---
_id: '61778'
abstract:
- lang: eng
  text: "Understanding the entanglement structure of local Hamiltonian ground spaces\r\nis
    a physically motivated problem, with applications ranging from tensor\r\nnetwork
    design to quantum error-correcting codes. To this end, we study the\r\ncomplexity
    of estimating ground state entanglement, and more generally entropy\r\nestimation
    for low energy states and Gibbs states. We find, in particular, that\r\nthe classes
    qq-QAM [Kobayashi, le Gall, Nishimura, SICOMP 2019] (a quantum\r\nanalogue of
    public-coin AM) and QMA(2) (QMA with unentangled proofs) play a\r\ncrucial role
    for such problems, showing: (1) Detecting a high-entanglement\r\nground state
    is qq-QAM-complete, (2) computing an additive error approximation\r\nto the Helmholtz
    free energy (equivalently, a multiplicative error\r\napproximation to the partition
    function) is in qq-QAM, (3) detecting a\r\nlow-entanglement ground state is QMA(2)-hard,
    and (4) detecting low energy\r\nstates which are close to product states can range
    from QMA-complete to\r\nQMA(2)-complete. Our results make progress on an open
    question of [Bravyi,\r\nChowdhury, Gosset and Wocjan, Nature Physics 2022] on
    free energy, and yield\r\nthe first QMA(2)-complete Hamiltonian problem using
    local Hamiltonians (cf. the\r\nsparse QMA(2)-complete Hamiltonian problem of [Chailloux,
    Sattath, CCC 2012])."
author:
- first_name: Sevag
  full_name: Gharibian, Sevag
  last_name: Gharibian
- first_name: Jonas
  full_name: Kamminga, Jonas
  last_name: Kamminga
citation:
  ama: Gharibian S, Kamminga J. On the complexity of estimating ground state entanglement
    and free  energy. <i>arXiv:251006796</i>. Published online 2025.
  apa: Gharibian, S., &#38; Kamminga, J. (2025). On the complexity of estimating ground
    state entanglement and free  energy. In <i>arXiv:2510.06796</i>.
  bibtex: '@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} }'
  chicago: Gharibian, Sevag, and Jonas Kamminga. “On the Complexity of Estimating
    Ground State Entanglement and Free  Energy.” <i>ArXiv:2510.06796</i>, 2025.
  ieee: S. Gharibian and J. Kamminga, “On the complexity of estimating ground state
    entanglement and free  energy,” <i>arXiv:2510.06796</i>. 2025.
  mla: Gharibian, Sevag, and Jonas Kamminga. “On the Complexity of Estimating Ground
    State Entanglement and Free  Energy.” <i>ArXiv:2510.06796</i>, 2025.
  short: S. Gharibian, J. Kamminga, ArXiv:2510.06796 (2025).
date_created: 2025-10-10T13:45:28Z
date_updated: 2026-04-20T13:53:47Z
department:
- _id: '7'
- _id: '623'
external_id:
  arxiv:
  - '2510.06796'
language:
- iso: eng
publication: arXiv:2510.06796
status: public
title: On the complexity of estimating ground state entanglement and free  energy
type: preprint
user_id: '71541'
year: '2025'
...