Given a local Hamiltonian, how difficult is it to determine the entanglement structure of its ground state? We show that this problem is computationally intractable even if one is only trying to decide if the ground state is volume-law vs near area-law entangled. We prove this by constructing strong forms of pseudoentanglement in a public-key setting, where the circuits used to prepare the states are public knowledge. In particular, we construct two families of quantum circuits which produce volume-law vs near area-law entangled states, but nonetheless the classical descriptions of the circuits are indistinguishable under the Learning with Errors (LWE) assumption. Indistinguishability of the circuits then allows us to translate our construction to Hamiltonians. Our work opens new directions in Hamiltonian complexity, for example whether it is difficult to learn certain phases of matter.

翻译：给定一个局域哈密顿量，确定其基态纠缠结构有多困难？我们证明，即使仅试图判断基态是满足体积律还是近似面积律纠缠，该问题在计算上也是难解的。我们通过在公钥设置中构造强形式的伪纠缠来证明这一点——在此设置中，制备这些态所用的电路是公开已知的。具体而言，我们构造了两个量子电路族，分别产生体积律和近似面积律纠缠态，但这两个电路族的经典描述在带误差学习（LWE）假设下是不可区分的。电路间的不可区分性使我们能够将构造推广到哈密顿量。我们的工作为哈密顿量复杂性理论开辟了新方向，例如确定学习某些物相是否具有困难性。