schroedinger equation - What are the primary obstacles to solve the many-body problem in quantum mechanics?

(This is a simple question, with likely a rather involved answer.)

What are the primary obstacles to solve the many-body problem in quantum mechanics?

Specifically, if we have a Hamiltonian for a number of interdependent particles, why is solving for the time-independent wavefunction so hard? Is the problem essentially just mathematical, or are there physical issues too? The many-body problem of Newtonian mechanics (for example gravitational bodies) seems to be very difficult, with no solution for $n > 3$. Is the quantum mechanical case easier or more difficult, or both in some respects?

In relation to this, what sort of approximations/approaches are typically used to solve a system composed of many bodies in arbitrary states? (We do of course have perturbation theory which is sometimes useful, though not in the case of high coupling/interaction. Density functional theory, for example, applies well to solids, but what about arbitrary systems?)

Finally, is it theoretically and/or practically impossible to simulate high-order phenomena such as chemical reactions and biological functions precisely using Schrodinger's quantum mechanics, over even QFT (quantum field theory)?

(Note: this question is largely intended for seeding, though I'm curious about answers beyond what I already know too!)

Answer

First let me start by saying that the $N$-body problem in classical mechanics is not computationally difficult to approximate a solution to. It is simply that in general there is not a closed form analytic solution, which is why we must rely on numerics.

For quantum mechanics, however, the problem is much harder. This is because in quantum mechanics, the state space required to represent the system must be able to represent all possible superpositions of particles. While the number of orthogonal states is exponential in the size of the system, each has an associated phase and amplitude, which even with the most coarse grain discretization will lead to a double exponential in the number of possible states required to represent it. Thus in quantum systems you need $O(2^{2^n})$ variables to reasonable approximate any possible state of the system, versus only $O(2^n)$ required to represent an analogous classical system. Since we can represent $2^m$ states with $m$ bits, to represent the classical state space we need only $O(n)$ bits, versus $O(2^n)$ bits required to directly represent the quantum system. This is why it is believed to be impossible to simulate a quantum computer in polynomial time, but Newtonian physics can be simulated in polynomial time.

Calculating ground states is even harder than simulating the systems. Indeed, in general finding the ground state of a classical Hamiltonian is NP-complete, while finding the ground state of a quantum Hamiltonian is QMA-complete. (On the other hand, ground states are to some extent less relevant because the systems for which is is computationally hard to calculate the ground state of (at least on a QC) don't cool efficiently either.)