Resource tradeoffs can often be established by solving an appropriate robust optimization problem for a variety of scenarios involving constraints on optimization variables and uncertainties. Using an approach based on sequential convex programming, we demonstrate that quantum gate transformations can be made substantially robust against uncertainties while simultaneously using limited resources of control amplitude and bandwidth. Achieving such a high degree of robustness requires a quantitative model that specifies the range and character of the uncertainties. Using a model of a controlled one-qubit system for illustrative simulations, we identify robust control fields for a universal gate set and explore the tradeoff between the worst-case gate fidelity and the field fluence. Our results demonstrate that, even for this simple model, there exist a rich variety of control design possibilities. In addition, we study the effect of noise represented by a stochastic uncertainty model.

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