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With my VQE algorithm, I want to obtain both the ground state energy and its corresponding ground eigenstate. However, the state I reach using VQE is usually very different from the true ground eigenstate, though its energy is still typically nearly exactly equal to the ground state energy. It seems to me that, depending on the initial state, a system can have multiple states which correspond to the ground state energy - however these usually aren't eigenstates. Does this mean I'm doing something wrong? Is there a way to guarantee that I reach the ground eigenstate?

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    $\begingroup$ Do you have any more precise details of the implementation of the algorithm and a Hamiltonian you are experiencing this with? The lowest energy level can be degenerate and also an imperfect approximation naturally won't quite be an eigenstate but should be close. $\endgroup$ Jun 16, 2023 at 2:25
  • $\begingroup$ @JosephGeipel I am using the Hardcore Bose-Hubbard Hamiltonian and the BFGS minimizer. The degenerate states would still be eigenstates right? I am pretty consistently getting states which are not eigenstates, and their energies seems to be effectively equal to the ground state (to multiple significant figures ~6). Do you think this means I have some error? $\endgroup$
    – user22395
    Jun 16, 2023 at 3:56

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