Tunable inductive coupling of superconducting qubits in the strongly nonlinear regime

For a variety of superconducting qubits, tunable interactions are achieved through mutual inductive coupling to a coupler circuit containing a nonlinear Josephson element. In this paper we derive the general interaction mediated by such a circuit under the Born-Oppenheimer approximation. This interaction naturally decomposes into a classical part with origin in the classical circuit equations and a quantum part associated with the zero-point energy of the coupler. Our result is non-perturbative in the qubit-coupler coupling strengths and circuit nonlinearities, leading to significant departures from previous treatments in the nonlinear or strong coupling regimes. Specifically, it displays no divergences for large coupler nonlinearities, and it can predict k-body and non-stoquastic interactions that are absent in linear theories. Our analysis provides explicit and efficiently computable series for any term in the interaction Hamiltonian and can be applied to any superconducting qubit type.