Towards N$^3$LO PDFs and their implications for Higgs production cross-sections

We review the recently published PDF sets MSHT20xNNPDF40_an3lo and MSHT20xNNPDF40_an3lo_qed, which have been obtained by combining the approximate N$^3$LO (aN$^3$LO) sets of parton distributions (PDFs) from the MSHT20 and NNPDF4.0 series. We investigate the impact of the most recent results on N$^3$LO splitting functions onto the two PDF sets. Finally, we demonstrate the impact of N$^3$LO PDFs on the computation of Higgs production cross-sections at the LHC.