When quantum thermal states look classical

Impact Assessment

Core Contribution. This paper establishes a unified "hierarchy of classical-to-quantum transitions" for quantum Gibbs states of long-range Pauli Hamiltonians, characterizing at exactly which inverse-temperature scales distinct notions of classicality break down: separability (death of entanglement) at β = Θ(1/sk), stabilizerness (death of magic) at β = Θ(log(1/ε)/sk), and the thermodynamic infinite-temperature phase (zero-free disk) at β = Θ(1/s√k). Crucially, it works in the physically salient long-range setting (bounded ℓ₁ on-site strength, including power-law interactions with α > D), where prior techniques based on bounded degree and Lieb-Robinson bounds fail. The paper resolves several concrete open questions: (i) it shows long-range systems still exhibit a constant-temperature death of entanglement (open question of RFA25), (ii) it resolves the correlation-decay conjecture of HMS20 in dimensions above 1D, and (iii) it provides polynomial-time classical algorithms that rule out the proposed superpolynomial quantum advantage of SSTSMA25 for long-range Pauli systems. It also tightens BLMT24 (FOCS'24) to be optimal up to constants and reaches colder temperatures than known quantum Gibbs samplers.

Methodological Rigor. The work is technically rigorous, with matching upper and lower bounds establishing tightness (up to constants) for each transition. The proofs are constructive and detailed, combining new long-range propagator expansions, an adaptive site/term-pinning strategy, cluster expansions controlled by the Kotecky-Preiss criterion, and a novel randomized polymer-sampling scheme that upgrades quasipolynomial Barvinok-type algorithms to polynomial time. The "repair trick" exploiting the Pauli basis to improve the combinatorial growth from (sk)^m to (s√k)^m is elegant and yields the improved 1/√k scaling. Appendices supply explicit entangled/magical constructions demonstrating tightness. The authors disclose AI assistance (ChatGPT) on some appendices while asserting human verification — a minor reproducibility caveat but the proofs appear self-contained.

Potential Impact. This sits at the center of an active, competitive line of work (FOCS'24, STOC'20, STOC'25, PRL'26) on classical simulability of quantum thermal states and the boundary of quantum advantage. By ruling out a proposed superpolynomial advantage and sharply characterizing where classicality fails, it directly shapes how the community reasons about high-temperature quantum systems and Gibbs sampling. The introduced techniques (long-range cluster expansions, pinned zero-freeness, polymer sampling) are reusable primitives that the authors themselves connect to open problems in fast mixing of quantum Gibbs samplers, Dobrushin-Shlosman conditions, and structural properties of Gibbs states. The "pinned zero-freeness" concept in particular is flagged as a plausibly load-bearing tool for future mixing-time analyses.

Timeliness & Relevance. Extremely timely. It addresses very recent open questions (STOC'25, PRL'26 preprints) and the central question of quantum advantage in thermal-state tasks. The long-range extension is exactly the setting where prior positive quantum-advantage conjectures were located, making the negative (classical-easiness) results here especially consequential.

Strengths & Limitations. Strengths: tightness of all bounds, breadth of resolved open problems, conceptual clarity of the transition hierarchy, and genuinely new proof machinery that overcomes the failure of Lieb-Robinson-based methods. The exposition is unusually clear for such a dense theoretical work, with a strong technical overview. Limitations: results are confined to Pauli Hamiltonians with bounded ℓ₁ strength — physically important fermionic systems (electronic structure, SYK) with diverging ℓ₁ strength are explicitly outside scope. Constants are not optimized, the conjectured zero-free strip (needed to reach β ~ 1/s) remains open, and the relationship between zero-freeness and quantum mixing times is unresolved. It is pure theory with no experimental component and limited direct translational value beyond classical simulation tooling.

Additional observations. The refutation content is notable: the paper explicitly overturns a recently conjectured superpolynomial quantum advantage, which raises its refutation value above the typical paper. Its foundationality is high — the techniques are clearly designed as building blocks, and the framing may become a reference point for the subfield. Resource intensity is minimal (pure theory), lowering the barrier for others to engage, though the required expertise is substantial.