The direct determination of parton distribution functions (PDFs) from lattice QCD is an unsolved problem. The traditional approach expresses moments of PDFs in terms of matrix elements of local operators, but these operators suffer from power-divergent mixing that stems from the hypercubic symmetry of the lattice regulator. I will introduce the gradient flow as a means to suppress this power-divergent mixing, at the expense of introducing a new physical scale in the continuum. I will highlight two approaches to dealing with this new scale that could ultimately lead to ab initio calculations of PDFs on the lattice.