  Thermodynamic properties of liquid water as well as hexagonal (Ih) and cubic
(Ic) ice are predicted based on density functional theory at the
hybrid-functional level, rigorously taking into account quantum nuclear motion,
anharmonic fluctuations and proton disorder. This is made possible by combining
advanced free energy methods and state-of-the-art machine learning techniques.
The ab initio description leads to structural properties in excellent agreement
with experiments, and reliable estimates of the melting points of light and
heavy water. We observe that nuclear quantum effects contribute a crucial 0.2
meV/H$_2$O to the stability of ice Ih, making it more stable than ice Ic. Our
computational approach is general and transferable, providing a comprehensive
framework for quantitative predictions of ab initio thermodynamic properties
using machine learning potentials as an intermediate step.
