On extrema of the objective functional for the problem of ultra-fast generation of single qubit quantum gates

We consider the control problem of a qubit (a two-level quantum system) by using coherent control, e.g., electromagnetic field. A qubit is the basic element for the implementation of quantum computing. We consider the problem of existence of traps — local but not global maxima of the objective functional. We prove that the objective functional which described generation of a single qubit quantum gate has no traps for almost all parameters of the problem for arbitrarily small control durations. For the remained set of parameters a new estimate for the lower boundary of the control duration which ensures the absence of traps is obtained.