Qutrit Dichromatic Calculus and Its Universality

TL;DR

This paper introduces a dichromatic calculus for qutrit systems, demonstrating its universality for quantum mechanics and providing new insights into qutrit gate decompositions and quantum algorithms.

Contribution

It presents a new dichromatic calculus for qutrits, proves the universality of qudit ZX calculus, and offers a quantum algorithm example with a single qutrit.

Findings

Decomposition of qutrit Hadamard gate is non-unique.

Counterexample to Ranchin's universality proof.

Qudit ZX calculus is universal for quantum mechanics.

Abstract

We introduce a dichromatic calculus (RG) for qutrit systems. We show that the decomposition of the qutrit Hadamard gate is non-unique and not derivable from the dichromatic calculus. As an application of the dichromatic calculus, we depict a quantum algorithm with a single qutrit. Since it is not easy to decompose an arbitrary d by d unitary matrix into Z and X phase gates when d > 2, the proof of the universality of qudit ZX calculus for quantum mechanics is far from trivial. We construct a counterexample to Ranchin's universality proof, and give another proof by Lie theory that the qudit ZX calculus contains all single qudit unitary transformations, which implies that qudit ZX calculus, with qutrit dichromatic calculus as a special case, is universal for quantum mechanics.