  We analyze the gauge fixing precision dependence of some non-local
quark-blinear lattice operators interesting in computing parton physics for
several measurements, using 5 lattice spacings ranging from 0.032 fm to 0.121
fm. Our results show that gauge dependent non-local measurements are
significantly more sensitive to the precision of gauge fixing than anticipated.
The impact of imprecise gauge fixing is significant for fine lattices and long
distances. For instance, even with the typically defined precision of Landau
gauge fixing of $10^{-8}$, the deviation caused by imprecise gauge fixing can
reach 12 percent, when calculating the trace of Wilson lines at 1.2 fm with a
lattice spacing of approximately 0.03 fm. Similar behavior has been observed in
$\xi$ gauge and Coulomb gauge as well. For both quasi PDFs and quasi TMD-PDFs
operators renormalized using the RI/MOM scheme, convergence for different
lattice spacings at long distance is only observed when the precision of Landau
gauge fixing is sufficiently high. To describe these findings quantitatively,
we propose an empirical formula to estimate the required precision.
