On the complexity of estimating ground state entanglement and free energy

TL;DR

This work investigates the computational complexity of estimating ground-state entanglement and finite-temperature free energy for $k$-local Hamiltonians. It introduces three problem families—HELES, FEA, and LELES/LEAPS—and leverages two main tools: an entropy-verification protocol within the $\qqQAM$ framework and a channel-to-Hamiltonian construction that embeds channel outputs into Hamiltonian ground spaces, enabling channel- or PP-like hardness to transfer to local Hamiltonians. The results reveal a nuanced landscape: HELES is $\qqQAM$-complete for $k\ge5$, FEA lies in $\qqQAM$, LELES is $QMAt$-hard (even for 2D physically motivated models), and LEAPS can be $QMA$- or $QMAt$-complete depending on parameter regimes, demonstrating a jump in complexity between unentangled and entangled proofs. The analysis extends to physically relevant Hamiltonians via universal Hamiltonian simulation, yielding the first $QMAt$-complete local Hamiltonian problem and showing the hardness persists for common models like Heisenberg and XY lattices. Together, these results advance understanding of when and how ground-state entanglement and free-energy estimation become computationally intractable, with implications for tensor-network design, quantum error correction, and phase-diagram studies.

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]).

Figures (1)

• Figure 1: Schematic representation of the entropy verification protocol. If the proof is such that the $E$ register after applying $U_T^\dagger$ is approximately $|0^{2n_A}\rangle$, then the contents of the $A$ register $\sigma_A$ are such that $T(\sigma_A) \approx \tilI$. As $T$ is an extractor, this implies that the min-entropy of $\sigma_A$ is large.

Theorems & Definitions (53)

• definition 1: High Entropy Low Energy State ($\HELES$)
• theorem 1
• definition 2
• theorem 2
• definition 3
• theorem 3
• definition 4
• theorem 4
• theorem 5
• definition 5: $\qqQAM$-protocol