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 $\epsilon\rightarrow 0$.

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