Canonical forms for single-qutrit Clifford+T operators

TL;DR

This paper introduces unique, T-optimal canonical forms for single-qutrit Clifford+T circuits, along with an efficient algorithm to construct these forms, extending prior qubit-focused work to higher dimensions.

Contribution

It establishes the existence and uniqueness of canonical forms for single-qutrit Clifford+T operators and provides a linear-time algorithm for their construction.

Findings

Canonical forms are unique for each operator.

Canonical forms minimize T gates among all implementations.

Algorithm runs in linear time relative to T gates.

Abstract

We introduce canonical forms for single qutrit Clifford+T circuits and prove that every single-qutrit Clifford+T operator admits a unique such canonical form. We show that our canonical forms are T-optimal in the sense that among all the single-qutrit Clifford+T circuits implementing a given operator our canonical form uses the least number of T gates. Finally, we provide an algorithm which inputs the description of an operator (as a matrix or a circuit) and constructs the canonical form for this operator. The algorithm runs in time linear in the number of T gates. Our results provide a higher-dimensional generalization of prior work by Matsumoto and Amano who introduced similar canonical forms for single-qubit Clifford+T circuits.