  Learning dynamics from repeated observation of the time evolution of an open
quantum system, namely, the problem of quantum process tomography is an
important task. This task is difficult in general, but, with some additional
constraints could be tractable. This motivates us to look at 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. The
resultant noisy estimate of the channel can then be denoised by diagonalizing
the nominal Choi matrix, truncating some eigenvalues, and altering it to a
genuine Choi matrix. This processed Choi matrix is then compared to the
original one. We see that as the number of samples increases, our
reconstruction becomes more accurate. We also use tools from random matrix
theory to understand the effect of estimation noise in the eigenspectrum of the
estimated Choi matrix.
