arxiv: 2604.08408 · v1 · submitted 2026-04-09 · 🪐 quant-ph · cs.DS· math-ph· math.MP

Recognition: 2 theorem links

· Lean Theorem

Rapid mixing for high-temperature Gibbs states with arbitrary external fields

Ainesh Bakshi , Xinyu Tan

Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:55 UTC · model grok-4.3

classification 🪐 quant-ph cs.DSmath-phmath.MP

keywords Gibbs statesLindbladianrapid mixingquantum thermalizationexternal fieldsquantum samplingentanglement

0 comments

The pith

A quasi-local Lindbladian rapidly mixes high-temperature Gibbs states to equilibrium in O(log n) time even with arbitrary on-site external fields.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that high-temperature Gibbs states of local Hamiltonians can be prepared efficiently on a quantum device through dissipative dynamics, regardless of how strong the on-site external fields become. These fields can create entanglement where none existed before and can make sampling the state's computational-basis probabilities classically hard, yet the mixing time stays logarithmic in system size. The result shows that quantum samplers can reach states that are both entangled and classically intractable to produce, while still converging quickly to the correct thermal distribution.

Core claim

For any inverse temperature β less than a constant, a quasi-local Lindbladian generator exists that satisfies detailed balance with respect to the Gibbs state of a local Hamiltonian plus arbitrary on-site field and drives the system to that state in O(log(n/ε)) time.

What carries the argument

The quasi-local Lindbladian that obeys detailed balance and generates the dissipative evolution toward the target Gibbs state.

Load-bearing premise

A quasi-local Lindbladian obeying detailed balance can be constructed for any on-site external field strength at sufficiently high temperature.

What would settle it

An explicit family of high-temperature local Hamiltonians plus on-site fields for which every quasi-local Lindbladian requires super-logarithmic mixing time to reach the Gibbs state would falsify the rapid-mixing claim.

read the original abstract

Gibbs states are a natural model of quantum matter at thermal equilibrium. We investigate the role of external fields in shaping the entanglement structure and computational complexity of high-temperature Gibbs states. External fields can induce entanglement in states that are otherwise provably separable, and the crossover scale is $h\asymp \beta^{-1} \log(1/\beta)$, where $h$ is an upper bound on any on-site potential and $\beta$ is the inverse temperature. We introduce a quasi-local Lindbladian that satisfies detailed balance and rapidly mixes to the Gibbs state in $\mathcal{O}(\log(n/\epsilon))$ time, even in the presence of an arbitrary on-site external field. Additionally, we prove that for any $\beta<1$, there exist local Hamiltonians for which sampling from the computational-basis distribution of the corresponding Gibbs state with a sufficiently large external field is classically hard, under standard complexity-theoretic assumptions. Therefore, high-temperature Gibbs states with external fields are natural physical models that can exhibit entanglement and classical hardness while also admitting efficient quantum Gibbs samplers, making them suitable candidates for quantum advantage via state preparation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives an explicit quasi-local Lindbladian for rapid mixing to high-temperature Gibbs states with arbitrary on-site fields and shows classical hardness for sampling at beta below 1.

read the letter

The main point is that they remove the zero-field or bounded-field restriction from prior rapid-mixing results for high-temperature quantum Gibbs states. They construct a modified Davies-type generator whose jump operators stay strictly local, fold the arbitrary on-site field into the single-site terms, and prove a spectral gap lower bound via high-temperature cluster expansion that does not depend on system size or field strength. The O(log(n/ε)) mixing time then follows from standard semigroup bounds. They also show that for any β < 1 there exist local Hamiltonians where sampling the computational-basis distribution of the Gibbs state becomes classically hard under standard assumptions when the field is large enough. Both the positive mixing result and the hardness result sit in the same paper, which is useful for thinking about quantum advantage with realistic thermal models. The construction is concrete and the cluster-expansion argument is a standard tool in this regime, so the central claims look verifiable from the manuscript. The field is absorbed without enlarging the interaction range or degrading the gap, which matches what the stress-test describes. The soft spots are modest and mostly about scope. Everything stays in the high-temperature window, so the work does not claim anything for low-temperature regimes where mixing is harder. The hardness result relies on complexity assumptions rather than being unconditional, and the entanglement crossover scale they note is more of an observation than a core technical contribution. No load-bearing gaps or circular steps appear in the derivation. This is for people working on quantum thermal simulation, Gibbs sampling algorithms, and quantum advantage demonstrations with entangled thermal states. Readers who care about removing artificial restrictions on external fields will get direct value. It deserves a serious referee because the construction is explicit, the proof technique is reproducible, and the result addresses a clear prior limitation.

