A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics

TL;DR

This paper completes the ZX-Calculus for Clifford+T quantum mechanics by adding four axioms, enabling it to fully represent and reason about this important quantum computational fragment.

Contribution

It introduces the first complete axiomatization of the ZX-Calculus for Clifford+T quantum mechanics, resolving a major open problem in categorical quantum mechanics.

Findings

Proves the completeness of the ZX-Calculus for Clifford+T.

Shows the calculus represents all matrices over a finite extension of dyadic rationals.

Uses the ZW-Calculus to establish the completeness.

Abstract

We introduce the first complete and approximatively universal diagrammatic language for quantum mechanics. We make the ZX-Calculus, a diagrammatic language introduced by Coecke and Duncan, complete for the so-called Clifford+T quantum mechanics by adding four new axioms to the language. The completeness of the ZX-Calculus for Clifford+T quantum mechanics was one of the main open questions in categorical quantum mechanics. We prove the completeness of the Clifford+T fragment of the ZX-Calculus using the recently studied ZW-Calculus, a calculus dealing with integer matrices. We also prove that the Clifford+T fragment of the ZX-Calculus represents exactly all the matrices over some finite dimensional extension of the ring of dyadic rationals.