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

TL;DR

This paper introduces new spider nest identities in ZX-calculus, proving a completeness theorem for quantum Boolean circuits and presenting an algorithm that outperforms previous T-count reduction methods.

Contribution

It derives novel spider nest identities and establishes a completeness theorem for QBCs, enabling a more effective T-count reduction algorithm.

Findings

New spider nest identities improve quantum circuit optimization.

Proved a completeness theorem for quantum Boolean circuits.

Developed an algorithm that surpasses previous T-count reduction methods.

Abstract

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