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 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.Part 4 (Corollary 1) Let D 1 , D 2 be a diagram of the ZX-Calculus. If ZX π/4 ⊢ D 1 XW W X = D 2 XW W X then ZX π/4 ⊢ D 1 = D 2 . Our main theorem is now obvious:Proof (Proof of Theorem 1). Let D 1 , D 2 be two diagrams of the ZX π/4 -Calculus s.t. D 1 = D 2 . By Part 2, D 1 XW = D 2 XW . By Part 1, the ZW 1/2 -calculus is complete and therefore ZW 1/2 ⊢ D 1 XW = D 2 XW . By Part 3, ZX π/4 ⊢ D 1 XW W X = D 2 XW W X . By Part 4 this implies ZX π/4 ⊢ D 1 = D 2 .⊓ ⊔ This approach gives a completion procedure. It gives a set of equalities between ZX π/4 -diagrams whose derivability proves the completeness of the language. Hence, the new rules of the ZX π/4 -Calculus we introduced have obviously been chosen for Parts 4 and 3 to hold. However they have been greatly simplified from what one can obtain using the approach naively.

The ZX-Calculus is a graphical language for diagrammatic reasoning in quantum mechanics and quantum information theory. An axiomatisation has recently been proven to be complete for an approximatively universal fragment of quantum mechanics, the so-called Clif-ford+T fragment. We focus here on the expressive power of this axiomatisation beyond Clifford+T Quantum mechanics. We consider the full pure qubit quantum mechanics, and mainly prove two results: (i) First, the axiomatisation for Clifford+T quantum mechanics is also complete for all equations involving some kind of linear diagrams. The linearity of the diagrams reflects the phase group structure, an essential feature of the ZX-calculus. In particular all the axioms of the ZX-calculus are involving linear diagrams. (ii) We also show that the axiomatisation for Clifford+T is not complete in general but can be completed by adding a single (non linear) axiom, providing a simpler axiomatisation of the ZXcalculus for pure quantum mechanics than the one recently introduced by Ng&Wang.