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We present a study of the principal deuterium Hugoniot for pressures up to$150$ GPa, using Machine Learning potentials (MLPs) trained with Quantum MonteCarlo (QMC) energies, forces and pressures. In particular, we adopted arecently proposed workflow based on the combination of Gaussian kernelregression and $\Delta$-learning. By fully taking advantage of this method, weexplicitly considered finite-temperature electrons in the dynamics, whoseeffects are highly relevant for temperatures above $10$ kK. The Hugoniot curveobtained by our MLPs shows an excellent agreement with the most recentexperiments, with an accuracy comparable to the best DFT functionals. Our workdemonstrates that QMC can be successfully combined with $\Delta$-learning todeploy reliable MLPs for complex extended systems across differentthermodynamic conditions, by keeping the QMC precision at the computationalcost of a mean-field calculation.

Principal deuterium Hugoniot via Quantum Monte Carlo and $\Delta$-learning