On the complexity of estimating ground state entanglement and free energy

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]).

💡 Research Summary

The paper investigates the computational complexity of estimating two fundamental physical quantities associated with local Hamiltonians: the entanglement structure of low‑energy (ground‑state) subspaces and the Helmholtz free energy of Gibbs states. By formulating precise decision problems that capture high‑entanglement, low‑entanglement, and free‑energy approximation, the authors map each problem to well‑studied quantum complexity classes, revealing a nuanced landscape of hardness and containment.

The first problem, High‑Entropy Low‑Energy State (HELES), asks whether a given k‑local Hamiltonian H admits a state of energy at most α whose reduced density matrix on subsystem A has von Neumann entropy at least s. The authors prove that HELES is complete for the class qq‑QAM, a quantum analogue of public‑coin AM introduced by Kobayashi, le Gall, and Nishimura. The proof hinges on a two‑turn interactive protocol where the verifier supplies polynomially many Bell‑pair “quantum coins” and the prover responds with a mixed Choi state of a quantum channel. An entropy‑verification subroutine based on a quantum extractor ensures that the verifier can distinguish high‑entropy from low‑entropy cases with only a small additive gap. This result shows that, unlike the circuit‑specified case (NIQSZK‑complete), entanglement estimation for Hamiltonian‑defined states jumps to qq‑QAM, which contains NIQSZK.

The second problem, Free‑Energy Approximation (FEA), asks to decide whether the free energy F(H)=−(1/β)log Z(H) of a k‑local Hamiltonian at inverse temperature β lies below a or above b, where b−a=1/poly(n). By exploiting the relation between free energy and the partition function Z, the authors embed a relative‑error approximation of Z into a qq‑QAM protocol. The verifier again uses quantum coins to sample thermal expectation values, while the prover supplies a suitable Choi state. Consequently, additive free‑energy approximation is shown to be in qq‑QAM, improving upon the previously known containment in P^#P.

The third problem, Low‑Entropy Low‑Energy State (LELES), is the natural counterpart of HELES: it asks whether there exists a low‑energy state whose subsystem A entropy is at most t. Here the complexity rises to QMA(2), the class of quantum Merlin‑Arthur proofs with two unentangled witnesses. The authors develop a “channel‑to‑Hamiltonian” reduction: given a QMA(2) verification circuit, they construct a k‑local Hamiltonian whose low‑energy space encodes the two unentangled witnesses via the Stinespring dilation of a channel. This yields QMA(2)‑hardness even for physically relevant Hamiltonians (e.g., 2‑D Heisenberg). Thus, bounding the complexity of LELES is at least as hard as understanding the power of QMA(2), an open problem in quantum complexity theory.

Finally, the paper studies Low‑Energy Approximate Product State (LEAPS), which asks whether a low‑energy state is close (in trace distance) to a product state. By tuning the energy gap and distance parameters, the problem is shown to be QMA‑complete in one regime and QMA(2)‑complete in another. This mirrors earlier work on separable Hamiltonians but now applies to strictly local Hamiltonians, providing a natural example where the same physical problem “leaps’’ from QMA to QMA(2) depending on quantitative thresholds.

Technically, the work relies on two core tools: (i) a qq‑QAM protocol for entropy verification based on quantum extractors, and (ii) a refined circuit‑to‑Hamiltonian construction that embeds the Stinespring representation of a channel into a local Hamiltonian. These tools are encapsulated in Lemmas 3.1 and 3.2 and are likely to be reusable for other Hamiltonian‑based complexity questions.

In summary, the paper establishes that (a) detecting high‑entanglement ground states is qq‑QAM‑complete, (b) additive free‑energy approximation lies in qq‑QAM, (c) detecting low‑entanglement ground states is QMA(2)‑hard, and (d) low‑energy product‑state approximation can be either QMA‑ or QMA(2)‑complete depending on parameters. These results answer an open question about the complexity of quantum partition‑function estimation, introduce the first QMA(2)‑complete Hamiltonian problem using strictly local terms, and deepen our understanding of how entanglement and thermodynamic quantities intertwine with quantum computational complexity.