  Dynamic control via optimized, piecewise-constant pulses is a common paradigm
for open-loop control to implement quantum gates. While numerous methods exist
for the synthesis of such controls, there are many open questions regarding the
robustness of the resulting control schemes in the presence of model
uncertainty; unlike in classical control, there are generally no analytical
guarantees on the control performance with respect to inexact modeling of the
system. In this paper a new robustness measure based on the differential
sensitivity of the gate fidelity error to parametric (structured) uncertainties
is introduced, and bounds on the differential sensitivity to parametric
uncertainties are used to establish performance guarantees for optimal
controllers for a variety of quantum gate types, system sizes, and control
implementations. Specifically, it is shown how a maximum allowable perturbation
over a set of Hamiltonian uncertainties that guarantees a given fidelity error,
can be reliably computed. This measure of robustness is inversely proportional
to the upper bound on the differential sensitivity of the fidelity error
evaluated under nominal operating conditions. Finally, the results show that
the nominal fidelity error and differential sensitivity upper bound are
positively correlated across a wide range of problems and control
implementations, suggesting that in the high-fidelity control regime, rather
than there being a trade-off between fidelity and robustness, higher nominal
gate fidelities are positively correlated with increased robustness of the
controls in the presence of parametric uncertainties.
