  Computing the electronic structure of molecules with high precision is a
central challenge in the field of quantum chemistry. Despite the enormous
success of approximate methods, tackling this problem exactly with conventional
computers is still a formidable task. This has triggered several theoretical
and experimental efforts to use quantum computers to solve chemistry problems,
with first proof-of-principle realizations done in a digital manner. An
appealing alternative to the digital approach is analog quantum simulation,
which does not require a scalable quantum computer, and has already been
successfully applied in condensed matter physics problems. However, all
available or planned setups cannot be used in quantum chemistry problems, since
it is not known how to engineer the required Coulomb interactions with them.
Here, we present a new approach to the simulation of quantum chemistry problems
in an analog way. Our method relies on the careful combination of two
technologies: ultra-cold atoms in optical lattices and cavity QED. In the
proposed simulator, fermionic atoms hopping in an optical potential play the
role of electrons, additional optical potentials provide the nuclear
attraction, and a single spin excitation over a Mott insulator mediates the
electronic Coulomb repulsion with the help of a cavity mode. We also provide
the operational conditions of the simulator and benchmark it with a simple
molecule. Our work opens up the possibility of efficiently computing electronic
structures of molecules with analog quantum simulation.
