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

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