We present the first constraints on primordial magnetic fields from the Lyman-$\alpha$ forest using full cosmological hydrodynamic simulations. At the scales and redshifts probed by the data, the flux power spectrum is extremely sensitive to the extra power induced by primordial magnetic fields in the linear matter power spectrum, at a scale that we parametrize with $k_{\mathrm{peak}}$. We rely on a set of more than a quarter million flux models obtained by varying thermal and reionization histories and cosmological parameters. We find a hint of extra power that is well fitted by the primordial magnetic field model with $B\sim 0.2\mathrm{nG}$, corresponding to $k_{\mathrm{peak}}\sim 20{\mathrm{Mpc}}^{-1}$. However, when applying very conservative assumptions on the modeling of the noise, we obtain a $3\sigma$ C.L. lower limit $k_{\mathrm{peak}}>30{\mathrm{Mpc}}^{-1}$, which translates into the tightest bounds on the strength of primordial intergalactic magnetic fields: $B<0.30\mathrm{nG}$ (for a fixed, nearly scale-invariant $n_{\mathrm{B}}=-2.9$).

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• Large scale structure of the Universe
• Primordial magnetic fields
• Astrophysical & cosmological simulations