Normal form for single-qutrit Clifford+T operators and synthesis of single-qutrit gates

TL;DR

This paper introduces a normal form for single-qutrit Clifford+T gates, proving its optimality and uniqueness, and provides an efficient algorithm for exact gate synthesis.

Contribution

It extends the normal form concept to qutrits, enabling optimal and unique representation of Clifford+T operators with an efficient synthesis algorithm.

Findings

Normal form for single-qutrit Clifford+T gates is established.

The normal form is proven to be optimal and unique.

An algorithm for exact synthesis of qutrit Clifford+T operators is provided.

Abstract

We study single-qutrit gates composed of Clifford and $T$ gates, using the qutrit version of the $T$ gate proposed by Howard and Vala. We propose a normal form for single-qutrit gates analogous to the Matsumoto-Amano normal form for qubits. We prove that the normal form is optimal with respect to the number of $T$ gates used and that any string of qutrit Clifford+$T$ operators can be put into this normal form in polynomial time. We also prove that this form is unique and provide an algorithm for exact synthesis of any single qutrit Clifford+$T$ operator.