  Hybrid qubits have recently drawn intensive attention in quantum computing.
We here propose a method to implement a universal controlled-phase gate of two
hybrid qubits via two three-dimensional (3D) microwave cavities coupled to a
superconducting flux qutrit. For the gate considered here, the control qubit is
a microwave photonic qubit (particle-like qubit), whose two logic states are
encoded by the vacuum state and the single-photon state of a cavity, while the
target qubit is a cat-state qubit (wave-like qubit), whose two logic states are
encoded by the two orthogonal cat states of the other cavity. During the gate
operation, the qutrit remains in the ground state; therefore decoherence from
the qutrit is greatly suppressed. The gate realization is quite simple, because
only a single basic operation is employed and neither classical pulse nor
measurement is used. Our numerical simulations demonstrate that with current
circuit QED technology, this gate can be realized with a high fidelity. The
generality of this proposal allows to implement the proposed gate in a wide
range of physical systems, such as two 1D or 3D microwave or optical cavities
coupled to a natural or artificial three-level atom. Finally, this proposal can
be applied to create a novel entangled state between a particle-like photonic
qubit and a wave-like cat-state qubit.
