  Hybrid quantum gates have recently drawn considerable attention. They play
significant roles in connecting quantum information processors with qubits of
different encoding and have important applications in the transmission of
quantum states between a quantum processor and a quantum memory. In this work,
we propose a single-step implementation of a multi-target-qubit controlled-NOT
gate with one superconducting (SC) qubit simultaneously controlling $n$ target
cat-state qubits. In this proposal, the gate is implemented with $n$ microwave
cavities coupled to a three-level SC qutrit. The two logic states of the
control SC qubit are represented by the two lowest levels of the qutrit, while
the two logic states of each target cat-state qubit are represented by two
quasi-orthogonal cat states of a microwave cavity. This proposal operates
essentially through the dispersive coupling of each cavity with the qutrit. The
gate realization is quite simple because it requires only a single-step
operation. There is no need of applying a classical pulse or performing a
measurement. The gate operation time is independent of the number of target
qubits, thus it does not increase as the number of target qubits increases.
Moreover, because the third higher energy level of the qutrit is not occupied
during the gate operation, decoherence from the qutrit is greatly suppressed.
As an application of this hybrid multi-target-qubit gate, we further discuss
the generation of a hybrid Greenberger-Horne-Zeilinger (GHZ) entangled state of
SC qubits and cat-state qubits. As an example, we numerically analyze the
experimental feasibility of generating a hybrid GHZ state of one SC qubit and
three cat-state qubits within present circuit QED technology.
