# Example of Hamiltonian Simulation solving interesting problem?

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.

• Simulation on classical or on quantum computers? Time evolution or ground states? What do you mean by "a quantum algorithm is known"? Aug 29, 2018 at 21:34
• It is simulation on a quantum computer. No preference for time evolution or ground state as long as it is an interesting problem as defined in my question. With "a quantum algorithm is known" I mean that there is a known way to produce a quantum program that runs on a universal quantum computer that solves the problem. E.g. math.nist.gov/quantum/zoo gives many examples of quantum algorithms. Aug 29, 2018 at 22:17
• So you want both a quantum algorithm running on a full quantum computer and the possibility to run it on a quantum simulator? Or do you mean by "algorithm" a way to solve this on a simulator (which by no means needs to be universal). If the former, what's the point - would the simulator be faster? Aug 29, 2018 at 22:50
• I think you are mixing "quantum simulator" and "hamiltonian simulation". These are 2 different concepts. Forget about the "quantum simulator" (this term is also not used in my question). With Hamiltionian Simulation I mean a simulation of a quantum mechanical system (see also 2nd point in first answer to question quantumcomputing.stackexchange.com/questions/2399/…) Aug 30, 2018 at 6:44
• @JanVdA sure, but then what do you mean by "for which a quantum algorithm is known"? A quantum algorithm to solve the problem of simulating a quantum system? Isn't that essentially what Hamiltonian simulation is? So why are you writing as if they are two different things?
– glS
Aug 30, 2018 at 18:00