  Channel simulation involves generating a sample $Y$ from the conditional
distribution $P_{Y|X}$, where $X$ is a remote realization sampled from $P_X$.
This paper introduces a novel approach to approximate Gaussian channel
simulation using dithered quantization. Our method concurrently simulates $n$
channels, reducing the upper bound on the excess information by half compared
to one-dimensional methods. When used with higher-dimensional lattices, our
approach achieves up to six times reduction on the upper bound. Furthermore, we
demonstrate that the KL divergence between the distributions of the simulated
and Gaussian channels decreases with the number of dimensions at a rate of
$O(n^{-1})$.
