  We have studied mixed-state dissipation in epitaxial MgB$_2$ thin films by
measurements of resistive transition, current-voltage characteristics, Hall
effect, and point-contact tunnelling spectrum. We found that unlike single gap
superconductors with negligible vortex quantum fluctuations in which vortices
are frozen at T=0 K, finite zero-temperature dissipation due to vortex motion
exists in MgB$_2$ over a wide magnetic field range. This dissipation was found
to be associated with proliferation of quasiparticles from the $\pi$-band of
MgB$_2$. The result shows that the vortex fluctuations are enhanced by two-band
superconductivity in MgB$_2$ and we suggest that the vortex quantum fluctuation
is a possible cause of the non-vanishing zero-temperature dissipation.
