The rational fragment of the ZX-calculus

TL;DR

This paper presents a new, more natural axiomatisation of the rational fragment of the ZX-calculus, a diagrammatic language for quantum mechanics, which is complete for rational angles and characterized using diophantine geometry.

Contribution

It introduces a novel axiomatisation relying on the cyclotomic supplementarity rule, removing the need for metarules, and characterizes diagram equalities involving arbitrary angles.

Findings

Axiomatisation is complete for diagrams with rational angles.

Diagram equality involving arbitrary angles is a limit of rational angle equalities.

Complete characterization of provable Euler equations in the framework.

Abstract

We introduce here a new axiomatisation of the rational fragment of the ZX-calculus, a diagrammatic language for quantum mechanics. Compared to the previous axiomatisation introduced in [8], our axiomatisation does not use any metarule , but relies instead on a more natural rule, called the cyclotomic supplementarity rule, that was introduced previously in the literature. Our axiomatisation is only complete for diagrams using rational angles , and is not complete in the general case. Using results on diophantine geometry, we characterize precisely which diagram equality involving arbitrary angles are provable in our framework without any new axioms, and we show that our axiomatisation is continuous, in the sense that a diagram equality involving arbitrary angles is provable iff it is a limit of diagram equalities involving rational angles. We use this result to give a complete characterization of all Euler equations that are provable in this axiomatisation.