  In the present work the extrema of the objective functional for the problem
of generation of quantum gates (logical elements for quantum computations) for
two--level systems are investigated for short duration of the control. The
problem of existence of local but not global extrema, the so called traps, is
considered. In prior works the absence of traps was proved for a sufficiently
long control duration. In this paper we prove that for almost all target
unitary operators and system Hamiltonians traps are absent for an arbitrarily
small control duration. For the remainder target unitary operators and
Hamiltonians we obtain a new estimate for the lower boundary of the control
duration which guarantees the absence of traps.
