  We study the controllability of a central spin guided by a classical field
and interacting with a spin bath, showing that the central spin is fully
controllable independently of the number of bath spins. Additionally we find
that for unequal system-bath couplings even the bath becomes controllable by
acting on the central spin alone. We then analyze numerically how the time to
implement gates on the central spin scales with the number of bath spins and
conjecture that for equal system-bath couplings it reaches a saturation value.
We provide evidence that sometimes noise can be effectively suppressed through
control.
