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Quantum information processing with hybrid protocols making use of discrete-and continuous-variable currently attracts of great interest because of itspromising applications in scalable quantum computer and distant quantumnetwork. By inducing a giant cross-Kerr nonlinearity between two cavities, wepropose a general protocol for hybrid quantum gate and quantum entanglementwith high fidelity between a stationary, discrete photonic qubit and a flyingphotonic state. Interestingly, our protocol can be used to conduct acontrolled-Z quantum gate between a stationary microwave photon stored in aslowly-decaying microwave cavity and a flying optical photon, and thereforeenable to build quantum network for distant superconducting quantum circuits.

Hybrid quantum gate and entanglement between a "stationary" photonic qubit and a flying optical state via a giant cross Kerr nonlinear effect