Validity of resonant two-qubit gates in the ultrastrong coupling regime of circuit QED

We investigate theoretically the performance of two-qubit resonant gates in the crossover from the strong to the ultrastrong coupling (USC) regime of light-matter interaction in circuit QED. Two controlled-PHASE gate schemes---that works well within the rotating wave-approximation (RWA)---are analyzed while taking into account the effects of counter-rotating terms appearing in the Hamiltonian. Our numerical results show that the fidelity of the gate operation is above 96 % when the ratio between the coupling strength and the resonator frequency, g/\omega_r, is of about 10 %. Novel schemes are required in order to implement ultrafast quantum gates when increasing the ratio g/\omega_r.