Heisenberg evolution of matrix product operators by time-dependent variational principle

We apply the time-dependent variational principle to study Heisenberg evolution of matrix product operators (MPO). Compared to the analogous Shroedinger evolution approach based on matrix product states recently developed by Leviatan et.al., the MPO approach presents the following advantages: it avoids the ensemble averaging; starting from a local observable, the "entanglement" of the Heisenberg-picture operator grows less rapidly. As applications, we study the chaotic wave front and the hydrodynamic diffusion of local conserved quantities.