Deterministic two-photon C-Z gate with the two-photon quantum Rabi model

We propose a scheme for realizing a deterministic two-photon C-Z gate based on variants of the two-photon quantum Rabi model (QRM), which is feasible within the framework of circuit QED. We begin by utilizing the two-photon interaction to implement the nonlinear sign (NS) gate, and subsequently, we construct the C-Z gate following the KLM scheme. We consider three different regimes: the strong coupling regime, the perturbative ultrastrong coupling regime, and the large detuning regime. Our results indicate that the C-Z gate operates fast with high fidelity, and is robust against decoherence. We also show the photonic state in the waveguide can be input into the circuit QED system through a variable coupler, and released after interaction with almost the same waveform except for a $\pi$-phase shift. Our scheme offers a suitable approach for achieving fast and deterministic two-photon quantum gates via light-matter interactions.