  In view of the forthcoming High-Luminosity phase of the LHC,
next-to-next-to-next-to-leading (N$^3$LO) calculations for the most
phenomenologically relevant processes become necessary. In this work, we take
the first step towards this goal for H$+$jet production by computing the one-
and two-loop helicity amplitudes for the two contributing processes, $H\to
ggg$, $H\to q\bar{q}g$, in an effective theory with infinite top quark mass, to
higher orders in the dimensional regulator. We decompose the amplitude in
scalar form factors related to the helicity amplitudes and in a new basis of
tensorial structures. The form factors receive contributions from Feynman
integrals which were reduced to a novel canonical basis of master integrals. We
derive and solve a set of differential equations for these integrals in terms
of Multiple Polylogarithms (MPLs) of two variables up to transcendental weight
six.
