  Using synthetic Lyman-$\alpha$ forests from the Dark Energy Spectroscopic
Instrument (DESI) survey, we present a study of the impact of errors in the
estimation of quasar redshift on the Lyman-$\alpha$ correlation functions.
Estimates of quasar redshift have large uncertainties of a few hundred
$\text{km s}^{-1}\,$ due to the broadness of the emission lines and the
intrinsic shifts from other emission lines. We inject Gaussian random redshift
errors into the mock quasar catalogues, and measure the auto-correlation and
the Lyman-$\alpha$-quasar cross-correlation functions. We find a smearing of
the BAO feature in the radial direction, but changes in the peak position are
negligible. However, we see a significant unphysical correlation for small
separations transverse to the line of sight which increases with the amplitude
of the redshift errors. We interpret this contamination as a result of the
broadening of emission lines in the measured mean continuum, caused by quasar
redshift errors, combined with the unrealistically strong clustering of the
simulated quasars on small scales.
