  We study the approximation of expectations $\E(f(X))$ for Gaussian random
elements $X$ with values in a separable Hilbert space $H$ and Lipschitz
continuous functionals $f \colon H \to \R$. We consider restricted Monte Carlo
algorithms, which may only use random bits instead of random numbers. We
determine the asymptotics (in some cases sharp up to multiplicative constants,
in the other cases sharp up to logarithmic factors) of the corresponding $n$-th
minimal error in terms of the decay of the eigenvalues of the covariance
operator of $X$. It turns out that, within the margins from above, restricted
Monte Carlo algorithms are not inferior to arbitrary Monte Carlo algorithms,
and suitable random bit multilevel algorithms are optimal. The analysis of this
problem leads to a variant of the quantization problem, namely, the optimal
approximation of probability measures on $H$ by uniform distributions supported
by a given, finite number of points. We determine the asymptotics (up to
multiplicative constants) of the error of the best approximation for the
one-dimensional standard normal distribution, for Gaussian measures as above,
and for scalar autonomous SDEs.
