An Algebraic Axiomatisation of ZX-calculus

ZX-calculus is a graphical language for quantum computing which complete in the sense that calculation matrices can be done purely diagrammatic way. However, all previous universally axiomatisations of have included at least one rule involving trigonometric functions such as sin and cos makes it difficult application purpose. In this paper we give an algebraic axiomatisation instead there are only ring operations involved phases. With ZX-calculus, able to establish first time simple translation diagrams from another called ZH-calculus derive ZX-translated rules ZH-calculus. As consequence, great benefit techniques obtained transplanted can't by just using completeness ZX-calculus.