  The broad success of theoretical and experimental quantum optimal control is
intimately connected to the topology of the underlying control landscape. For
several common quantum control goals, including the maximization of an
observable expectation value, the landscape has been shown to lack local optima
if three assumptions are satisfied: (i) the quantum system is controllable,
(ii) the Jacobian of the map from the control field to the evolution operator
is full-rank, and (iii) the control field is not constrained. In the case of
the observable objective, this favorable analysis shows that the associated
landscape also contains saddles, i.e., critical points that are not local
suboptimal extrema. In this paper, we investigate whether the presence of these
saddles affects the trajectories of gradient-based searches for an optimal
control. We show through simulations that both the detailed topology of the
control landscape and the parameters of the system Hamiltonian influence
whether the searches are attracted to a saddle. For some circumstances with a
special initial state and target observable, optimizations may approach a
saddle very closely, reducing the efficiency of the gradient algorithm.
Encounters with such attractive saddles are found to be quite rare. Neither the
presence of a large number of saddles on the control landscape nor a large
number of system states increase the likelihood that a search will closely
approach a saddle. Even for applications that encounter a saddle, well-designed
gradient searches with carefully chosen algorithmic parameters will readily
locate optimal controls.
