Current efforts to build quantum computers focus mainly on the two-state qubit, which often involves suppressing readily-available higher states. In this work, we break this abstraction and synthesize short-duration control pulses for gates on generalized d-state qudits. We present Incremental Pulse Re-seeding, a practical scheme to guide optimal control software to the lowest-duration pulse by iteratively seeding the optimizer with previous results. We find a near-linear relationship between Hilbert space dimension and gate duration through explicit pulse optimization for one- and two-qudit gates on transmons. Our results suggest that qudit operations are much more efficient than previously expected in the practical regime of interest and have the potential to significantly increase the computational power of current hardware.
* indicates equal contribution