  We extend the non-Hermitian one-dimensional quantum walk model [Phys. Rev.
Lett. 102, 065703 (2009)] by taking the dephasing effect into account. We prove
that the feature of topological transition does not change even when dephasing
between the sites within units is present. The potential experimental
observation of our theoretical results in the circuit QED system consisting of
superconducting qubit coupled to a superconducting resonator mode is discussed
and numerically simulated. The results clearly show a topological transition in
quantum walk and display the robustness of such a system to the decay and
dephasing of qubits. We also discuss how to extend this model to higher
dimension in the circuit QED system.
