  We present the computation of the contributions to N3LO inclusive Higgs boson
production due to two-loop amplitudes. Our result is a Laurent expansion in the
dimensional regulator, with coefficients that are linear combinations of
harmonic polylogarithms of the ratio of the Higgs boson mass and the partonic
center of mass energy. We outline our method of solving the differential
equations for master integrals appearing in the cross section. Solving these
differential equations requires the determination of boundary conditions and we
present a new technique for decomposing the boundary conditions into physical
contributions. We show how these boundary conditions can be calculated using
the method of expansion by regions.
