Tunable coupler for superconducting Xmon qubits: Perturbative nonlinear   model
  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. 
Michael R. Geller, Emmanuel Donate, Yu Chen, Charles Neill, Pedram   Roushan, John M. Martinis
quant-ph cond-mat.supr-con