The quantum PCP conjecture for local Hamiltonians
In plain words
Computing the ground energy of a quantum system exactly enough is known to be as hard as any problem a quantum computer can check. The conjecture says that even an estimate with a fixed fractional error stays that hard.
Precise statement
Let $H = \operatorname{sum}_{i=1}^m H_i$ be a k-local Hamiltonian on $n$ qubits with $\mid\mid H_i\mid\mid \le 1$ and each qubit in $O(1)$ terms. Is it QMA-hard to approximate its ground energy to additive error $\epsilon m$ for some constant $\epsilon > 0$? An answer is a proof of QMA-hardness, or a proof that the problem lies in NP (or a smaller class).
What would settle it
A QMA-hardness reduction for constant relative precision, or a classical-witness algorithm for it.
Status in the literature
The weaker NLTS conjecture was proved in 2022 (Anshu, Breuckmann and Nirkhe, arXiv:2206.13228); the full conjecture is open.