  We derive an integral expression for the filter-transfer function of an
arbitrary one-qubit gate through the use of dynamical invariant theory and
Hamiltonian reverse engineering. We use this result to define a cost function
which can be efficiently optimized to produce one-qubit control pulses that are
robust against specified frequency bands of the noise power spectral density.
We demonstrate the utility of our result by generating optimal control pulses
that are designed to suppress broadband detuning and pulse amplitude noise. We
report an order of magnitude improvement in gate fidelity in comparison with
known composite pulse sequences. More broadly, we also use the same theoretical
framework to prove the robustness of nonadiabatic geometric quantum gates under
specific error models and control constraints.
