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

Searching for an optimal control in the presence of saddles on the quantum mechanical observable landscape