Related papers: Completeness of the ZH-calculus

Completeness of the ZH-calculus

URL: http://arxiv.org/abs/2103.06610v2
Date: Tue, 11 Jul 2023 15:54:41 GMT
Title: Completeness of the ZH-calculus
Authors: Miriam Backens, Aleks Kissinger, Hector Miller-Bakewell, John van de Wetering, Sal Wolffs
Abstract summary: We study the ZH-calculus, an alternative graphical language of string diagrams. We find a set of simple rewrite rules for this calculus and show it is complete with respect to matrices over $mathbbZ[frac12]$. 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.
Score: 0.0
License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
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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