Completeness of the ZH-calculus

TL;DR

This paper introduces the ZH-calculus, a graphical language for quantum computation that effectively encodes complex Boolean operations like the Toffoli gate, and proves its completeness for matrices over certain rings.

Contribution

It establishes the completeness of the ZH-calculus for matrices over $\

Findings

ZH-calculus can encode Toffoli gates and complex Boolean logic.

Simple rewrite rules make the ZH-calculus complete.

Extended ZH-calculus is complete over rings where 1+1 is not a zero-divisor.

Abstract

There are various gate sets used for describing quantum computation. A particularly popular one consists of Clifford gates and arbitrary single-qubit phase gates. Computations in this gate set can be elegantly described by the ZX-calculus, a graphical language for a class of string diagrams describing linear maps between qubits. The ZX-calculus has proven useful in a variety of areas of quantum information, but is less suitable for reasoning about operations outside its natural gate set such as multi-linear Boolean operations like the Toffoli gate. In this paper we study the ZH-calculus, an alternative graphical language of string diagrams that does allow straightforward encoding of Toffoli gates and other more complicated Boolean logic circuits. We find a set of simple rewrite rules for this calculus and show it is complete with respect to matrices over $\mathbb{Z}[\frac12]$, which correspond to the approximately universal Toffoli+Hadamard gateset. Furthermore, we construct an extended version of the ZH-calculus that is complete with respect to matrices over any ring $R$ where $1+1$ is not a zero-divisor.