  The cross-Kerr nonlinearity (XKNL) effect can induce efficient photon
interactions in principle with which photonic multiqubit gates can be performed
using far fewer physical resources than linear optical schemes. Unfortunately,
it is extremely challenging to generate giant cross-Kerr nonlinearities. In
recent years, much effort has been made to perform multiqubit gates via weak
XKNLs. However, the required nonlinearity strengths are still difficult to
achieve in the experiment. We here propose an XKNL-based scheme for realizing a
two-photon polarization-parity gate, a universal two-qubit gate, in which the
required strength of the nonlinearity could be orders of magnitude weaker than
those required for previous schemes. The scheme utilizes a ring cavity fed by a
coherent state as a quantum information bus which interacts with a path mode of
the two polarized photons (qubits). The XKNL effect makes the bus pick up a
phase shift dependent on the photon number of the path mode. Even when the
potential phase shifts are very small they can be effectively measured using
photon-number resolving detectors, which accounts for the fact that our scheme
can work in the regime of tiny XKNL. The measurement outcome reveals the parity
(even parity or odd parity) of the two polarization qubits.
