Hexagonal ice ($\rm{I_h}$), the most common structure of ice, displays a variety of fascinating properties. Despite major efforts, a theoretical description of all its properties is still lacking. In particular, correctly accounting for its density and interatomic interactions is of utmost importance as a stepping stone for a deeper understanding of other properties. Deep potentials are a recent alternative to investigate the properties of {\iceIH}, which aims to match the accuracy of \textit{ab initio} simulations with the simplicity and scalability of classical molecular dynamics. This becomes particularly significant if one wishes to address nuclear quantum effects. In this work, we use machine learning potentials obtained for different exchange and correlation functionals to simulate the structural and vibrational properties of {\iceIH}. We show that most functionals overestimate the density of ice compared to experimental results. Furthermore, a quantum treatment of the nuclei leads to even further distancing from experiments. We understand this by highlighting how different inter-atomic interactions play a role in obtaining the equilibrium density. In particular, different from water clusters and bulk water, nuclear quantum effects lead to stronger H-bonds in {\iceIH}.