Phase gate of one qubit simultaneously controlling n qubits in a cavity

We propose how to realize a three-step controlled-phase gate of one qubit simultaneously controlling n qubits in a cavity or coupled to a resonator. The n two-qubit controlled-phase gates, forming this multiqubit phase gate, can be performed simultaneously. The operation time of this phase gate is independent of the number n of qubits. This phase gate controlling at once n qubits is insensitive to the initial state of the cavity mode and can be used to produce an analogous cnot gate simultaneously acting on n qubits. We present two alternative approaches to implement this gate. One approach is based on tuning the qubit frequency while in the other method the resonator frequency is tuned. Using superconducting qubits coupled to a resonator as an example, we show how to implement the proposed gate with one superconducting qubit simultaneously controlling n qubits selected from N qubits coupled to a resonator (1<n<N). We also give a discussion on realizing the proposed gate with atoms, by using one cavity initially in an arbitrary state.