Accurate quantum time-correlation functions for anharmonic liquids
In plain words
Infrared spectra and diffusion constants of a liquid come from how it changes in time, and for light nuclei that motion is quantum. Exact quantum simulation of it is exponentially costly, and the approximations in use produce known artifacts such as shifted or split spectral peaks.
Precise statement
For liquid water at 300 K on a given potential of ab initio quality, compute the Kubo-transformed velocity and dipole time-correlation functions, hence the diffusion coefficient and the infrared spectrum including the O-H stretch band near 3400 $\mathrm{cm}^{-1}$, with errors relative to exact quantum dynamics smaller than the spread among centroid MD, RPMD, Matsubara-based and semiclassical methods. An answer is a systematically improvable method with polynomial cost in particle number, or a proof that such cost is impossible for generic anharmonic potentials.
What would settle it
Comparison with numerically exact quantum dynamics on reduced models of increasing size, plus agreement with measured water spectra and diffusion on the full liquid.
Status in the literature
Matsubara dynamics (2015) identified the classical approximation that preserves the quantum Boltzmann distribution but carries a phase problem; practical methods still show artifacts in the O-H stretch region.