Implementing a Multi-Qubit Quantum Phase Gate Encoded by Photonic Qubit

A protocol is proposed to implement a three-qubit phase gate for photonic qubits in a three-mode cavity. The idea can be extended to directly implement a N-qubit phase gate. We also show that the interaction time remains unchanged with the increasing number of qubits. In addition, the influence of cavity decay and atomic spontaneous emission on the gate fidelity and photon loss probability is also discussed by numerical calculation.