TY - JOUR

T1 - Completeness of the ZH-calculus

AU - Backens, Miriam

AU - Kissinger, Aleks

AU - Miller-Bakewell, Hector

AU - van de Wetering, John

AU - Wolffs, Sal

PY - 2023/7/12

Y1 - 2023/7/12

N2 - 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[½], 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.

AB - 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[½], 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.

UR - https://arxiv.org/abs/2103.06610

UR - https://compositionality-journal.org/

U2 - 10.32408/compositionality-5-5

DO - 10.32408/compositionality-5-5

M3 - Article

SN - 2631-4444

VL - 5

JO - Compositionality

JF - Compositionality

IS - 5

ER -