What are hamiltonians in the context of quantum computing

Question

Sorry if this question is a bit too generic or basic, but my background lies only in mathematics and computer science. I am currently writing my thesis on the topic of simulating quantum computing circuits using tensor networks. I am currently reading into how the entanglement of a tensor network scales, and one paper I'm currently reading ("Entanglement and Tensor Networks" by Jens Eisert) deals with entanglement for local Hamiltonians. Another paper is talking about how "gapped Hamiltonians with local interactions" occupy only a small corner of the Hilbert space, meaning they follow an area law.

My question now is this: What exactly is a Hamiltonian in the context of quantum computing? Is it identical with the ground state and applied gates?

And if you're patient enough, I would love if you could also explain to me, what "gapped Hamiltonians with local interactions" and local Hamiltonians are.

Thank you very much in advance!

Answer 1

What exactly is a Hamiltonian in the context of quantum computing?

The Hamiltonian is an equation that describes how a physical system behaves. The Hamiltonian reflects the physical interactions within the system, including potential fields, particle interactions, and external forces. It is a mathematical operator 𝐻 that encapsulates the total energy of a system, including both kinetic and potential energy components.

Quantum computers are designed such way, that quantum mechanic is dominant factor of how they behave.

Is it identical with the ground state and applied gates?

Ground state is lowest energy state of Hamiltonian. Hamiltonian contains all possible energy states of the system.

Gates are operations that modify quantum states, which can be interpreted as the result of applying a Hamiltonian for a specific time.

local Hamiltonians and gapped Hamiltonians

Local Hamiltonian is such Hamiltonian, where interactions are local. Such as particles that have interactions between only nearest neighbours. That can be a good approximation in many times, and can simplify systems one need to deal with quite a bit.

The term "gapped" refers to the energy difference between the ground state and the first excited state. If this energy difference is non-zero, the Hamiltonian is said to be "gapped". The presence of a gap has significant implications for the physical properties of the system, such as stability against small perturbations, robustness of the ground state, and the behavior of excitations. In case of quantum computing those are needed to do quantum computing without errorness.