  We present methods to compute higher orders in the threshold expansion for
the one-loop production of a Higgs boson in association with two partons at
hadron colliders. This process contributes to the N$^3$LO Higgs production
cross section beyond the soft-virtual approximation. We use reverse unitarity
to expand the phase-space integrals in the small kinematic parameters and to
reduce the coefficients of the expansion to a small set of master integrals. We
describe two methods for the calculation of the master integrals. The first was
introduced for the calculation of the soft triple-real radiation relevant to
N$^3$LO Higgs production. The second uses a particular factorization of the
three body phase-space measure and the knowledge of the scaling properties of
the integral itself. Our result is presented as a Laurent expansion in the
dimensional regulator, although some of the master integrals are computed to
all orders in this parameter.
