Understanding the entanglement structure of local Hamiltonian ground spaces is a physically motivated problem, with applications ranging from tensor network design to quantum error-correcting codes. To this end, we study the complexity of estimating ground state entanglement, and more generally entropy estimation for low energy states and Gibbs states. We find, in particular, that the classes qq-QAM [Kobayashi, le Gall, Nishimura, SICOMP 2019] (a quantum analogue of public-coin AM) and QMA(2) (QMA with unentangled proofs) play a crucial role for such problems, showing: (1) Detecting a high-entanglement ground state is qq-QAM-complete, (2) computing an additive error approximation to the Helmholtz free energy (equivalently, a multiplicative error approximation to the partition function) is in qq-QAM, (3) detecting a low-entanglement ground state is QMA(2)-hard, and (4) detecting low energy states which are close to product states can range from QMA-complete to QMA(2)-complete. Our results make progress on an open question of [Bravyi, Chowdhury, Gosset and Wocjan, Nature Physics 2022] on free energy, and yield the first QMA(2)-complete Hamiltonian problem using local Hamiltonians (cf. the sparse QMA(2)-complete Hamiltonian problem of [Chailloux, Sattath, CCC 2012]).

翻译：理解局域哈密顿量基态空间的纠缠结构是一个具有物理动机的问题，其应用涵盖张量网络设计到量子纠错码。为此，我们研究了估计基态纠缠的复杂性，更一般地，还研究了低能态和吉布斯态的熵估计。我们发现，特别是qq-QAM类（公钥AM的量子类比）[Kobayashi, le Gall, Nishimura, SICOMP 2019]和QMA(2)（具有无纠缠证明的QMA）对此类问题起到关键作用，展示了：(1) 检测高纠缠基态是qq-QAM完全的；(2) 计算亥姆霍兹自由能的加性误差近似（等价于配分函数的乘性误差近似）属于qq-QAM；(3) 检测低纠缠基态是QMA(2)困难的；(4) 检测接近积态的低能态的范围可从QMA完全到QMA(2)完全。我们的结果推进了[Bravyi, Chowdhury, Gosset和Wocjan, Nature Physics 2022]关于自由能的一个开放问题，并利用局域哈密顿量首次提出了QMA(2)完全的哈密顿量问题（对比[Chailloux, Sattath, CCC 2012]的稀疏QMA(2)完全哈密顿量问题）。