  We introduce the notion of the relaxation time for noisy quantum maps on the
2d-dimensional torus - a generalization of previously studied dissipation time.
We show that relaxation time is sensitive to the chaotic behavior of the
corresponding classical system if one simultaneously considers the
semiclassical limit ($\hbar$ -> 0) together with the limit of small noise
strength ($\ep$ -> 0).
  Focusing on quantized smooth Anosov maps, we exhibit a semiclassical regime
$\hbar<\ep^{E}$ << 1 (where E>1) in which classical and quantum relaxation
times share the same asymptotics: in this regime, a quantized Anosov map
relaxes to equilibrium fast, as the classical map does. As an intermediate
result, we obtain rigorous estimates of the quantum-classical correspondence
for noisy maps on the torus, up to times logarithmic in $\hbar^{-1}$. On the
other hand, we show that in the ``quantum regime'' $\ep$ << $\hbar$ << 1,
quantum and classical relaxation times behave very differently. In the special
case of ergodic toral symplectomorphisms (generalized ``Arnold's cat'' maps),
we obtain the exact asymptotics of the quantum relaxation time and precise the
regime of correspondence between quantum and classical relaxations.
