A normal form for single-qudit Clifford+T operators

Abstract

Normal forms play a crucial role in the problem of gate synthesis. We propose a normal form for single-qudit gates composed of Clifford and T-gates for qudits of odd prime dimension $$p\ge 5$$ . We prove that any single-qudit Clifford+T operator can be re-expressed in this normal form in polynomial time. We obtained strong computational evidence that this normal form is unique. Assuming uniqueness, we are able to use this normal form to provide an efficient algorithm for exact synthesis of any single-qudit Clifford+T operator with minimal T-count.