  We present a study of the principal deuterium Hugoniot for pressures up to
$150$ GPa, using Machine Learning potentials (MLPs) trained with Quantum Monte
Carlo (QMC) energies, forces and pressures. In particular, we adopted a
recently proposed workflow based on the combination of Gaussian kernel
regression and $\Delta$-learning. By fully taking advantage of this method, we
explicitly considered finite-temperature electrons in the dynamics, whose
effects are highly relevant for temperatures above $10$ kK. The Hugoniot curve
obtained by our MLPs shows a good agreement with the most recent experiments,
particularly in the region below 60 GPa. At larger pressures, our Hugoniot
curve is slightly more compressible than the one yielded by experiments, whose
uncertainties generally increase, however, with pressure. Our work demonstrates
that QMC can be successfully combined with $\Delta$-learning to deploy reliable
MLPs for complex extended systems across different thermodynamic conditions, by
keeping the QMC precision at the computational cost of a mean-field
calculation.
