  It is well known that heat pumps, while being all limited by the same basic
thermodynamic laws, may find realization on systems as "small" and "quantum" as
a three-level maser. In order to quantitatively assess how the performance of
these devices scales with their size, we design generalized $N$-dimensional
ideal heat pumps by merging $N-2$ elementary three-level stages. We set them to
operate in the absorption chiller mode between given hot and cold baths, and
study their maximum achievable cooling power and the corresponding efficiency
as a function of $N$. While the efficiency at maximum power is roughly
size-independent, the power itself slightly increases with the dimension,
quickly saturating to a constant. Thus, interestingly, scaling up autonomous
quantum heat pumps does not render a significant enhancement beyond the optimal
double-stage configuration.
