  We provide a quantitative blueprint for realizing two-dimensional quantum
simulators employing ultracold dipolar molecules or magnetic atoms by studying
their accuracy in predicting ground state properties of lattice models with
long-range interactions. For experimentally relevant ranges of potential
depths, interaction strengths, particle fillings, and geometric configurations,
we map out the agreement between the state prepared in the quantum simulator
and the target lattice state. We do so by separately calculating numerically
exact many-body wave functions in the continuum and single- or multi-band
lattice representations, and building their many-body state overlaps. While the
agreement between quantum simulator and single-band models is good for deep
optical lattices with weaker interactions and low particle densities, the
higher band population rapidly increases for shallow lattices, stronger
interactions, and in particular above half filling. This induces drastic
changes to the properties of the simulated ground state, potentially leading to
false predictions. Furthermore, we show that the interplay between
commensurability and interactions can lead to quasidegeneracies, rendering a
faithful ground state preparation even more challenging.
