We present two deterministic compilation algorithms for single-qutrit unitaries with $O(log1/\varepsilon )$ gate depth. Each algorithm selects a nearby approximation to the target unitary and then exactly synthesizes the approximation over the Clifford $+$ $\mathbf{R}$ basis. The first algorithm exhaustively searches over the group; while the second algorithm searches only for Householder reflections. The exhaustive search algorithm yields an average $\mathbf{R}$ count of $2.193\left(11\right)+8.621\left(7\right){log}_{10}(1/\varepsilon )$, albeit with a time complexity of $O\left({\varepsilon }^{-4.4}\right)$. The Householder search algorithm results in a larger average $\mathbf{R}$ count of $3.20\left(13\right)+10.77\left(3\right){log}_{10}(1/\varepsilon )$ at a reduced time complexity of $O\left({\varepsilon }^{-0.42}\right)$, greatly extending the reach in $\varepsilon$. These costs correspond asymptotically to 35% and 69% more non-Clifford gates compared with synthesizing the same unitary with two qubits. Such initial results are encouraging for using the $\mathbf{R}$ gate as the nontransversal gate for qutrit-based computation.

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• Quantum circuits
• Quantum error correction