  We present a model for the cosmological evolution of quasars (QSOs) under the
assumption that they are powered by massive accreting black holes. Accretion
flows around massive black holes make a transition from high radiative
efficiency ($\sim 10%$) to low efficiency, advection-dominated flows when
${\dot M}/{\dot M}_{Edd}$ falls below the critical rate $\sim 0.3\alpha^2\sim
10^{-2}$ where ${\dot M}$ is the mass accretion rate, ${\dot M}_{Edd}\propto M$
is the usual Eddington rate with the nominal 10% efficiency, and $\alpha (\le
1)$ is the dimensionless viscosity parameter. We identify this transition with
the observed break at a redshift $\sim 2$ in the QSOs' X-ray luminosity
evolution. Growth of black holes through accretion could naturally lead to such
a transition at a critical redshift $z_c\sim 1-3$, provided that most of high
redshift QSOs appear with near Eddington luminosities at $z\sim 3-4$ and the
accretion rates decline over the Hubble time in a roughly synchronous manner.
Before the transition, the QSOs' luminosities (with a high efficiency) slowly
decrease and after the transition at $z_c$, the QSO luminosities evolve
approximately as $\propto (1+z)^{K(z)}$ where $K(z)$ gradually varies from
$z=z_c$ to $z\sim 0$ around $K\sim3$. The results depend on the details of the
QSO X-ray emission mechanism. We discuss some further implications.
