ABSTRACT. Categorical quantum mechanics places finite-dimensional quantum theory in the context of compact closed categories, with an emphasis on diagrammatic reasoning. In this framework, two equational diagrammatic calculi have been proposed for pure-state qubit quantum computing: the ZW calculus, developed by Coecke, Kissinger and the first author for the purpose of qubit entanglement classification, and the ZX calculus, introduced by Coecke and Duncan to give an abstract description of complementary observables. Neither calculus, however, provided a complete axiomatisation of their model.

In this paper, we solve this problem by presenting extended versions of ZW and ZX, and showing their completeness for pure-state qubit theory. First, we extend the original ZW calculus to represent states and linear maps with coefficients in an arbitrary commutative ring, and prove completeness by a strategy that rewrites all diagrams into a normal form. We then extend the language and axioms of the original ZX calculus, and show their completeness for pure-state qubit theory through a translation between ZX and ZW, specialised to the ring of complex numbers. Finally, after restricting the ring of complex numbers to a subring corresponding to the approximately universal Clifford+T fragment, we obtain a ZX calculus which axiomatises Clifford+T quantum computing, thus solving two major open problems in categorical quantum mechanics.