  We present the first measurement of the Ly$\alpha$ forest BAO shift parameter
from cosmological simulations. In particular, we generate a suite of $1000$
accurate effective field-level bias-based Ly$\alpha$ forest simulations of
volume $V=(1 \, h^{-1} \, {\rm Gpc})^3$ at $z=2$, both in real and redshift
space, calibrated upon two fixed-and-paired cosmological hydrodynamic
simulations. To measure the BAO, we stack the three-dimensional power spectra
of the $1000$ different realizations, compute the average, and use a model
accounting for a proper smooth-peak component decomposition of the power
spectrum, to fit it via an efficient Markov Chain Monte Carlo scheme estimating
the covariance matrices directly from the simulations. We report the BAO shift
parameters to be $\alpha=0.9969^{+0.0014}_{-0.0014}$ and
$\alpha=0.9905^{+0.0027}_{-0.0027}$ in real and redshift space, respectively.
We also measure the bias $b_{\rm lya}$ and the BAO broadening parameter
$\Sigma_{\rm nl}$, finding $b_{\rm lya}=-0.1786^{+0.0001}_{-0.0001}$ and
$\Sigma_{\rm nl}=3.87^{+0.20}_{-0.20}$ in real space, and $b_{\rm
lya}=-0.073^{+0.005}_{-0.004}$ and $\Sigma_{\rm nl}=6.55^{+0.23}_{-0.22}$ in
redshift space. Moreover, we measure the linear Kaiser factor $\beta_{\rm
lya}=1.39^{+0.24}_{-0.18}$ from the isotropic redshift space fit. Overall, we
find evidence for a negative shift of the BAO peak at the $\sim 2.2\sigma$ and
$\sim 3.5\sigma$ level in real and redshift space, respectively. This work sets
new important theoretical constraints on the Ly$\alpha$ forest BAO scale and
offers a potential solution to the tension emerging from previous observational
analysis, in light of ongoing and upcoming Ly$\alpha$ forest spectroscopic
surveys, such as DESI, PFS, and WEAVE-QSO.
