  The quantum speed limit sets the minimum time required to transfer a quantum
system completely into a given target state. At shorter times the higher
operation speed has to be paid with a loss of fidelity. Here we quantify the
trade-off between the fidelity and the duration in a system driven by a
time-varying control. The problem is addressed in the framework of Hilbert
space geometry offering an intuitive interpretation of optimal control
algorithms. This approach is applied to non-uniform time variations which leads
to a necessary criterion for control optimality applicable as a measure of
algorithm convergence. The time fidelity trade-off expressed in terms of the
direct Hilbert velocity provides a robust prediction of the quantum speed limit
and allows to adapt the control optimization such that it yields a predefined
fidelity. The results are verified numerically in a multilevel system with a
constrained Hamiltonian, and a classification scheme for the control sequences
is proposed based on their optimizability.
