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Conference Papers Year : 2023

Compositionality of planar perfect matchings

Abstract

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.
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Dates and versions

hal-04002282 , version 1 (23-02-2023)

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Cite

Titouan Carette, Etienne Moutot, Thomas Perez, Renaud Vilmart. Compositionality of planar perfect matchings: A universal and complete fragment of ZW-calculus. 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023), Jul 2023, Paderborn, Germany. pp.120:1--120:17, ⟨10.4230/LIPIcs.ICALP.2023.120⟩. ⟨hal-04002282⟩
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