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Naturally, in general, ground state preparation is QMA-complete. There exists a paper by Andrew Childs, David Gosset & Zak Webb, which shows that ground state preparation for the Bose-Hubbard model is QMA-complete.

However, is it known what the complexity of ground state preparation for the Fermi-Hubbard model is? I could not find this in the literature directly, does this follow trivially from a general result which I am missing?

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In this paper, Schuch and Verstraete determined the computational complexity of finding the ground state of the Fermi-Hubbard model, showing that it is among the hardest problems in the complexity class QMA, Quantum Merlin Arthur.

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  • $\begingroup$ This is very helpful, thank you! (will upvote as soon as I have the reputation) But if I understand correctly, strictly speaking, this completeness proof works only for the Fermi-Hubbard Hamiltonian with local magnetic fields $H_{\mathrm{Hubb}}=-t \sum_{\langle i, j\rangle, s} a_{i, s}^{\dagger} a_{j, s}+U \sum_i n_{i, \uparrow} n_{i, \downarrow}-\sum_i \boldsymbol{\sigma}_i \cdot \mathbf{B}_i$ - so this doesn't quite proof that an efficient ground state preparation for the Fermi-Hubbard model without magnetic fields is impossible, if I am not mistaken. $\endgroup$
    – lm1909
    Feb 3, 2023 at 16:27
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    $\begingroup$ Yes, indeed the proof by Schuch and Verstraete does include the coupling of local magnetic fields to the spin degrees of freedom. Moreover, the proof involves going from the spin-1/2 Heisenberg model to the Fermi-Hubbard model, which is only strictly valid at half-filling in the limit $U/t \to \infty$, since charge fluctuations are frozen. I have looked for other papers discussing the complexity class of the Fermi-Hubbard model, but could not find any that were more relevant than this one. $\endgroup$
    – bm442
    Feb 4, 2023 at 10:48
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    $\begingroup$ Let me also note that, even if the Fermi-Hubbard model is in the QMA class, it does not mean it is impossible to achieve quantum advantage in the determination of its ground state properties relative to state-of-the-art numerical methods (e.g., PEPS-based Corner Transfer Matrix, Quantum Monte Carlo) implemented on conventional hardware for system sizes of the order of $30-50$ sites in two-dimensional lattices. It simply means that the resources required to scale up the simulations grow exponentially with the number of electrons. $\endgroup$
    – bm442
    Feb 4, 2023 at 10:55
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    $\begingroup$ Worth pointing out that the magnetic field loophole appears to be closed, see this (open-access) journals.aps.org/prxquantum/abstract/10.1103/… $\endgroup$ Jul 8, 2023 at 20:03
  • $\begingroup$ @Dr.T.Q.Bit thank you, this is indeed exactly what I was looking for! In case you prefer to turn this into an answer, I am very happy to mark it as accepted. $\endgroup$
    – lm1909
    Sep 26, 2023 at 15:38

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