No low-energy sampleable states conjecture
In plain words
Some quantum systems are now known to have no simple low-energy states that a shallow circuit can make. A stronger claim, that no low-energy state can even be sampled by a classical computer, is unproved.
Precise statement
Does there exist a family of $O(1)$-local Hamiltonians on $n$ qubits and a constant $\epsilon > 0$ such that every state with energy below $E_{0} + \epsilon m$ fails to be classically sampleable in the sense of Gharibian and Le Gall (2022), i.e. no $\operatorname{poly}(n)$ classical algorithm samples its computational-basis distribution with the required query access?
What would settle it
An explicit Hamiltonian family with a proof that no low-energy state is classically sampleable.
Status in the literature
Unverified note
2025 work excluded restricted classes (Clifford, near-Clifford and fermionic Gaussian low-energy states, e.g. arXiv:2502.15368).