  In the last decade, quantum simulators, and in particular cold atoms in
optical lattices, have emerged as a valuable tool to study strongly correlated
quantum matter. These experiments are now reaching regimes that are numerically
difficult or impossible to access. In particular they have started to fulfill a
promise which has contributed significantly to defining and shaping the field
of cold atom quantum simulations, namely the exploration of doped and
frustrated quantum magnets and the search for the origins of high-temperature
superconductivity in the fermionic Hubbard model. Despite many future
challenges lying ahead, such as the need to further lower the experimentally
accessible temperatures, remarkable studies have already emerged. Among them,
spin-charge separation in one-dimensional systems has been demonstrated,
extended-range antiferromagnetism in two-dimensional systems has been observed,
connections to modern day large-scale numerical simulations were made, and
unprecedented comparisons with microscopic trial wavefunctions have been
carried out at finite doping. In many regards, the field has acquired new
realms, putting old ideas to a new test and producing new insights and
inspiration for the next generation of physicists. In the first part of this
paper, we review the results achieved in cold atom realizations of the
Fermi-Hubbard model in recent years. In the second part of this paper, with the
stage set and the current state of the field in mind, we propose a new
direction for cold atoms to explore: namely mixed-dimensional bilayer systems,
where the charge motion is restricted to individual layers which remain coupled
through spin-exchange. We propose a novel, strong pairing mechanism in these
systems, which puts the formation of hole pairs at experimentally accessible,
elevated temperatures within reach.
