Understanding the errors for native gates of a Rydberg quantum computer is key to bring demonstrators closer to real-world applications. In our new preprint, we perform a theoretical study of a high fidelity controlled-phase gate taking into account fundamental error sources such as the photon recoil, harmonic trapping of the qubits, and the specifics of the Rydberg atom interaction potential. We identify the dominant contribution to the gate infidelity, and study optimal laser pulse shapes aiming to reduce this error. We demonstrate a remarkable robustness of the gate protocol and possible gate fidelities above 99.9%.