Compositionality of Planar Perfect Matchings: A Universal and Complete Fragment of ZW-Calculus

Authors Titouan Carette , Etienne Moutot , Thomas Perez, Renaud Vilmart

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Author Details

Titouan Carette
  • Centre for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia
Etienne Moutot
  • CNRS, I2M, Aix-Marseille Université, Marseille, France
Thomas Perez
  • Université de Lyon, ENS de Lyon, France
Renaud Vilmart
  • Université Paris-Saclay, ENS Paris-Saclay, Inria, CNRS, LMF, 91190, Gif-sur-Yvette, France

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Titouan Carette, Etienne Moutot, Thomas Perez, and Renaud Vilmart. Compositionality of Planar Perfect Matchings: A Universal and Complete Fragment of ZW-Calculus. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 120:1-120:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We exhibit a strong connection between the matchgate formalism introduced by Valiant and the ZW-calculus of Coecke and Kissinger. This connection provides a natural compositional framework for matchgate theory as well as a direct combinatorial interpretation of the diagrams of ZW-calculus through the perfect matchings of their underlying graphs. We identify a precise fragment of ZW-calculus, the planar W-calculus, that we prove to be complete and universal for matchgates, that are linear maps satisfying the matchgate identities. Computing scalars of the planar W-calculus corresponds to counting perfect matchings of planar graphs, and so can be carried in polynomial time using the FKT algorithm, making the planar W-calculus an efficiently simulable fragment of the ZW-calculus, in a similar way that the Clifford fragment is for ZX-calculus. This work opens new directions for the investigation of the combinatorial properties of ZW-calculus as well as the study of perfect matching counting through compositional diagrammatical technics.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Theory of computation → Equational logic and rewriting
  • Mathematics of computing → Matchings and factors
  • Perfect Matchings Counting
  • Quantum Computing
  • Matchgates
  • ZW-Calculus
  • String Diagrams
  • Completeness


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