2026-07-22T14:20:41Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:18172 2024-03-06T11:01:13Z ddc:004 open_access

Compositionality of Planar Perfect Matchings: A Universal and Complete Fragment of ZW-Calculus Carette, Titouan Moutot, Etienne Perez, Thomas Vilmart, Renaud Perfect Matchings Counting Quantum Computing Matchgates ZW-Calculus String Diagrams Completeness 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Titouan Carette and Etienne Moutot and Thomas Perez and Renaud Vilmart 2023 Is Part Of LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2023.120 urn:nbn:de:0030-drops-181726 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.120 eng https://creativecommons.org/licenses/by/4.0/legalcode