The efficient compiling of arbitrary single qubit gates into a sequence of gates from an inverse-closed finite gate set is of fundamental importance in quantum computation. The exact bounds of this compiling are given by the Solovay-Kitaev theorem, which serves as a powerful tool in compiling quantum algorithms that require many qubits. However, the inverse-closure condition it imposes on the gate set adds a certain complexity to the experimental compilation, making the process less-efficient. This was recently resolved by a version of the Solovay-Kitaev theorem for inverse-free gate sets, yielding a significant gain.
Considering the recent progress in the direction of three-level quantum systems, in which qubits are replaced by qutrits, it is possible to achieve the quantum speedup guaranteed by the Solovay-Kitaev theorem simply from orthogonal gates. Nevertheless, it has not been investigated previously whether the condition of inverse-closure can be relaxed for these qutrit-based orthogonal compilations as well. In this work we answer this positively, by obtaining improved Solovay-Kitaev approximations to an arbitrary orthogonal qutrit gate, to an accuracy ε from a sequence of O(log8.62(1/ε)) orthogonal gates taken from an inverse-free set.