AND-gates in ZX-calculus: Spider Nest Identities and QBC-completeness

In this paper we exploit the utility of triangle symbol which has a complicated expression in terms spider diagrams ZX-calculus, and its role within ZX-representation AND-gates particular. First, derive nest identities are key importance to recent developments quantum circuit optimisation T-count reduction Then, using same rule set, prove completeness theorem for Boolean circuits (QBCs) whose rewriting rules can be directly used new method reduction. We give an algorithm based on show that results our outperform all previous best non-probabilistic algorithms.