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arXiv:2607.28536

When quantum thermal states look classical

Harald Putterman, Alexander Zlokapa, Jordan Cotler

quant-phSubmitted Jul 30, 2026• Announced Jul 31, 2026• 0 citations

v1 · Jul 30, 2026 v2 · Oct 5, 2026

Abstract

At high temperature, quantum Gibbs states retain several classical features of the maximally mixed state: the absence of entanglement, the absence of magic, analyticity of the partition function, correlation decay, and algorithmic tractability. We prove new and sharp bounds showing that these features persist down to finite temperatures independent of system size, but fail at distinct inverse-temperature scales, forming a hierarchy of classical-to-quantum transitions. Our results hold for long-range Pauli interactions with bounded strength at every site. Despite such all-to-all interactions, we show that the death of entanglement occurs at constant temperature, resolving an open question of Rouze, Franca and Alhambra (STOC'25). We give a polynomial-time classical algorithm that prepares Gibbs states up to the death of entanglement transition. Notably, this is asymptotically colder than temperatures at which quantum Gibbs samplers are known to mix quickly, as well as the original separability temperature of Bakshi et al. (FOCS'24), which we improve to be tight up to constants. At asymptotically even colder temperatures, we show that the Gibbs state remains in the thermodynamic infinite-temperature phase. This leads to polynomial-time classical algorithms for estimating thermal expectations despite both entanglement and magic, and the resolution of a correlation decay conjecture of Harrow, Mehraban and Soleimanifar (STOC'20).

Comments: 97 pages, 3 figures

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Estimating thermal expectations of local observables is a natural target for quantum advantage. We give a simple classical algorithm that approximates thermal expectations for Gibbs states of local Hamiltonians, and we show it has quasi-polynomial cost

n^{O(\log (n/ε))}

for all temperatures above a phase transition in the free energy. For many natural models, this coincides with the entire fast-mixing, quantumly easy phase. Our results apply to the Sachdev-Ye-Kitaev (SYK) model at any constant temperature due to its absence of a phase transition -- despite its entanglement, sign problem, and polynomial quantum circuit lower bounds. Beyond SYK, we rigorously establish a universal classically easy high-temperature phase for all local, bounded-degree Hamiltonians and show that it extends to temperatures strictly colder than the death of entanglement transition.

We show that thermal states of local Hamiltonians are separable above a constant temperature. Specifically, for a local Hamiltonian

H

on a graph with degree

\mathfrak{g}

, its Gibbs state at inverse temperature

\beta

, denoted by

\rho=e^{-\beta H}/\text{tr}(e^{-\beta H})

, is a classical distribution over product states for all

\beta < 1/ (c \mathfrak{{g}})

, where

c

is a constant. This sudden death of thermal entanglement upends conventional wisdom about the presence of short-range quantum correlations in Gibbs states. Moreover, we show that we can efficiently sample from the distribution over product states. In particular, for any

\beta < 1/(c\mathfrak{g}^{3})

, we can prepare a state

\varepsilon

-close to

\rho

in trace distance with a depth-one quantum circuit and

\text{poly}(n)\log(1/\varepsilon)

classical overhead.11In independent and concurrent work, Rouzé, França, and Alhambra [37] obtain an efficient quantum algorithm for preparing high-temperature Gibbs states via a dissipative evolution.

The Sachdev-Ye-Kitaev model is a strongly interacting fermionic system that has been well-studied in condensed matter and high energy physics. It is highly quantum: Gaussian states are far from the thermal state (Hastings and O'Donnell, STOC'22) and representing the thermal state requires large polynomial-size quantum circuits (Anschuetz et al., QIP'25). Very recently, it was nonetheless proven that classical algorithms can estimate local thermal expectations at sufficiently high temperature in quasipolynomial time (Zlokapa, FOCS'26). We show that classical algorithms can in fact estimate local observables at all constant temperatures in polynomial time. Our techniques also extend straightforwardly to classical systems: we resolve an open question about computing thermal expectations of a classical spin glass up to its phase transition (Bencs et al., STOC'26). Our proof develops a fully rigorous quantum cavity method. Due to the success of the classical cavity method in optimization, sampling, inference and learning, we expect the quantum cavity method to find further applications of independent interest. As an example, we give a quantum algorithm that learns SYK Hamiltonians from the Gibbs state at any constant temperature with polynomial time and sample complexity.

Above some temperature every thermal state of a Hamiltonian family is separable, and above another every thermal state is a mixture of stabilizer states, which carry no magic. Putterman, Zlokapa and Cotler conjectured that the second temperature is the lower one, so that heating removes magic from a whole family before entanglement. We prove this ordering for Pauli Hamiltonians of bounded interaction degree or bounded neighbourhood strength, the two families whose separability threshold is known exactly. A Clifford reduction shows that a Gibbs state is a stabilizer mixture whenever that of a canonical all-anticommuting Hamiltonian on the anticommutation graph of its terms is, with equivalence for independent terms, so commuting overlaps drop out of the guarantee. A cluster expansion exploiting anticommutation then shows that every Gibbs state is a stabilizer mixture for βΛ ≤ 0.2183, strictly beyond the sharp separability constant 1/(2e) ≈ 0.1839, and an exactly solvable graph caps the constant at 0.6232. A single Hamiltonian can still order its own transitions the other way. Because the guarantee counts only anticommuting neighbours, it certifies magic-free Gibbs states of the transverse-field Ising, Heisenberg, Kitaev and perturbed toric-code models at temperatures 1.1 to 5.6 times lower than those at which separability is guaranteed.

It is of great interest to understand the thermalization of open quantum many-body systems, and how quantum computers are able to efficiently simulate that process. A recently introduced dissipative evolution, inspired by existing models of open system thermalization, has been shown to be efficiently implementable on a quantum computer. Here, we prove that, at high enough temperatures, this evolution reaches the Gibbs state in time scaling logarithmically with system size. The result holds for Hamiltonians that satisfy the Lieb-Robinson bound, such as local Hamiltonians on a lattice, and includes long-range systems. To the best of our knowledge, these are the first results rigorously establishing the rapid mixing property of high-temperature quantum Gibbs samplers, which is known to give the fastest possible speed for thermalization in the many-body setting. We then apply our result to the problem of estimating partition functions at high temperature, showing an improved performance over previous classical and quantum algorithms.

We determine the exact universal high-temperature separability threshold for Pauli Hamiltonians of bounded degree

Δ\ge2

. If every coefficient in the Hamiltonian has magnitude at most one and each term has overlapping support with at most

Δ

other terms, the Gibbs state is a mixture of product Pauli eigenstates whenever

β\le z_Δ:= \operatorname{arctanh}\left[\max_{0\le x \le 1}x\left(\frac{1-x}{1+x}\right)^{Δ-1} \right].

For every

β>z_Δ

, a finite commuting Hamiltonian with maximum overlap degree at most

Δ

has an entangled Gibbs state. At any fixed

β<z_Δ

strictly below the threshold, a classical polynomial-time algorithm produces samples from a distribution over product Pauli eigenstates approximating the Gibbs state in trace distance.