Compositionality of planar perfect matchings

TL;DR

This paper establishes a connection between matchgate formalism and ZW-calculus, introducing a planar W-calculus fragment that is complete, universal, and efficiently simulable for perfect matchings in planar graphs.

Contribution

It identifies a specific fragment of ZW-calculus that is complete and universal for matchgates, providing a combinatorial interpretation and polynomial-time computability.

Findings

The planar W-calculus is complete and universal for matchgates.

Counting scalars in the planar W-calculus corresponds to perfect matching enumeration.

The planar W-calculus can be efficiently simulated using the FKT algorithm.

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.