  The phase diagram of water harbours many mysteries: some of the phase
boundaries are fuzzy, and the set of known stable phases may not be complete.
Starting from liquid water and a comprehensive set of 50 ice structures, we
compute the phase diagram at three hybrid density-functional-theory levels of
approximation, accounting for thermal and nuclear fluctuations as well as
proton disorder. Such calculations are only made tractable because we combine
machine-learning methods and advanced free-energy techniques. The computed
phase diagram is in qualitative agreement with experiment, particularly at
pressures $\lesssim$8000 bar, and the discrepancy in chemical potential is
comparable with the subtle uncertainties introduced by proton disorder and the
spread between the three hybrid functionals. None of the hypothetical ice
phases considered is thermodynamically stable in our calculations, suggesting
the completeness of the experimental water phase diagram in the region
considered. Our work demonstrates the feasibility of predicting the phase
diagram of a polymorphic system from first principles and provides a
thermodynamic way of testing the limits of quantum-mechanical calculations.
