I think this is not correct. A fundamental problem in quantum simulation of quantum systems is the Hamiltonian problem, where you give me a Hamiltonian H, which

I think this is not correct. A fundamental problem in quantum simulation of quantum systems is the Hamiltonian problem, where you give me a Hamiltonian H, which is the sum of polynomially many little local Hamiltonians H_i each of which acts on at most k subsystems of your big total system. Then you give me two numbers a and b (with some technical constraints) and ask me whether the ground state energy of H is between a and b. This problem is QMA-complete as long as k >= 2, and known to be QMA-hard even for some pretty nice looking Hamiltonians. Here QMA stands for Quantum Merlin-Arthur which is the complexity class equivalent to MA (Merlin-Arthur) for classical computers. You can think of MA as being related to BPP in the same way that NP is related to P, and this is the same way that QMA is related to BQP. Basically QMA problems are not expected to be solved efficiently by a quantum computer.