  We study a recently demonstrated design for a high-performance tunable
coupler suitable for superconducting Xmon and planar transmon qubits [Y. Chen
et al., arXiv:1402.7367]. The coupler circuit uses a single flux-biased
Josephson junction and acts as a tunable current divider. We calculate the
effective qubit-qubit interaction Hamiltonian by treating the nonlinearity of
the qubit and coupler junctions perturbatively. We find that the qubit
nonlinearity has two principal effects: The first is to suppress the magnitude
of the transverse XX coupling from that obtained in the harmonic approximation
by about 15%. The second is to induce a small diagonal ZZ coupling. The effects
of the coupler junction nonlinearity are negligible in the parameter regime
considered.
