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Effective Hamiltonian models of the cross-resonance gate

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.