  We present the PRIYA suite of cosmological simulations, based on the code and
hydrodynamic model of the ASTRID simulation, and designed for cosmological
analyses of the Lyman-$\alpha$ forest. Our simulation suite spans a
$9$-dimensional parameter space, including $4$ cosmological parameters and $5$
astrophysical/thermal parameters. We have run $48$ low fidelity simulations
with $1536^3$ particles in a $120$ Mpc/h box and $3$ high fidelity simulations
with $3072^3$ particles in a $120$ Mpc/h box. All our simulations include a
full physics model for galaxy formation, including supernova and AGN feedback,
and thus also contain a realistic population of DLAs. We advance on earlier
simulations suites by larger particle loads, by incorporating new physical
models for patchy hydrogen and helium reionization, and by self-consistently
incorporating a model for AGN feedback. We show that patchy helium reionization
imprints an excess in the 1D flux power spectrum on large scales, which may
allow future measurements of helium reionization bubble sizes. Simulation
parameters are chosen based on a Latin hypercube design and a Gaussian process
is used to interpolate to arbitrary parameter combinations. We build a
multi-fidelity emulator for the 1D flux power spectrum and the mean IGM
temperature. We show that our final interpolation error is $< 1\%$ and that our
simulations produce a flux power spectrum converged at the percent level for
$z=5.4$ - $2.2$. Our simulation suite will be used to interpret Lyman-$\alpha$
forest 1D flux power spectra from SDSS and future DESI data releases.
