  Quantum optimal control has enjoyed wide success for a variety of theoretical
and experimental objectives. These favorable results have been attributed to
advantageous properties of the corresponding control landscapes, which are free
from local optima if three conditions are met: (1) the quantum system is
controllable, (2) the Jacobian of the map from the control field to the
evolution operator is full rank, and (3) the control field is not constrained.
This paper explores how gradient searches for globally optimal control fields
are affected by deviations from assumption (2). In some quantum control
problems, so-called singular critical points, at which the Jacobian is
rank-deficient, may exist on the landscape. Using optimal control simulations,
we show that search failure is only observed when a singular critical point is
also a second-order trap, which occurs if the control problem meets additional
conditions involving the system Hamiltonian and/or the control objective. All
known second-order traps occur at constant control fields, and we also show
that they only affect searches that originate very close to them. As a result,
even when such traps exist on the control landscape, they are unlikely to
affect well-designed gradient optimizations under realistic searching
conditions.
