  A closed quantum system is defined as completely controllable if an arbitrary
unitary transformation can be executed using the available controls. In
practice, control fields are a source of unavoidable noise. Can one design
control fields such that the effect of noise is negligible on the time-scale of
the transformation? Complete controllability in practice requires that the
effect of noise can be suppressed for an arbitrary transformation. The present
study considers a paradigm of control, where the Lie-algebraic structure of the
control Hamiltonian is fixed, while the size of the system increases,
determined by the dimension of the Hilbert space representation of the algebra.
We show that for large quantum systems, generic noise in the controls dominates
for a typical class of target transformations i.e., complete controllability is
destroyed by the noise.
