  Quantum computers and simulators promise to enable the study of strongly
correlated quantum systems. Yet, surprisingly, it is hard for them to compute
ground states. They can, however, efficiently compute the dynamics of closed
quantum systems. We propose a method to study the quantum thermodynamics of
strongly correlated electrons from quantum dynamics. We define time-averaged
classical shadows (TACS) and prove it is a classical shadow(CS) of the von
Neumann ensemble, the time-averaged density matrix. We then show that the
diffusion maps, an unsupervised machine learning algorithm, can efficiently
learn the phase diagram and phase transition of the one-dimensional transverse
field Ising model both for ground states using CS \emph{and state trajectories}
using TACS. It does so from state trajectories by learning features that appear
to be susceptibility and entropy from a total of 90,000 shots taken along a
path in the microcanonical phase diagram. Our results suggest a low number of
shots from quantum simulators can produce quantum thermodynamic data with a
quantum advantage.
