Completeness of the ZX-calculus for Pure Qubit Clifford+T Quantum Mechanics

Abstract

Recently, we gave a complete axiomatisation of the ZX-calculus for the overall pure qubit quantum mechanics. Based on this result, here we also obtain a complete axiomatisation of the ZX-calculus for the Clifford+T quantum mechanics by restricting the ring of complex numbers to its subring corresponding to the Clifford+T fragment resting on the completeness theorem of the ZW-calculus for arbitrary commutative ring. In contrast to the first complete axiomatisation of the ZX-calculus for the Clifford+T fragment, we have two new generators as features rather than novelties: the triangle can be employed as an essential component to construct a Toffoli gate in a very simple form, while the lambda box can be slightly extended to a generalised phase so that the generalised supplementarity (cyclotomic supplementarity ) is naturally seen as a special case of the generalised spider rule.