This is sort of true. The fact that it turns into the n-body problem prevents us from being able to do quantum mechanics analytically. Once we're stuck doing it numerically, then all the issues of sampling density of the wave function et al. crop up, and they make it very difficult to solve numerically.

Thanks for pointing this out. These numerical difficulties are also a big part of the problem, albeit less accessible to people who aren't comfortable with the concept of high-dimensional Hilbert spaces. A friend of mine had a really nice write-up in his thesis on this difficulty. I'll see if I can dig it up.