  Active galactic nuclei (AGN) feedback models are generally calibrated to
reproduce galaxy observables such as the stellar mass function and the
bimodality in galaxy colors. We use variations of the AGN feedback
implementations in the IllustrisTNG (TNG) and Simba cosmological hydrodynamic
simulations to show that the low redshift Lyman-$\alpha$ forest can provide
constraints on the impact of AGN feedback. We show that TNG over-predicts the
number density of absorbers at column densities $N_{\rm HI} < 10^{14}$
cm$^{-2}$ compared to data from the Cosmic Origins Spectrograph (in agreement
with previous work), and we demonstrate explicitly that its kinetic feedback
mode, which is primarily responsible for galaxy quenching, has a negligible
impact on the column density distribution (CDD) of absorbers. In contrast, we
show that the fiducial Simba model which includes AGN jet feedback is the
preferred fit to the observed CDD of the $z = 0.1$ Lyman-$\alpha$ forest across
five orders of magnitude in column density. We show that the Simba results with
jets produce a quantitatively better fit to the observational data than the
Simba results without jets, even when the UVB is left as a free parameter. AGN
jets in Simba are high speed, collimated, weakly-interacting with the
interstellar medium (via brief hydrodynamic decoupling) and heated to the halo
virial temperature. Collectively these properties result in stronger long-range
impacts on the IGM when compared to TNG's kinetic feedback mode, which drives
isotropic winds with lower velocities at the galactic radius. Our results
suggest that the low redshift Lyman-$\alpha$ forest provides plausible evidence
for long-range AGN jet feedback.
