Suppressing leakage and maintaining robustness in transmon qubits: Signatures of a trade-off relation

We study the problem of optimally generating quantum gates in a logical subspace embedded in a larger Hilbert space, where the dynamics is also affected by unknown static imperfections. This general problem is widespread across various emergent quantum technology architectures. We derive the fidelity susceptibility in the computational subspace as a measure of robustness to perturbations, and define a cost function that quantifies leakage out of the subspace. We tackle both effects using a two-stage optimization where two cost functions are minimized in series. Specifically, we apply this framework to the generation of single-qubit gates in a superconducting transmon system, and find high-fidelity solutions robust to detuning and amplitude errors across various parameter regimes. We also show control pulses which maximize fidelity while minimizing leakage at all times during the evolution. However, finding control solutions that address both effects simultaneously is shown to be much more challenging, indicating the presence of a trade-off relation.