Tags:Categorical Quantum Mechanics, Diagrammatic Reasoning and Quantum Computing

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 two 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 ZX-Calculus using the recently studied ZW-Calculus, a calculus dealing with integer matrices. We also prove that this fragment of the ZX-Calculus represents exactly all the matrices over some finite dimensional extension of the ring of dyadic rationals.

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