Simulation of every physical quantum system on quantum computer Let me quote from the section 9.3 of Classical and Quantum Computation by Kitaev, Shen and Vyalyi. 


  With high confidence, we may claim that every physical quantum system
  can be efficiently simulated on a quantum computer, but we can never
  prove this statement.


Why can't we ever prove this statement?
We will never be able to prove this statement, because we can never be able to know for sure whether we have the exact laws of physics, or just a very good approximation to them. Even if we had a satisfactory theory of everything which we could use to make good predictions about every experimentally measurable physical system, there would be no way to tell whether it was correct or a very good approximation.

Having said that, we are still quite far from even coming close to a proof of this statement. For example, we can't even prove that the Standard Model is simulable by a quantum computer. Jordan, Lee, and Preskill have two papers showing how to use a quantum computer to simulate a quantum field theory which is much simpler than the Standard Model. This turns out to be harder than it might at first appear.