  A variety of physically relevant bilinear Schr\"odinger equations are known
to be globally approximately controllable in large times. There are however
examples which are globally approximately controllable in large times, but not
in small times. This obstruction happens e.g. in the presence of (sub)quadratic
potentials, because Gaussian states are preserved, at least for small times.
  In this work, we provide the first examples of small-time globally
approximately controllable bilinear Schr\"odinger equations. In particular, we
show that a control on the frequency of a quadratic potential permits to
construct approximate solutions that evolve arbitrarily fast along specific
transport flows, namely, space-dilations. Once we have access to
space-dilations, we can exploit them to generate time-contractions. In this
way, we build on previous results of large-time control, to obtain global
approximate controllability in small times.
