  Time-dependent density matrix renormalization group method with a matrix
product ansatz is employed for explicit computation of non-equilibrium steady
state density operators of several integrable and non-integrable quantum spin
chains, which are driven far from equilibrium by means of Markovian couplings
to external baths at the two ends. It is argued that even though the
time-evolution can not be simulated efficiently due to fast entanglement
growth, the steady states in and out of equilibrium can be typically accurately
approximated, so that chains of length of the order n ~ 100 are accessible. Our
results are demonstrated by performing explicit simulations of steady states
and calculations of energy/spin densities/currents in several problems of heat
and spin transport in quantum spin chains. Previously conjectured relation
between quantum chaos and normal transport is re-confirmed with high acurracy
on much larger systems.
