  We analyze analytically, semi-analytically, and numerically the operation of
Cross-Resonance (CR) gate for superconducting qubits (transmons). We find that
a relatively simple semi-analytical method gives accurate results for the
CNOT-equivalent gate duration and compensating single-qubit rotations. It also
allows us to minimize the CNOT gate duration over the amplitude of the applied
microwave drive and find dependence on the detuning between the qubits.
However, full numerical simulations are needed to calculate intrinsic fidelity
of the CR gate. We decompose numerical infidelity into contributions from
various physical mechanisms, thus finding the intrinsic error budget. In
particular, at small drive amplitudes the CR gate fidelity is limited by
imperfections of the target-qubit rotations, while at large amplitudes it is
limited by leakage. The gate duration and fidelity are analyzed numerically as
functions of the detuning between qubits, their coupling, drive frequency,
relative duration of pulse ramps, and microwave crosstalk. The effect of the
echo sequence is also analyzed numerically. Our results show that the CR gate
can provide intrinsic infidelity of less than $10^{-3}$ when a simple pulse
shape is used.
