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

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, Phys. Rev. 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.