Hamiltionian Simulation (= simulation of quantum mechanical systems) is claimed to be one of the most promising applications of a quantum computer in the future.
One of the earliest – and most important – applications of a quantum computer is likely to be the simulation of quantum mechanical systems. There are quantum systems for which no efficient classical simulation is known, but which we can simulate on a universal quantum computer. What does it mean to “simulate” a physical system? According to the OED, simulation is “the technique of imitating the behaviour of some situation or process (whether economic, military, mechanical, etc.) by means of a suitably analogous situation or apparatus”. What we will take simulation to mean here is approximating the dynamics of a physical system. Rather than tailoring our simulator to simulate only one type of physical system (which is sometimes called analogue simulation), we seek a general simulation algorithm which can simulate many different types of system (sometimes called digital simulation)
For the details, check chapter 7 of the lecture notes by Ashley Montaro.
Question: Assuming tomorrow we have such a powerful universal quantum computer: which interesting problem (1) based on simulating a quantum system (2) for which a quantum algorithm is known, can we solve ?
Note that it is important that a quantum algorithm is already known to solve this problem or at least that there is good evidence supporting that such quantum algorithm can be found.
With interesting I mean that it should have substantial impact beyond the field of quantum computing and quantum chemistry.
Note that interesting problem definitely includes finding molecules that can cure diseases, designing materials with specific characteristics.