This talk focuses on the high energy limit of the amplitudes for the production of a Higgs boson in association with a QCD jet. In this regime, the amplitudes factorise into universal components, which control the pattern of the large logarithms involving the center-of-mass energy squared of the process. I will describe the determination of one such components, the two-loop impact factor of the Higgs boson, which enters the amplitudes at next-to-next-to-leading logarithmic accuracy.