A classically hard equilibrium observable an analog simulator can reach
In plain words
Hot quantum systems turn out to be easy for classical computers. The question is whether any cooled simulator can reach a regime that is both hard for classical methods and accessible to the hardware.
Precise statement
Identify a family of local Hamiltonians and a temperature $T$ for which estimating a local observable in the Gibbs state is classically hard (under standard complexity assumptions), while an analog simulator with realistic cooling can prepare that Gibbs state to accuracy $\epsilon$. Answer: an explicit family with a hardness proof and a preparation scheme.
What would settle it
A complexity-theoretic hardness proof for a specific Hamiltonian family plus a cooling scheme with proven or measured accuracy.
Status in the literature
Unverified note
2024 results showed Gibbs states above a constant temperature are efficiently preparable classically (Bakshi, Liu, Moitra, Tang), so any advantage must be at low T.