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We analyze analytically, semi-analytically, and numerically the operation ofCross-Resonance (CR) gate for superconducting qubits (transmons). We find thata relatively simple semi-analytical method gives accurate results for theCNOT-equivalent gate duration and compensating single-qubit rotations. It alsoallows us to minimize the CNOT gate duration over the amplitude of the appliedmicrowave drive and find dependence on the detuning between the qubits.However, full numerical simulations are needed to calculate intrinsic fidelityof the CR gate. We decompose numerical infidelity into contributions fromvarious physical mechanisms, thus finding the intrinsic error budget. Inparticular, at small drive amplitudes the CR gate fidelity is limited byimperfections of the target-qubit rotations, while at large amplitudes it islimited by leakage. The gate duration and fidelity are analyzed numerically asfunctions of the detuning between qubits, their coupling, drive frequency,relative duration of pulse ramps, and microwave crosstalk. The effect of theecho sequence is also analyzed numerically. Our results show that the CR gatecan provide intrinsic infidelity of less than $10^{-3}$ when a simple pulseshape is used.

Operation and intrinsic error budget of a two-qubit cross-resonance gate