Deterministic two-photon controlled-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Z</mml:mi></mml:math> gate with the two-photon quantum Rabi model

We propose a scheme for realizing a deterministic two-photon controlled-$Z$ (cz) gate based on variants of the two-photon quantum Rabi model, which is feasible within the framework of circuit QED. We begin by utilizing the two-photon interaction to implement the nonlinear sign gate and subsequently we construct the cz gate following the Knill-Laflamme-Milburn scheme [Nature (London) 409, 46 (2001)]. We consider three different regimes: the strong-coupling regime, the perturbative ultrastrong-coupling regime, and the large-detuning regime. Our results indicate that the cz gate operates fast with high fidelity. We also show that 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 $\ensuremath{\pi}$-phase shift. Our scheme offers a suitable approach for achieving fast and deterministic two-photon quantum gates via light-matter interactions.