A ZX-Calculus with Triangles for Toffoli-Hadamard, Clifford+T, and Beyond

TL;DR

This paper introduces an extended ZX-calculus with triangle nodes, providing a complete and simplified axiomatisation for Toffoli-Hadamard and Clifford+T quantum mechanics, enhancing reasoning capabilities in quantum computing.

Contribution

It develops a new ZX-calculus with triangles, proves its completeness for specific quantum fragments, and simplifies the axiomatisation for Clifford+T quantum mechanics.

Findings

The calculus precisely characterizes matrices in the Toffoli-Hadamard fragment.

A single axiom suffices for completeness in Clifford+T quantum mechanics.

The new axiomatisation simplifies reasoning in quantum computing.

Abstract

We consider a ZX-calculus augmented with triangle nodes which is well-suited to reason on the so-called Toffoli-Hadamard fragment of quantum mechanics. We precisely show the form of the matrices it represents, and we provide an axiomatisation which makes the language complete for the Toffoli-Hadamard quantum mechanics. We extend the language with arbitrary angles and show that any true equation involving linear diagrams which constant angles are multiple of Pi are derivable. We show that a single axiom is then necessary and sufficient to make the language equivalent to the ZX-calculus which is known to be complete for Clifford+T quantum mechanics. As a by-product, it leads to a new and simple complete axiomatisation for Clifford+T quantum mechanics.