  Quantum optimal control experiments and simulations have successfully
manipulated the dynamics of systems ranging from atoms to biomolecules.
Surprisingly, these collective works indicate that the effort (i.e., the number
of algorithmic iterations) required to find an optimal control field appears to
be essentially invariant to the complexity of the system. The present work
explores this matter in a series of systematic optimizations of the
state-to-state transition probability on model quantum systems with the number
of states $N$ ranging from 5 through 100. The optimizations occur over a
landscape defined by the transition probability as a function of the control
field. Previous theoretical studies on the topology of quantum control
landscapes established that they should be free of sub-optimal traps under
reasonable physical conditions. The simulations in this work include nearly
5000 individual optimization test cases, all of which confirm this prediction
by fully achieving optimal population transfer of at least 99.9% upon careful
attention to numerical procedures to ensure that the controls are free of
constraints. Collectively, the simulation results additionally show invariance
of required search effort to system dimension $N$. This behavior is
rationalized in terms of the structural features of the underlying control
landscape. The very attractive observed scaling with system complexity may be
understood by considering the distance traveled on the control landscape during
a search and the magnitude of the control landscape slope. Exceptions to this
favorable scaling behavior can arise when the initial control field fluence is
too large or when the target final state recedes from the initial state as $N$
increases.
