  We propose a method to simulate the real time evolution of one dimensional
quantum many-body systems at finite temperature by expressing both the density
matrices and the observables as matrix product states. This allows the
calculation of expectation values and correlation functions as scalar products
in operator space. The simulations of density matrices in inverse temperature
and the local operators in the Heisenberg picture are independent and result in
a grid of expectation values for all intermediate temperatures and times.
Simulations can be performed using real arithmetics with only polynomial growth
of computational resources in inverse temperature and time for integrable
systems. The method is illustrated for the XXZ model and the single impurity
Anderson model.
