Quantum nondemolition measurements of a flux qubit coupled to a noisy detector

We theoretically study the quantum nondemolition measurements of a flux qubit coupled to a noisy superconduct- ing quantum interference device （SQUID）. The obtained analytical results indicate that the measurement probability is frequency-dependent in a short time scale and has a close relationship with the measurement-induced dephasing. Furthermore, when the detuning between the driven and bare resonator equals the coupling strength, we can obtain the maximum measurement rate that is determined by the character of the noise in the SQUID. Finally, we analysed the mixed effect caused by coupling between the non-diagonal term and the external variable. It is found that the initial information of the qubit is destroyed due to quantum tunneling between the qubit states.