  This paper discusses the important role of controllability played on the
complexity of optimizing quantum mechanical control systems. The study is based
on a topology analysis of the corresponding quantum control landscape, which is
referred to as the optimization objective as a functional of control fields. We
find that the degree of controllability is closely relevant with the ruggedness
of the landscape, which determines the search efficiency for global optima.
This effect is demonstrated via the gate fidelity control landscape of a system
whose controllability is restricted on a SU(2) dynamic symmetry group. We show
that multiple local false traps (i.e., non-global suboptima) exist even if the
target gate is realizable and that the number of these traps is increased by
the loss of controllability, while the controllable systems are always devoid
of false traps.
