# 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"? – Norbert Schuch Aug 29 '18 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. – JanVdA Aug 29 '18 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? – Norbert Schuch Aug 29 '18 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/…) – JanVdA Aug 30 '18 at 6:44
• "Hamiltonian simulation" is typically used to refer to the task of simulating a specific dynamics, that is, a Hamiltonian. The straightforward use of this is to simulate a given quantum system. Do you have a reference for how knowledge of a quantum algorithm solving a given problem can be used to build a Hamiltonian which would allow solving said problem via Hamiltonian simulation? It doesn't sound straightforward to me.. – glS Aug 30 '18 at 9:35

## 3 Answers

It's not an area I personally know much about, but I know that many of my physicist friends are excited about being able to investigate the Hubbard Model on larger lattices than we can simulate today. There are known and published algorithms for finding the ground state energy, computing Green's functions, and other important characteristics of the model.

The hope is that understanding the Hubbard model on larger, more realistic lattices might help us understand superconductivity better, and in particular lead to materials that stay superconducting at higher and higher temperatures.

Several graph theory problems such as Graph Coloring (which is NP-complete) can be cleverly mapped to finding ground states of some classes of Hamiltonians.

Graph Partitioning using Quantum Annealing on the D-Wave System

I consider the leading candidate (considering you explicitly excluded quantum chemistry and hence all of biochemistry) the calculation of nuclear properties: This will, with suitable quantum computers, one day allow to compare experimental, atomic physics with theoretical (quantum computer calculated) values from the standard model. It's not entirely science fiction either, see this viewpoint about such a calculation.

• Note that I didn't exclude the finding/designing of a drug or material with specific characteristics as interesting problem. I added a paragraph to the question to make this clear. I only wanted to exclude things that are only interesting for the people working in the domain of quantum chemistry. Sorry for the confusion. – JanVdA Aug 30 '18 at 11:34