  We analyze the fidelity of a deterministic quantum phase gate for two photons
counterpropagating as polaritons through a cloud of Rydberg atoms under the
condition of electromagnetically induced transparency (EIT). We provide
analytical results for the phase shift of the quantum gate, and provide an
estimation for all processes leading to a reduction to the gate fidelity.
Especially, the influence of losses form the intermediate level, dispersion of
the photon wave packet, scattering into additional polariton channels, finite
lifetime of the Rydberg state, as well as effects of transverse size of the
wave packets are accounted for. We show that the flatness of the effective
interaction, caused by the blockade phenomena, suppresses the corrections due
to the finite transversal size. This is a strength of Rydberg-EIT setup
compared to other approaches. Finally, we provide the experimental requirements
for the realization of a high fidelity quantum phase gate using Rydberg
polaritons.
