  We study a class of fully-discrete schemes for the numerical approximation of
solutions of stochastic Cahn--Hilliard equations with cubic nonlinearity and
driven by additive noise. The spatial (resp. temporal) discretization is
performed with a spectral Galerkin method (resp. a tamed exponential Euler
method). We consider two situations: space-time white noise in dimension $d=1$
and trace-class noise in dimensions $d=1,2,3$. In both situations, we prove
weak error estimates, where the weak order of convergence is twice the strong
order of convergence with respect to the spatial and temporal discretization
parameters. To prove these results, we show appropriate regularity estimates
for solutions of the Kolmogorov equation associated with the stochastic
Cahn--Hilliard equation, which have not been established previously and may be
of interest in other contexts.
