  For a family of 1-d quantum harmonic oscillator with a perturbation which is
$C^2$ parametrized by $E\in{\mathcal I}\subset{\Bbb R}$ and quadratic on $x$
and $-{\rm i}\partial_x$ with coefficients quasi-periodically depending on time
$t$, we show the reducibility (i.e., conjugation to time-independent) for a.e.
$E$. As an application of reducibility, we describe the behaviors of solution
in Sobolev space:
  -- Boundedness w.r.t. $t$ is always true for "most" $E\in{\mathcal I}$.
  -- For "generic" time-dependent perturbation, polynomial growth and
exponential growth to infinity w.r.t. $t$ occur for $E$ in a "small" part of
${\mathcal I}$.
  Concrete examples are given for which the growths of Sobolev norm do occur.
