Completeness of the Phase-free ZH-calculus

TL;DR

This paper proves the completeness of a simplified, phase-free version of the ZH-calculus, a graphical language for quantum maps, making it more practical for quantum computing applications.

Contribution

It introduces a modified rule-set for the phase-free ZH-calculus and proves its completeness, connecting it to existing calculus frameworks.

Findings

The phase-free ZH-calculus is complete with the new rule-set.

The rule-set is minimal and intuitively interpretable.

Completeness is shown via reduction to Vilmart's ΔZX-calculus.

Abstract

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 $\Delta$ZX-calculus.