Miriam Backens ; Aleks Kissinger ; Hector Miller-Bakewell ; John van de Wetering ; Sal Wolffs - Completeness of the ZH-calculus

compositionality:13524 - Compositionality, July 12, 2023, Volume 5 (2023) - https://doi.org/10.32408/compositionality-5-5
Completeness of the ZH-calculusArticle

Authors: Miriam Backens 1; Aleks Kissinger 2; Hector Miller-Bakewell 2; John van de Wetering 2; Sal Wolffs 3

  • 1 School of Computer Science, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK
  • 2 Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, UK
  • 3 Institute for Computing and Information Sciences, Radboud Universiteit, Toernooiveld 212, 6525 EC Nijmegen, NL

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.


Volume: Volume 5 (2023)
Published on: July 12, 2023
Imported on: May 2, 2024
Keywords: Quantum Physics
Funding:
    Source : OpenAIRE Graph
  • Diagrammatic Quantum Computation; Funder: European Commission; Code: 101018390

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