  We numerically study two-dimensional quantum turbulence with a
Gross--Pitaevskii model. With the energy initially accumulated at large scale,
quantum turbulence with many quantized vortex points is generated. Due to the
lack of enstrophy conservation in this model, direct energy cascade with a
Kolmogorov--Obukhov energy spectrum $E(k) \propto k^{-5/3}$ is observed, which
is quite different from two-dimensional incompressible classical turbulence in
the decaying case. A positive value for the energy flux guarantees a
\emph{direct} energy cascade in the inertial range (from large to small
scales). After almost all the energy at the large scale cascades to the small
scale, the compressible kinetic energy realizes the thermodynamic equilibrium
state without quantized vortices.
