Digital quantum simulations with superconducting circuits

Superconducting circuits are a promising quantum technology for the implementation of quantum information protocols. In particular, digital quantum simulations are an efficient method for reproducing dynamics that are not produced naturally in the simulating system. We propose a method for simulating efficiently the dynamics of prototypical spin and fermionic models in circuit quantum electrodynamics architectures with either qubit-qubit pairwise interactions or resonators acting as quantum buses. We show how to implement Ising and Heisenberg spin models, and the Fermi-Hubbard model, making use of the Jordan-Wigner mapping and M\o lmer-S\o rensen gates.