Mixed-state dynamics in one-dimensional quantum lattice systems: a   time-dependent superoperator renormalization algorithm
  We present an algorithm to study mixed-state dynamics in one-dimensional quantum lattice systems. The algorithm can be used, e.g., to construct thermal states or to simulate real time evolutions given by a generic master equation. Its two main ingredients are (i) a superoperator renormalization scheme to efficiently describe the state of the system and (ii) the time evolving block decimation (TEBD) technique to efficiently update the state during a time evolution. The computational cost of a simulation increases significantly with the amount of correlations between subsystems but it otherwise depends only linearly in the system size. We present simulations involving quantum spins and fermions in one spatial dimension. 
Michael Zwolak, Guifre Vidal