  Subject of this contribution is to demonstrate that the observed faintness of
the supernovae at the high redshift can be considered as a manifestation of
quantum effects at cosmological scales. We show that observed redshift
distribution of coordinate distances to the type Ia supernovae can be explained
by the local manifestations of quantum fluctuations of the cosmological scale
factor about its average value. These fluctuations can arise in the early
universe, grow with time, and produce observed accelerating or decelerating
expansions of space subdomains containing separate supernovae with high
redshift whereas the universe as a whole expands at a steady rate.
