  The dynamical cluster approximation (DCA) and its DCA$^+$ extension use
coarse-graining of the momentum space to reduce the complexity of quantum
many-body problems, thereby mapping the bulk lattice to a cluster embedded in a
dynamical mean-field host. Here, we introduce a new form of an interlaced
coarse-graining and compare it with the traditional coarse-graining. While it
gives a more localized self-energy for a given cluster size, we show that it
leads to more controlled results with weaker cluster shape and smoother cluster
size dependence, which converge to the results obtained from the standard
coarse-graining with increasing cluster size. Most importantly, the new
coarse-graining reduces the severity of the fermionic sign problem of the
underlying quantum Monte Carlo cluster solver and thus allows for calculations
on larger clusters. This enables the treatment of longer-ranged correlations
than those accessible with the standard coarse-graining and thus can allow for
the evaluation of the exact infinite cluster size result via finite size
scaling. As a demonstration, we study the hole-doped two-dimensional Hubbard
model and show that the interlaced coarse-graining in combination with the
extended DCA$^+$ algorithm permits the determination of the superconducting
$T_c$ on cluster sizes for which the results can be fit with a
Kosterlitz-Thouless scaling law.
