  The impact of cosmic reionization on the Ly$\alpha$ forest power spectrum has
recently been shown to be significant even at low redshifts ($z \sim 2$). This
memory of reionization survives cosmological time scales because high-entropy
mean-density gas is heated to $\sim 3\times10^4$ K by reionization, which is
inhomogeneous, and subsequent shocks from denser regions. In the near future,
the first measurements of the Ly$\alpha$ forest 3D power spectrum will be very
likely achieved by upcoming observational efforts such as the Dark Energy
Spectroscopic Instrument (DESI). In addition to abundant cosmological
information, these observations have the potential to extract the astrophysics
of reionization from the Ly$\alpha$ forest. We forecast, for the first time,
the accuracy with which the measurements of Ly$\alpha$ forest 3D power spectrum
can place constraints on the reionization parameters with DESI. Specifically,
we demonstrate that the constraints on the ionization efficiency, $\zeta$, and
the threshold mass for haloes that host ionizing sources, $m_{\rm turn}$, will
have the $1\sigma$ error at the level of $\zeta = 25.0 \pm 11.6$ and $\log_{10}
(m_{\rm turn}/{\rm M}_\odot) = 8.7^{+0.36}_{-0.70}$, respectively. The
Ly$\alpha$ forest 3D power spectrum will thus provide an independent probe of
reionization, probably even earlier in detection with DESI, with a sensitivity
only slightly worse than the upcoming 21 cm power spectrum measurement with the
Hydrogen Epoch of Reionization Array (HERA), i.e.\ $\sigma_{\rm DESI} /
\sigma_{\rm HERA} \approx 1.5$ for $\zeta$ and $\sigma_{\rm DESI}/\sigma_{\rm
HERA} \approx 2.0$ for $\log_{10}(m_{\rm turn} / $M$_\odot)$. Nevertheless, the
Ly$\alpha$ forest constraint will be improved about three times tighter than
the current constraint from reionization observations with high-z galaxy
priors.
