  After the discovery of a Higgs boson in 2012 at the CERN Large Hadron
Collider (LHC) the detailed study of its properties, and most importantly its
couplings to other particles, has started. This is a very important task to be
completed, in particular to test whether it is indeed the Higgs boson predicted
by the Standard Model (SM). The precise study of the Higgs couplings to gauge
bosons is of particular importance and requires as much information as
possible. In this view this paper provides the next-to-leading order (NLO) QCD
corrections to the production cross sections and differential distributions of
a SM Higgs boson in association with a pair of weak bosons $W^+W^-$, $W^\pm Z$
and $ZZ$, matched with parton shower (PS) in the POWHEG-BOX framework. The NLO
QCD corrections are found to be significant and PS effects are sizable at low
$p_T$ in the jet differential distributions, as expected, while these effects
are negligible in other distributions. We will also provide a detailed study of
the theoretical uncertainties affecting the total production rates at the LHC
and at the Future Circular Collider in hadron-hadron mode, the potential 100
TeV follow-up of the LHC machine: the scale uncertainty calculated by the
variation of the renormalization and factorization scales, the parton
distribution function and related $\alpha_s$ errors as well as the parametric
uncertainties on the input weak boson masses.
