We develop a deep variational free energy framework to compute the equation of state of hydrogen in the warm dense matter region. This method parameterizes the variational density matrix of hydrogen nuclei and electrons at finite temperature using three deep generative models: a normalizing flow model for the Boltzmann distribution of the classical nuclei, an autoregressive transformer for the distribution of electrons in excited states, and a permutational equivariant flow model for the unitary backflow transformation of electron coordinates in Hartree-Fock states. By jointly optimizing the three neural networks to minimize the variational free energy, we obtain the equation of state and related thermodynamic properties of dense hydrogen for the temperature range where electrons occupy excited states. We compare our results with other theoretical and experimental results on the deuterium Hugoniot curve, aiming to resolve existing discrepancies. Our results bridge the gap between the results obtained by path-integral Monte Carlo calculations at high temperature and ground-state electronic methods at low temperature, thus providing a valuable benchmark for hydrogen in the warm dense matter region.

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• Strongly correlated systems
• Warm-dense matter
• Machine learning
• Monte Carlo methods