A Normal Form for Single-Qudit Clifford+$T$ Operators

TL;DR

This paper introduces a polynomial-time normal form for single-qudit Clifford+$T$ operators in odd prime dimensions, enabling efficient and potentially unique representations for exact synthesis with minimal T-count.

Contribution

It presents a novel normal form for single-qudit Clifford+$T$ operators and demonstrates its polynomial-time computability and potential uniqueness, aiding optimal quantum circuit synthesis.

Findings

Normal form can be computed in polynomial time.

Numerical evidence suggests the normal form is unique.

Enables exact synthesis with minimal T-count.

Abstract

We propose a normal form for single-qudit gates composed of Clifford and $T$-gates for qudits of odd prime dimension $p\geq 5$. We prove that any single-qudit Clifford+$T$ operator can be re-expressed in this normal form in polynomial time. We also provide strong numerical evidence that this normal form is unique. Assuming uniqueness, we are able to use this normal form to provide an algorithm for exact synthesis of any single-qudit Clifford+$T$ operator with minimal $T$-count.