It is known that classical spin-glass problems are NP-hard, just like their quantum counterparts. Does this truly mean that finding their Ground-State is equally hard in both cases?

The most straightforward idea of solving the classical XY-model of $N$ sites exactly, for example, would be discretizing the angle $\theta_j$ at each site $j$ into $M_\theta$ possible values for example; try out all the possible combinations and take the many-body state with the lowest energy. This leads to $M_\theta^N$ states to be considered. Such a discretization approach is proposed, e.g. here for time-evolution, but I think that's easier than the ground state. Or is doing imaginary time-evolution a reliable approach that avoids metastabilities?

Taking the quantum limit on the other hand, leads to a Hilbert space spanned by 'only' $2^N$ basisvectors, in contrast. Or am I missing something?

EDIT added: it seems that my premise that the classical XY model can straightforwardly be studied as a quantum XY model seems wrong because the latter features entanglement while the former shouldn't. So the classical case rather matches some kind of Gutzwiller approch

  • $\begingroup$ No answer in 22 days? Seems rough, maybe try here where there's specific tags for ising-model, spin-models, heisenberg-model, model-hamiltonians, and condensed-matter: mattermodeling.stackexchange.com $\endgroup$ – user1271772 2 days ago

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