It is often said that interacting electron systems (say on a lattice, like the Hubbard model for example) are difficult to solve because of the exponential size of the Hilbert space.
However, I am not sure I find this argument very compelling. Non-interacting systems live in the same Hilbert space and yet are easily solved. There are also cases of interacting models that can be solved and it is not by brute force, I'm thinking about the Bethe ansatz for example.
I can also quote this interesting paper : Quantum Simulation of Time-Dependent Hamiltonians and the Convenient Illusion of Hilbert Space:
As an application, we showed that the set of quantum states that can be reached from a product state with a polynomial-time evolution of an arbitrary time-dependent quantum Hamiltonian is an exponentially small fraction of the Hilbert space. This means that the vast majority of quantum states in a many-body system are unphysical, as they cannot be reached in any reasonable time. As a con- sequence, all physical states live on a tiny submanifold, and that manifold can easily be parametrized by all poly-sized quantum circuits.
So, are these models difficult to solve because no one has found a suitable ansatz yet, or is there a fundamental reason to give up the hope of finding such an ansatz? Or at least a more convincing argument than the size of the Hilbert space?