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For a variety of superconducting qubits, tunable interactions are achievedthrough mutual inductive coupling to a coupler circuit containing a nonlinearJosephson element. In this paper we derive the general interaction mediated bysuch a circuit under the Born-Oppenheimer Approximation. This interactionnaturally decomposes into a classical part, with origin in the classicalcircuit equations, and a quantum part, associated with the coupler's zero-pointenergy. Our result is non-perturbative in the qubit-coupler coupling strengthsand in the coupler nonlinearity. This can lead to significant departures fromprevious, linear theories for the inter-qubit coupling, includingnon-stoquastic and many-body interactions. Our analysis provides explicit andefficiently computable series for any term in the interaction Hamiltonian andcan be applied to any superconducting qubit type. We conclude with a numericalinvestigation of our theory using a case study of two coupled flux qubits, andin particular study the regime of validity of the Born-OppenheimerApproximation.

Tunable inductive coupling of superconducting qubits in the strongly nonlinear regime