2023/07/12 by Miriam Backens, Aleks Kissinger, Hector Miller-Bakewell +2 · 7 citations
Computer Science · #Algorithm #Calculus (dental) #Computer science #Logic, programming, and type systems #Physics #Quantum #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum gate #Quantum mechanics #Qubit #Toffoli gate
paper · pdf · doi:10.32408/compositionality-5-5
published in Compositionality 5, 5
openalex publication_date 2023/07/12 · rss pubdate 2023/07/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 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.