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I am reading about adiabatic quantum computing- specifically, about how it can find the lowest energy configuration of the Ising model. It is said that the initial state is a superposition of all the possible values, and then we slowly change to a final "desired" Hamiltonian, $H_f$. How do we know what this Hamiltonian is, since the lowest energy configuration is what we are looking for? And how, more precisely, do we know when we reached the absolute minimum, and not just a local one?

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Just as with the case of algorithms like QAOA and VQE, the Hamiltonians are actually already known (for example, a Hamiltonian encoding a graph problem); in fact, the use case for any of these algorithms is finding unknown information about a known Hamiltonian (given some matrix/operator, you generally don't know the eigenvalues).

As for reaching the global minimum, there really isn't a guarantee that you aren't stuck in a local minimum.

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