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What is the goal of Hamiltonian simulation?

Is it going to simulate a quantum system on a classical computer or quantum computer or none of them?

What is the relationship between a quantum algorithm and the Hamiltonian simulation?

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    $\begingroup$ Hi Ramim! Welcome to QCSE! What have you researched, what is the background of your question? Can you edit your question to provide motivation for why you are asking, even if it's idle curiosity? $\endgroup$ – Mark S Jun 22 at 12:56
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Hamiltonian simulation is usually talking about simulating a quantum system on another quantum system.

In particular, a digital simulation would involve implementing the simulation on a quantum computer using, e.g. the gate model. In this setting, there are several different strategies, i.e. algorithms, which one can use to perform the simulation. These have different performance characteristics, such as the level of accuracy.

This simulation is BQP-complete. That means it's the hardest sort of problem that a quantum computer can perform. This is because you can perform simulation on a quantum computer, and you can embed any arbitrary quantum computation in a Hamiltonian evolution.

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If we have a certain quantum system which is in a state $|\psi\rangle$ and has a Hamiltonian $H$, then if we look at the state after a time $t$ it will have evolved to $|\psi_t\rangle = e^{it H}|\psi\rangle$. The goal of Hamiltonian simulation is to answer some questions about $|\psi_t\rangle$ given a (usually simple) input state $|psi\rangle$, Hamiltonian $H$ and time $t$.

This was the original problem suggested by Feynman where quantum computers could reasonably do better than classical computers, as a quantum computer can naturally represent the unitary evolution.

This raises another question: Why would one want to do Hamiltonian simulation? There are many reasons, but generally, if we can efficiently do Hamiltonian simulation (for instance on a quantum computer), then we can study directly how structures at the smallest scales (such as atoms or molecules, but also certain properties of Quantum Field Theory) interact and evolve. This is important to the field of quantum chemistry where one for instance wants to know how the physical structure of a molecule evolves trough time. At a larger scale it could also prove useful to for instance pharmaceutical companies who can then directly simulate the effect of their drugs on the cells in our bodies (although this is very much pie in the sky at the moment).

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