  We present an exact synthesis algorithm for qutrit unitaries in
$\mathcal{U}_{3^n}(\mathbb{Z}[1/3,e^{2\pi i/3}])$ over the Clifford$+T$ gate
set with at most one ancilla. This extends the already known result of qutrit
metaplectic gates being a subset of Clifford$+T$ gate set with one ancilla. As
an intermediary step, we construct an algorithm to convert 3-level unitaries
into multiply-controlled gates, analogous to Gray codes converting 2-level
unitaries into multiply-controlled gates. Finally, using catalytic embeddings,
we present an algorithm to exactly synthesize unitaries
$\mathcal{U}_{3^n}(\mathbb{Z}[1/3,e^{2\pi i/9}])$ over the Clifford$+T$ gate
set with at most 2 ancillas. This, in particular, gives an exact synthesis
algorithm of single-qutrit Clifford$+\mathcal{D}$ over the multi-qutrit
Clifford$+T$ gate set with at most two ancillas.
