  The simulation of high-temperature superconducting materials by implementing
strongly correlated fermionic models in optical lattices is one of the major
objectives in the field of analog quantum simulation. Here we show that local
control and optical bilayer capabilities create a versatile toolbox to study
both nickelate and cuprate high-temperature superconductors. On the one hand,
we present a scheme to implement a mixed-dimensional (mixD) bilayer model that
has been proposed to capture the essential pairing physics of pressurized
bilayer nickelates. This allows for the long-sought realization of a state with
long-range superconducting order in current lattice quantum simulation
machines. In particular, we show how coherent pairing correlations can be
accessed in a partially particle-hole transformed and rotated basis. On the
other hand, we demonstrate that control of local gates enables the observation
of $d$-wave pairing order in the two-dimensional (single-layer) repulsive
Fermi-Hubbard model through the simulation of a system with attractive
interactions. Lastly, we introduce a scheme to measure momentum-resolved dopant
densities, providing access to observables complementary to solid-state
experiments -- which is of particular interest for future studies of the
enigmatic pseudogap phase appearing in cuprates.
