  Control of quantum systems via time-varying external fields optimized to
maximize a fidelity measure at a given time is a mainstay in modern quantum
control. However, save for specific systems, current analysis techniques for
such quantum controllers provide no analytical robustness guarantees. In this
letter we provide analytical bounds on the differential sensitivity of the gate
fidelity error to structured uncertainties for a closed quantum system
controlled by piecewise-constant, optimal control fields. We additionally
determine those uncertainty structures that result in this worst-case maximal
sensitivity. We then use these differential sensitivity bounds to provide
conditions that guarantee performance, quantified by the fidelity error, in the
face of parameter uncertainty.
