  We consider the stochastic Cahn-Hilliard equation with additive space-time
white noise $\epsilon^{\gamma}\dot{W}$ in dimension $d=2,3$, where $\epsilon>0$
is an interfacial width parameter. We study numerical approximation of the
equation which combines a structure preserving implicit time-discretization
scheme with a discrete approximation of the space-time white noise. We derive a
strong error estimate for the considered numerical approximation which is
robust with respect to the inverse of the interfacial width parameter
$\epsilon$. Furthermore, by a splitting approach, we show that for sufficiently
large scaling parameter $\gamma$, the numerical approximation of the stochastic
Cahn-Hilliard equation converges uniformly to the deterministic
Hele-Shaw/Mullins-Sekerka problem in the sharp interface limit
$\epsilon\rightarrow 0$.
