Universal pulses for superconducting qudit ladder gates

Abstract: Qudits, generalizations of qubits to multi-level quantum systems, offer enhanced computational efficiency by encoding more information per lattice cell, avoiding costly swap operations and providing even exponential speedup in some cases. Utilizing the $d$-level manifold, however, requires high-speed gate operations because of the stronger decoherence at higher levels. While analytical control methods have proven effective for qubits in achieving fast gates with minimal control errors, their extension to qudits is nontrivial due to the increased complexity of the energy level structure arising from additional ancillary states. In this work, we present a universal pulse construction for generating rapid, high-fidelity unitary rotations between adjacent qudit levels, thereby providing a prescription for any gate in $SU(d)$. Control errors in these operations are effectively analyzed within a four-level subspace, including two leakage levels with approximately opposite detuning. By identifying the optimal degrees of freedom, we derive concise analytical pulse schemes that suppress multiple control errors and outperform existing methods. Remarkably, our approach achieves consistent coherent error scaling across all levels, approaching the quantum speed limit independently of parameter variations between levels. Validation on transmon circuits demonstrates significant improvements in gate fidelity for various qudit sizes aiming for $10^{-4}$ error. This method provides a scalable solution for improving qudit control and can be broadly applied to other quantum systems with ladder structures or operations involving multiple ancillary levels.