On extrema of the objective functional for short-time generation of single-qubit quantum gates

We study the extrema of the objective functional in the problem of generation of quantum gates (logical elements for quantum computations) for two-level systems with short duration of the control. We consider the problem of the existence of local but not global extrema, the so-called traps. The absence of traps was previously proved for a sufficiently long duration of the control. We prove that traps are absent for an arbitrarily small duration of the control for almost all target unitary operators and Hamiltonians. For the remaining target unitary operators and Hamiltonians we obtain a new lower bound for the duration of the control which guarantees the absence of traps.