Learning quantum dynamics: Lindblad operators from classical shadows

Learning dynamics from repeated observation of time evolution of an open quantum system, namely, the problem of quantum process tomography, is an important but, in general, difficult task. The exploration of additional constraints that make the problem tractable motivates us to consider the problem of Lindblad operator discovery from observations. We point out that for moderate-size Hilbert spaces, low Kraus rank of the channel, and short time steps, the eigenvalues of the Choi matrix corresponding to the channel have a special structure. We use the least square method for the estimation of a channel where, for fixed inputs, we estimate the outputs by classical shadows. We then denoise the resultant noisy estimate of the channel by diagonalizing the nominal Choi matrix, truncating some eigenvalues, and altering it to a genuine Choi matrix. We use tools from random matrix theory to understand the effect of estimation noise in the eigenspectrum of the estimated Choi matrix. We, further, verify that our method of channel reconstruction gets more accurate as the sample size increases.