An Algebraic Axiomatisation of ZX-calculus

TL;DR

This paper presents an algebraic, trigonometry-free complete axiomatisation of ZX-calculus, enabling simpler diagram translation from ZH-calculus and broadening its practical applications in quantum computing.

Contribution

It introduces the first algebraic complete axiomatisation of ZX-calculus using only ring operations, avoiding trigonometric rules.

Findings

Established a simple translation from ZH-calculus to ZX-calculus.

Derived all ZX-translated rules of ZH-calculus.

Enabled techniques from ZH-calculus to be applied in ZX-calculus.

Abstract

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