  Effective Hamiltonian methods are utilized to model the two-qubit
cross-resonance gate for both the ideal two-qubit case and when higher levels
are included. Analytic expressions are obtained in the qubit case and the
higher-level model is solved both perturbatively and numerically with the
solutions agreeing well in the weak drive limit. The methods are applied to
parameters from recent experiments and accounting for classical cross-talk
effects results in good agreement between theory and experimental results.
