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Track B: Automata, Logic, Semantics, and Theory of Programming
Compositionality of Planar Perfect Matchings: A Universal and Complete Fragment of ZW-Calculus

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

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)


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.

Cite as

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)


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@InProceedings{carette_et_al:LIPIcs.ICALP.2023.120,
  author =	{Carette, Titouan and Moutot, Etienne and Perez, Thomas and Vilmart, Renaud},
  title =	{{Compositionality of Planar Perfect Matchings: A Universal and Complete Fragment of ZW-Calculus}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{120:1--120:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.120},
  URN =		{urn:nbn:de:0030-drops-181726},
  doi =		{10.4230/LIPIcs.ICALP.2023.120},
  annote =	{Keywords: Perfect Matchings Counting, Quantum Computing, Matchgates, ZW-Calculus, String Diagrams, Completeness}
}
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