Characterizing the statistics of the qubit frequency noise*

Measurements of the qubit state brings in randomness, which is superposed on the noise in the qubit parameters. This complicates characterizing the noise. We show that not only the spectrum of the noise, but also its statistics can be revealed using periodically repeated Ramsey measurements. The method is based on studying two- and three-time correlators of the measurement outcomes. It allows identifying noise from a periodic frequency modulation and distinguishing a Gaussian from a non-Gaussian noise. It also identifies a characteristic footprint of the telegraph noise. We provide analytical expressions for the two-and three-time correlators for different kinds of noise. The results for commonly encountered types of noise, the Gaussian 1/f and  exponentially correlated noise, and for a telegraph noise are verified by numerical simulations.