  The origin of fast radio bursts (FRBs) is still unknown. Multi-wavelength
observations during or shortly after the FRB phase would be essential to
identify the counterpart of an FRB and to constrain its progenitor and
environment. In this work, we investigate the brightness of the "fast optical
bursts" (FOBs) associated with FRBs and the prospects of detecting them. We
investigate several inverse Compton (IC) scattering processes that might
produce an FOB, including both the one-zone and two-zone models. We also
investigate the extension of the same mechanism of FRB emission to the optical
band. We find that a detectable FOB with the current and forthcoming telescopes
is possible under the IC scenarios with very special conditions. In particular,
the FRB environment would need to invoke a neutron star with an extremely
strong magnetic field and an extremely fast spin, or an extremely young
supernova remnant (SNR) surrounding the FRB source. Furthermore, most electrons
in the source are also required to have a fine-tuned energy distribution such
that most of the IC energy is channeled in the optical band. We conclude that
the prospect of detecting FOBs associated with FRBs is low. On the other hand,
if FOBs are detected from a small fraction of FRBs, these FOBs would reveal
extreme physical conditions in the FRB environments.
