  There is a strong interest in optimal manipulating of quantum systems by
external controls. Traps are controls which are optimal only locally but not
globally. If they exist, they can be serious obstacles to the search of
globally optimal controls in numerical and laboratory experiments, and for this
reason the analysis of traps attracts considerable attention. In this paper we
prove that for a wide range of control problems for two-level quantum systems
all locally optimal controls are also globally optimal. Hence we conclude that
two-level systems in general are trap-free. In particular, manipulating
qubits---two-level quantum systems forming a basic building block for quantum
computation---is free of traps for fundamental problems such as the state
preparation and gate generation.
