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

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 π/4-fragment of the ZX-Calculus using the recently studied ZW-Calculus, a calculus dealing with integer matrices. We also prove that the π/4-fragment of the ZX-Calculus represents exactly all the matrices over some finite dimensional extension of the ring of dyadic rationals.
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Emmanuel Jeandel, Simon Perdrix, Renaud Vilmart. A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics. The 33rd Annual Symposium on Logic in Computer Science, 2018, Jul 2018, Oxford, United Kingdom. pp.559--568, ⟨10.1145/3209108.3209131⟩. ⟨hal-01529623v2⟩

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