  We propose a simple method to realize a hybrid controlled-controlled-Z (CCZ)
gate with two photonic qubits simultaneously controlling a superconducting (SC)
target qubit, by employing two microwave cavities coupled to a SC ququart (a
four-level quantum system). In this proposal, each control qubit is a photonic
qubit, which is encoded by two arbitrary orthogonal eigenstates (with
eigenvalues 1 and -1, respectively) of the photon-number parity operator. Since
the two arbitrary encoding states can take various quantum states, this
proposal can be applied to realize the hybrid CCZ gate, for which the two
control photonic qubits can have various encodings. The gate realization is
quite simple because only a basic operation is needed. During the gate
operation, the higher energy intermediate levels of the ququart are not
occupied, and, thus, decoherence from these levels is greatly suppressed. We
further discuss how to apply this gate to generate a hybrid
Greenberger-Horne-Zeilinger (GHZ) entangled state of a SC qubit and two
photonic qubits, which takes a general form. As an example, our numerical
simulation demonstrates that high-fidelity generation of a cat-cat-spin hybrid
GHZ state is feasible within current circuit QED technology. This proposal is
quite general, which can be applied to realize the hybrid CCZ gate as well as
to prepare various hybrid GHZ states of a matter qubit and two photonic qubits
in other physical systems, such as two microwave or optical cavities coupled to
a four-level natural or artificial atom.
