Fermionic quantum simulation on Andreev bound state superlattices

Arrays of superconducting qubits and cavities offer a promising route for realizing artificial materials. However, many analog simulations on superconducting circuit hardware have focused on bosonic systems. Fermionic simulations, in contrast, have largely relied on digital approaches. Here, we propose and study an alternative approach for analog fermionic quantum simulation based on arrays of 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 wave-function overlap, mediate quasiparticles hoppings. By developing a Wannier function approach, we show that these Andreev bound state arrays form an all-superconducting platform for emulating lattice models of fermionic quasiparticles that are phase and gate programmable. Interestingly, the junction lattices can also 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 topological materials on Andreev bound state quantum simulators.