On the complexity of estimating ground state entanglement and free energy

2025-10-08

AI Summary

This work investigates the computational complexity of estimating entanglement entropy and free energy for ground states, low-energy states, and Gibbs states of local Hamiltonians. Using techniques from quantum interactive proofs and multi-prover verification complexity theory, the authors construct the first QMA(2)-complete problem based on local Hamiltonians, thereby establishing QMA(2)-hardness for detecting low-entanglement states and QMA(2)-completeness for detecting low-energy states close to product states. They further prove that detecting highly entangled ground states and approximating the free energy are both qq-QAM-complete. This is the first result to reveal the fundamental roles of qq-QAM and QMA(2) in estimating quantum many-body entropies and thermodynamic quantities. It advances the long-standing open problem of free energy computation and establishes a deep connection between entanglement detection and quantum interactive proof systems.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Natural Language Processing: Question Answering

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

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]).

Problem

• Estimating ground state entanglement complexity in local Hamiltonians
• Computing Helmholtz free energy approximation in qq-QAM
• Detecting low-entanglement ground states using QMA(2)

Innovation

• Using qq-QAM for free energy approximation
• QMA(2) detects low-entanglement ground states
• Local Hamiltonians enable QMA(2)-complete problems