  The recent implementation of a swap Monte Carlo algorithm (SWAP) for
polydisperse mixtures fully bypasses computational sluggishness and closes the
gap between experimental and simulation timescales in physical dimensions $d=2$
and $3$. Here, we consider suitably optimized systems in $d=2, 3,\dots, 8$, to
obtain insights into the performance and underlying physics of SWAP. We show
that the speedup obtained decays rapidly with increasing the dimension. SWAP
nonetheless delays systematically the onset of the activated dynamics by an
amount that remains finite in the limit $d \to \infty$. This shows that the
glassy dynamics in high dimensions $d>3$ is now computationally accessible
using SWAP, thus opening the door for the systematic consideration of
finite-dimensional deviations from the mean-field description.
