  Circuit cavity quantum electrodynamics (QED) is proving to be a powerful
platform to implement quantum feedback control schemes due to the ability to
control superconducting qubits and microwaves in a circuit. Here, we present a
simple and promising quantum feedback control scheme for deterministic
generation and stabilization of a three-qubit $W$ state in the superconducting
circuit QED system. The control scheme is based on continuous joint Zeno
measurements of multiple qubits in a dispersive regime, which enables us not
only to infer the state of the qubits for further information processing but
also to create and stabilize the target $W$ state through adaptive quantum
feedback control. We simulate the dynamics of the proposed quantum feedback
control scheme using the quantum trajectory approach with an effective
stochastic maser equation obtained by a polaron-type transformation method and
demonstrate that in the presence of moderate environmental decoherence, the
average state fidelity higher than $0.9$ can be achieved and maintained for a
considerable long time (much longer than the single-qubit decoherence time).
This control scheme is also shown to be robust against measurement inefficiency
and individual qubit decay rate differences. Finally, the comparison of the
polaron-type transformation method to the commonly used adiabatic elimination
method to eliminate the cavity mode is presented.
