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arXiv · 2103.06610

Completeness of the ZH-calculus

Miriam Backens·Aleks Kissinger·Hector Miller-Bakewell·John van de Wetering·Sal Wolffs
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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.

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Miriam Backens, Aleks Kissinger, Hector Miller-Bakewell, John van de Wetering, Sal Wolffs. 2021-03-11. Completeness of the ZH-calculus. https://doi.org/10.32408/compositionality-5-5

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