Optimal conditions for high fidelity dispersive readout of a qubit with a photon number resolving detector
We determine the optimal parameters for a simple and efficient scheme of dispersive readout of a qubit. Depending on the qubit state (ground or excited) the resonance of a cavity is shifted either to the red or to the blue side. Qubit state is inferred by detecting the photon number transmitted through the cavity. It turns out that this kind of detection provides better measurement fidelity than the detection of the presence or absence of photons only. We show that radiating the cavity on either of the frequencies it shifts to results in a suboptimal measurement. The optimal frequency of the probe photons is determined, as well as the optimal ratio of the shift to the resonator leakage. It is shown that to maximize the fidelity of a long lasting measurement, it is sufficient to use the parameters optimizing the signal to noise ratio in the photon count. One can reach 99% fidelity for a single shot measurement in various physical realizations of the scheme.