  We obtain for the first time the two-loop amplitudes for Higgs plus three
gluons in Higgs effective field theory including dimension-seven operators.
This provides the S-matrix elements for the top mass corrections for Higgs plus
a jet production at LHC. The computation is based on the on-shell unitarity
method combined with integration by parts reduction. We work in conventional
dimensional regularization and obtain analytic expressions renormalized in the
MS-bar scheme. The two-loop anomalous dimensions present operator mixing
behavior. The infrared divergences agree with that predicted by Catani and the
finite remainders take remarkably simple forms, where the maximally
transcendental parts are identical to the corresponding results in N=4
super-Yang-Mills theory. The parts of lower transcendentality turn out to be
also largely determined by the N=4 results.
