A note on operator calculus in quantum mechanics

Gonzalo Marco Gimeno, Mercedes Xipell Gómez del Moral, M. Baig

R. Bruce Lindsay, Raymond J. Seeger

Rolf Hagedorn

Titouan Carette

This note describes how the the scalable ZXH calculus can be used to represent in a compact way the quantum gates that are diagonal in the computational basis. This includes controlled and multi-controlled Z gates, their generalizations, respectively graph and hypergraph operators, and also phase gadgets.

Miriam Backens

The ZX-calculus is a graphical calculus for reasoning about quantum systems and processes. It is known to be universal for pure state qubit quantum mechanics (QM), meaning any pure state, unitary operation and post-selected pure projective measurement can be expressed in the ZX-calculus. The calculus is also sound , i.e. any equality that can be derived graphically can also be derived using matrix mechanics. Here, we show that the ZX-calculus is complete for pure qubit stabilizer QM, meaning any equality that can be derived using matrices can also be derived pictorially. The proof relies on bringing diagrams into a normal form based on graph states and local Clifford operations.