  We present a comprehensive theoretical study of the cross-resonance gate
operation covering estimates for gate parameters and gate error as well as
analyzing spectator qubits and multi-qubit frequency collisions. We start by
revisiting the derivation of effective Hamiltonian models following Magesan et
al. (arXiv:1804.04073). Transmon qubits are commonly modeled as a weakly
anharmonic Kerr oscillator. Kerr theory only accounts for qubit frequency
renormalization, while adopting number states as the eigenstates of the bare
qubit Hamiltonian. Starting from the Josephson nonlinearity and by accounting
for the eigenstates renormalization, due to counter-rotating terms, we derive a
new starting model for the cross-resonance gate with modified qubit-qubit
interaction and drive matrix elements. Employing time-dependent
Schrieffer-Wolff perturbation theory, we derive an effective Hamiltonian for
the cross-resonance gate with estimates for the gate parameters calculated up
to the fourth order in drive amplitude. The new model with renormalized
eigenstates lead to 10-15 percent relative correction of the effective gate
parameters compared to Kerr theory. We find that gate operation is strongly
dependent on the ratio of qubit-qubit detuning and anharmonicity. In
particular, we characterize five distinct regions of operation, and propose
candidate parameter choices for achieving high gate speed and low coherent gate
error when the cross-resonance tone is equipped with an echo pulse sequence.
Furthermore, we generalize our method to include a third spectator qubit and
characterize possible detrimental multi-qubit frequency collisions.
