  Quantum information processing with hybrid protocols making use of discrete-
and continuous-variable currently attracts of great interest because of its
promising applications in scalable quantum computer and distant quantum
network. By inducing a giant cross-Kerr nonlinearity between two cavities, we
propose a general protocol for hybrid quantum gate and quantum entanglement
with high fidelity between a stationary, discrete photonic qubit and a flying
photonic state. Interestingly, our protocol can be used to conduct a
controlled-Z quantum gate between a stationary microwave photon stored in a
slowly-decaying microwave cavity and a flying optical photon, and therefore
enable to build quantum network for distant superconducting quantum circuits.
