  Arrays of superconducting qubits and cavities offer a promising route for
realizing highly controllable artificial materials. However, many analog
simulations of superconducting circuit hardware have focused on bosonic
systems. Fermionic simulations, on the other hand, have largely relied on
digital approaches that require non-local qubit couplings, which could limit
their scalability. Here, we propose and study an alternative approach for
analog fermionic quantum simulation based on arrays of coherently coupled
mesoscopic Josephson junctions. These Josephson junction arrays implement an
effective superlattice of Andreev bound state "atoms" that can trap individual
fermionic quasiparticles and, due to their wavefunction overlap, mediate
quasiparticle hoppings. By developing a Wannier function approach, we show that
these Andreev bound state arrays form an all-superconducting and circuit
QED-compatible platform for emulating lattice models of fermionic
quasiparticles that are phase- and gate-programmable. Interestingly, we also
find that the junction lattices can undergo a topological transition and host
fermionic boundary modes that can be probed by conductance measurements. We
hope our results will inspire the realization of artificial and possibly
topological materials on Andreev bound state quantum simulators.
