Completeness of the Phase-free ZH-calculus

Understand

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.