In Quantum Computing, the default language for expressing computations is the Quantum Circuit, comparable to the assembly language in classical computing. As quantum operations are unitary, the entire circuit represents a unitary transformation. This means it can be expressed using different arrangements and counts of gates. As quantum computers are still quite susceptible to errors, minimizing the number of gates and therefore operations to be performed is crucial for obtaining precise results. This can be achieved by different means, one of them being the use of graphical (diagrammatic) representations and respective calculi.

The ZX-Calculus is a graphical notation for representing linear maps between qubits, referred to as diagrams. It is therefore a generalization of the quantum circuit which is only capable of unitary maps. Being a calculus, it also contains rewrite rules allowing the transformation of one diagram into a different one representing the same linear map. A circuit can be optimized, by first converting it into a ZX diagram. We then optimize the diagram using the rewrite rules of the calculus, preserving the equality of the underlying map. Afterwards, we extract a now-optimized circuit from the resulting diagram.