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We know that with unfriendly energy gap circumstances, adiabatic quantum computing takes exponential time to find a ground state of some hamiltonian. But if some local minimum state is all we look for, then would/can adiabatic quantum computing discover local minimum states in polynomial time?

For sure this would require definining what local minimum would be for quantum states, which depends on perturbation models. But if we think of some Hamiltonian as reflecting some function f(x), and we aim to discover some local minimum of f(x), the question is whether adiabatic quantum computing can discover a local minimum easily or not, even if finding a ground state would take exponential time.

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