When quantum thermal states look classical

Harald Putterman, Alexander Zlokapa of MIT, and Jordan Cotler of Harvard University show that quantum Gibbs states can remain classically tractable across several distinct finite-temperature regimes, even for long-range Pauli Hamiltonians whose terms act on at most k qubits and have total incident strength at most s. Their hierarchy begins with separability: the Gibbs state is guaranteed unentangled for inverse temperature β ≤ 1/(72sk), and this Θ(1/(sk)) threshold is tight, including for commuting Hamiltonians. A randomized polynomial-time algorithm prepares the state as a mixture of pure product stabilizer states up to β ≤ 1/(4096e sk), ruling out the proposed superpolynomial quantum advantage in this regime. For Hamiltonians ε-close to commuting, stabilizerness persists to the parametrically colder scale Θ(log(1/ε)/(sk)), separating the disappearance of entanglement from the disappearance of magic. Independently, a cluster expansion based on Pauli-specific polymer counting establishes a zero-free partition-function disk of radius Θ(1/(s√k)); within it, local thermal expectations and log Z are estimable in randomized polynomial time, and geometrically local systems exhibit exponential correlation decay. The proofs combine Kotecký–Preiss cluster expansions, interaction-picture Dyson expansions, adaptive pinning, and importance sampling of polymers. The paper also acknowledges that ChatGPT Pro contributed to technical appendices and proof development, while the authors retained responsibility for correctness.