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We study single-qutrit gates composed of Clifford and T gates, using the qutrit version of the T gate proposed by Howard and Vala [M. Howard and J. Vala, Physical Review A 86, 022316 (2012)].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. * sprakash@dei.ac.in

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We propose a normal form for single-qudit gates composed of Clifford and T -gates for qudits of odd prime dimension p ≥ 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.