Operation and error budget of the Cross-Resonance gate

We analyze operation of the Cross-Resonance (CR) gate, in which control qubit is driven at the frequency of target qubit to realize the CNOT gate after additional single-qubit rotations. Numerical simulations for multi-level transmons and particular pulse shapes are used to find the CNOT time and parameters of single-qubit rotations as functions of the drive amplitude ε. To understand these dependences, including saturation of the CNOT time at large ε, we develop analytical and semi-analytical theories, which agree well with the numerics. We also calculate intrinsic infidelity of the CR gate as a function of ε and find a minimum, created by the interplay between leakage and imperfect unitary evolution. The error budget of the CR gate can be approximately described analytically or semi-analytically.