  Production of the Higgs boson, $H$ in association with a massive vector
boson, $V$, i.e., the $VH$ process, plays an important role in the explorations
of Higgs physics at the Large Hadron Collider, both for a precise study of
Higgs' Standard Model couplings and for probing New Physics. In this
publication we present the two-loop corrections in massless quantum
chromodynamics (QCD) to the amplitude of the Higgs production associated with a
$Z$ boson via the bottom quark-antiquark annihilation channel with a
non-vanishing bottom-quark Yukawa coupling, which is a necessary ingredient of
the full next-to-next-to-leading-order QCD corrections to the $VH$ process in
the five-flavour scheme. The computation is performed by projecting the
D-dimensional scattering amplitude directly onto an appropriate set of Lorentz
structures related to the linear polarisation states of the $Z$ boson. We
provide analytic expressions of the complete set of renormalised polarised
amplitudes in terms of polylogarithms of maximum weight four. To give an
estimation of the size of contributions from amplitudes considered in this
work, we compute numerically the resulting cross sections under the
soft-virtual approximation. We also take the opportunity to make a dedicated
discussion regarding an interesting subtlety appearing in the conventional form
factor decomposition of amplitudes involving axial currents regularised in D
dimensions.
