  We present an iterative optimal control method of quantum systems, aimed at
an implementation of a desired operation with optimal fidelity. The update step
of the method is based on the linear response of the fidelity to the control
operators, and its projection onto the mode functions of the corresponding
operator. Our method extends methods such as gradient ascent pulse engineering
and variational quantum algorithms, by determining the fidelity gradient in a
hyperparameter-free manner, and using it for a multi-parameter update,
capitalizing on the multi-mode overlap of the perturbation and the mode
functions. This directly reduces the number of dynamical trajectories that need
to be evaluated in order to update a set of parameters. We demonstrate this
approach, and compare it to the standard GRAPE algorithm, for the example of a
quantum gate on two qubits, demonstrating a clear improvement in convergence
and optimal fidelity of the generated protocol.
