  We present a way to realize a multiplex-controlled phase gate of n-1 control
qubits simultaneously controlling one target qubit, with n qubits distributed
in n different cavities. This multiqubit gate is implemented by using n qutrits
(three-level natural or artificial atoms) placed in n different cavities, which
are coupled to an auxiliary qutrit. Here, the two logic states of a qubit are
represented by the two lowest levels of a qutrit placed in a cavity. We show
that this n-qubit controlled phase gate can be realized using only 2n+2 basic
operations, i.e., the number of required basic operations only increases
linearly with the number n of qubits. Since each basic operation employs the
qutrit-cavity or qutrit-pulse resonant interaction, the gate can be fast
implemented when the number of qubits is not large. Numerical simulations show
that a three-qubit controlled phase gate, which is executed on three qubits
distributed in three different cavities, can be high-fidelity implemented by
using a circuit QED system. This proposal is quite general and can be applied
to a wide range of physical systems, with atoms, NV centers, quantum dots, or
various superconducting qutrits distributed in different cavities. Finally,
this method can be applied to implement a multiqubit controlled phase gate with
atoms using a cavity. A detailed discussion on implementing a three-qubit
controlled phase gate with atoms and one cavity is presented.
