  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 coupler's zero-point
energy. Our result is non-perturbative in the qubit-coupler coupling strengths
and in the coupler nonlinearity. This can lead to significant departures from
previous, linear theories for the inter-qubit coupling, including
non-stoquastic and many-body interactions. Our analysis provides explicit and
efficiently computable series for any term in the interaction Hamiltonian and
can be applied to any superconducting qubit type. We conclude with a numerical
investigation of our theory using a case study of two coupled flux qubits, and
in particular study the regime of validity of the Born-Oppenheimer
Approximation.
