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