Quantum Set Calculus

Satoko Titani, Haruhiko Kozawa

John van de Wetering, Sal Wolffs

The ZH-calculus is a graphical calculus for linear maps between qubits that allows a natural representation of the Toffoli+Hadamard gate set. The original version of the calculus, which allows every generator to be labelled by an arbitrary complex number, was shown to be complete by Backens and Kissinger. Even though the calculus is complete, this does not mean it allows one to easily reason in restricted settings such as is the case in quantum computing. In this paper we study the fragment of the ZH-calculus that is phase-free, and thus is closer aligned to physically implementable maps. We present a modified rule-set for the phase-free ZH-calculus and show that it is complete. We further discuss the minimality of this rule-set and we give an intuitive interpretation of the rules. Our completeness result follows by reducing to Vilmart's rule-set for the phase-free $Δ$ZX-calculus.

David Finkelstein

Jarosław Pykacz, Luigi Accardi, Guillaume Adenier +5 more

Representation of quantum logics by families of fuzzy sets allows to build fuzzy set‐theoretic models of quantum probability spaces in a way fully analogous to the Kolmogorov construction of a classical probability space. Therefore, quantum probability calculus on the one hand may be seen as a special kind of fuzzy probability calculi, and on the other hand it proves to be more similar to the classical probability calculus than it is generally believed.

Miriam Backens, Aleks Kissinger, Hector Miller-Bakewell +2 more

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[12] , 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.

Ernesto Rodríguez