  We study the dynamics of a quantum rotator kicked according to the
almost-periodic Fibonacci sequence. A special numerical technique allows us to
carry on this investigation for as many as $10^{12}$ kicks. It is shown that
above a critical kick strength the excitation of the system is well described
by regular diffusion, while below this border it becomes anomalous, and
sub-diffusive. A law for the dependence of the exponent of anomalous
sub-diffusion on the system parameters is established numerically. The analogy
between these results and quantum diffusion in models of quasi-crystal and in
the kicked Harper system is discussed.
