  We describe a quantum simulator for the Hubbard--Holstein model (HHM),
comprising two dressed Rydberg atom species held in a monolayer by independent
painted potentials, predicting that boson-mediated preformed pairing, and
Berezinskii--Kosterlitz--Thouless (BKT) transition temperatures are
experimentally accessible. The HHM is important for modeling the essential
physics of unconventional superconductors. Experimentally realizable quantum
simulators for HHMs are needed: (1) since HHMs are difficult to solve
numerically and analytically, (2) to explore how competition between
electron-phonon interactions and strong repulsion affects pairing in
unconventional superconductors, (3) to understand the role of boson-mediated
local pairing in pseudogaps and fermion condensates. We propose and study a
quantum simulator for the HHM, using optical lattices, painted using zeros in
the AC stark shift, to control two Rydberg atom species independently within a
monolayer. We predict that interactions are sufficiently tunable to probe: (1)
both HHMs and highly unconventional phonon-mediated repulsions, (2) the
competition between intermediate-strength phonon and Coulomb mediated
interactions, (3) BKT transitions, and preformed pairing that could be used to
examine key hypotheses related to the pseudogap. We discuss how the quantum
simulator can be used to investigate boson-mediated pairing and condensation of
fermions in unconventional superconductors.
