  The full-shape correlations of the Lyman alpha (Ly$\alpha$) forest contain a
wealth of cosmological information through the Alcock-Paczy\'{n}ski effect.
However, these measurements are challenging to model without robustly testing
and verifying the theoretical framework used for analyzing them. Here, we
leverage the accuracy and volume of the $N$-body simulation suite
\textsc{AbacusSummit} to generate high-resolution Ly$\alpha$ skewers and
quasi-stellar object (QSO) catalogs. One of the main goals of our mocks is to
aid in the full-shape Ly$\alpha$ analysis planned by the Dark Energy
Spectroscopic Instrument (DESI) team. We provide optical depth skewers for six
of the fiducial cosmology base-resolution simulations ($L_{\rm box} =
2\,h^{-1}{\rm Gpc}$, $N = 6912^3$) at $z = 2.5$. We adopt a simple recipe based
on the Fluctuating Gunn-Peterson Approximation (FGPA) for constructing these
skewers from the matter density in an $N$-body simulation and calibrate it
against the 1D and 3D Ly$\alpha$ power spectra extracted from the
hydrodynamical simulation IllustrisTNG (TNG; $L_{\rm box} = 205\,h^{-1}{\rm
Mpc}$, $N = 2500^3$). As an important application, we study the non-linear
broadening of the baryon acoustic oscillation (BAO) peak and show the
cross-correlation between DESI-like QSOs and our Ly$\alpha$ forest skewers. We
find differences on small scales between the Kaiser approximation prediction
and our mock measurements of the Ly$\alpha$$\times$QSO cross-correlation, which
would be important to account for in upcoming analyses. The
\textsc{AbacusSummit} Ly$\alpha$ forest mocks open up the possibility for
improved modelling of cross correlations between Ly$\alpha$ and cosmic
microwave background (CMB) lensing and Ly$\alpha$ and QSOs, and for forecasts
of the 3-point Ly$\alpha$ correlation function. Our catalogues and skewers are
publicly available on Globus.
