Multi-qutrit exact synthesis

We present an exact synthesis algorithm for qutrit unitaries in ${\mathcal{U}}_{3^n}(\mathbb{Z}[1\mathrm{/}3,e^{2\pi i\mathrm{/}3}])$ 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 ${\mathcal{U}}_{3^n}(\mathbb{Z}[1\mathrm{/}3,e^{2\pi i\mathrm{/}9}])$ 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.