Completeness of the ZX-Calculus

Quanlong Wang (University of Oxford), KangFeng Ng (Radboud University)

ZX-calculus is a graphical language introduced by Bob Coecke and Ross Duncan for reasoning about processes in quantum theory, especially the interaction between a pair of complementary quantum observables. Although being a graphical language, it is mathematically formalized within the framework of compact closed categories. An important aspect of ZX-calculus is the so-called completeness, which basically means quantum computing can be done purely diagrammatically in the ZX-calculus. This tutorial is to give a thorough survey on various completeness results of the ZX-calculus. Some related completeness results and their relations to the ZX-calculus will be discussed as well, including the completeness of ZW-calculus and ZH-calculus.