  Confronting measurements of the Lyman-$\alpha$ forest with cosmological
hydrodynamical simulations has produced stringent constraints on models of
particle dark matter and the thermal and ionization state of the intergalactic
medium. We investigate the robustness of such models of the Lyman-$\alpha$
forest, focussing on the effect of particle initial conditions on the
Lyman-$\alpha$ forest statistics in cosmological SPH simulations. We study
multiple particle initialization algorithms in simulations that are designed to
be identical in other respects. In agreement with the literature, we find that
the correct linear theory evolution is obtained when a glass-like configuration
is used for initial unperturbed gas particle positions alongside a regular grid
configuration for dark matter particles and the use of non-identical initial
density perturbations for gas and dark matter. However, we report that this
introduces a large scale-dependent distortion in the one-dimensional
Lyman-$\alpha$ transmission power spectrum at small scales ($k > 0.05$ s/km).
The effect is close to $50\%$ at $k\sim 0.1$ s/km, and persists at higher
resolution. This can severely bias inferences in parameters such as the dark
matter particle mass. By considering multiple initial conditions codes and
their variations, we also study the impact of a variety of other assumptions
and algorithmic choices, such as adaptive softening, background radiation
density, particle staggering, and perturbation theory accuracy, on the matter
power spectrum, the Lyman-$\alpha$ flux power spectrum, and the Lyman-$\alpha$
flux PDF. This work reveals possible pathways towards more accurate theoretical
models of the Lyman-$\alpha$ forest to match the quality of upcoming
measurements.
