Approximating the long time average of the density operator: Diagonal ensemble

For an isolated generic quantum system out of equilibrium, the long time average observables is given by diagonal ensemble, i.e. mixed state with same probability for energy eigenstates as initial but without coherences between different energies. In this work we present a method to approximate ensemble using tensor networks. Instead simulating real evolution, adapt filtering scheme introduced earlier in [Phys. Rev. B 101, 144305 (2020)] problem. We analyze performance on non-integrable spin chain, which observe that local converge towards thermal values polynomially inverse width filter.