preprint Sevag Gharibian author Jonas Kamminga author 7 department 623 department 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]). 2025 eng 2510.06796 Gharibian, S., &#38; Kamminga, J. (2025). On the complexity of estimating ground state entanglement and free  energy. In <i>arXiv:2510.06796</i>. S. Gharibian, J. Kamminga, ArXiv:2510.06796 (2025). @article{Gharibian_Kamminga_2025, title={On the complexity of estimating ground state entanglement and free  energy}, journal={arXiv:2510.06796}, author={Gharibian, Sevag and Kamminga, Jonas}, year={2025} } Gharibian, Sevag, and Jonas Kamminga. “On the Complexity of Estimating Ground State Entanglement and Free  Energy.” <i>ArXiv:2510.06796</i>, 2025. Gharibian S, Kamminga J. On the complexity of estimating ground state entanglement and free  energy. <i>arXiv:251006796</i>. Published online 2025. S. Gharibian and J. Kamminga, “On the complexity of estimating ground state entanglement and free  energy,” <i>arXiv:2510.06796</i>. 2025. Gharibian, Sevag, and Jonas Kamminga. “On the Complexity of Estimating Ground State Entanglement and Free  Energy.” <i>ArXiv:2510.06796</i>, 2025. 617782025-10-10T13:45:28Z2026-04-20T13:53:47Z