Well-tempered ZX and ZH Calculi

TL;DR

This paper introduces renormalised versions of the ZX and ZH calculi that form exact bialgebras, simplifying diagrammatic reasoning in quantum computing by eliminating scalar gadgets.

Contribution

It presents new renormalised generators for ZX and ZH calculi that form precise bialgebras, improving the simplicity and accuracy of quantum diagram transformations.

Findings

No scalar gadgets needed for common unitaries

Simpler diagram transformations achieved

Exact bialgebra structure established

Abstract

The ZX calculus is a mathematical tool to represent and analyse quantum operations by manipulating diagrams which in effect represent tensor networks. Two families of nodes of these networks are ones which commute with either Z rotations or X rotations, usually called "green nodes" and "red nodes" respectively. The original formulation of the ZX calculus was motivated in part by properties of the algebras formed by the green and red nodes: notably, that they form a bialgebra -- but only up to scalar factors. As a consequence, the diagram transformations and notation for certain unitary operations involve "scalar gadgets" which denote contributions to a normalising factor. We present renormalised generators for the ZX calculus, which form a bialgebra precisely. As a result, no scalar gadgets are required to represent the most common unitary transformations, and the corresponding diagram transformations are generally simpler. We also present a similar renormalised version of the ZH calculus. We obtain these results by an analysis of conditions under which various "idealised" rewrites are sound, leveraging the existing presentations of the ZX and ZH calculi.