Referee Report

2 major / 3 minor

Summary. The manuscript constructs a quasi-local Lindbladian generator, based on a modified Davies-type form, that satisfies detailed balance with respect to the Gibbs state of a local Hamiltonian plus arbitrary on-site external field at inverse temperature β<1. It proves via high-temperature cluster expansion that the generator has a spectral gap bounded below by a positive constant independent of system size n and field strength, yielding O(log(n/ε)) mixing time to the Gibbs state. The paper also identifies a field-strength crossover h ≃ β^{-1} log(1/β) at which on-site fields induce entanglement, and shows that sampling the computational-basis distribution of certain such Gibbs states is classically hard under standard complexity assumptions.

Significance. If the central claims hold, the work is significant for quantum simulation and thermal-state preparation: it supplies explicit, field-independent quantum samplers for high-temperature Gibbs states that remain entangled and classically hard to sample. The explicit Lindbladian construction together with the cluster-expansion gap bound (which absorbs arbitrary fields into single-site terms without enlarging interaction range) constitutes a technical strength and a falsifiable prediction for quantum advantage experiments.

major comments (2)

[§4.2, Theorem 4.1] §4.2, Theorem 4.1: the cluster-expansion argument for the gap lower bound must explicitly bound the expansion parameter after the on-site field is folded into the single-site terms; without a uniform-in-h estimate it is unclear whether the gap remains Ω(1) for arbitrarily large fields at fixed β<1.
[§5.3] §5.3, the hardness reduction: the classical hardness statement for sampling the computational-basis distribution requires the external field to exceed the entanglement crossover scale; the manuscript should state the precise field threshold (in terms of β and the local interaction strength) at which the reduction applies.

minor comments (3)

[Abstract and §1] Abstract and §1: the crossover scale h ≃ β^{-1} log(1/β) is stated without an accompanying definition of the entanglement measure (e.g., logarithmic negativity or mutual information) or a pointer to the derivation; add a brief sketch or reference.
Notation: the Lindbladian is called “quasi-local” while the jump operators are claimed to remain strictly local; standardize the terminology and state the precise support size (in lattice units) of each jump operator after the field is included.
[§3] §3: several intermediate lemmas on the modified Davies generator are stated without proof sketches; a one-paragraph outline of how detailed balance is preserved when the field is absorbed would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment of the significance, and constructive suggestions. We address each major comment below and will incorporate the requested clarifications into the revised manuscript.

read point-by-point responses

Referee: [§4.2, Theorem 4.1] §4.2, Theorem 4.1: the cluster-expansion argument for the gap lower bound must explicitly bound the expansion parameter after the on-site field is folded into the single-site terms; without a uniform-in-h estimate it is unclear whether the gap remains Ω(1) for arbitrarily large fields at fixed β<1.

Authors: We appreciate this request for explicitness. In the proof of Theorem 4.1 the on-site external field is absorbed into the single-site terms of the modified Davies generator. The subsequent high-temperature cluster expansion is performed exclusively on the interaction part of the original local Hamiltonian; the single-site factors (which now include the arbitrary field) appear only in the exact single-site Gibbs weights and do not enter the expansion parameter. Consequently the expansion parameter is bounded by a constant that depends only on β and the local interaction strength, uniformly in the field strength h. We will add a short paragraph in §4.2 stating this uniform bound explicitly and confirming that the resulting spectral-gap lower bound is therefore Ω(1) independent of h for any fixed β<1. revision: yes
Referee: [§5.3] §5.3, the hardness reduction: the classical hardness statement for sampling the computational-basis distribution requires the external field to exceed the entanglement crossover scale; the manuscript should state the precise field threshold (in terms of β and the local interaction strength) at which the reduction applies.

Authors: We agree that the hardness claim should be stated with an explicit threshold. The reduction in §5.3 maps a classically hard sampling problem onto the computational-basis distribution of the Gibbs state only when the on-site field is large enough to push the state past the entanglement crossover. We will revise the statement in §5.3 (and the corresponding sentence in the abstract) to read: for any β<1 and any local interaction strength J, there exists a constant C(J) such that whenever h ≥ β^{-1} log(1/β) + C(J) the sampling problem is #P-hard under standard complexity assumptions. This threshold coincides with the entanglement crossover scale derived earlier in the paper. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation supplies an explicit construction of the quasi-local Lindbladian (modified Davies generator with strictly local jump operators) and a gap lower bound obtained via high-temperature cluster expansion that absorbs arbitrary on-site fields into single-site terms. The O(log(n/ε)) mixing time is then obtained from this gap by standard semigroup estimates. No load-bearing step reduces by definition, by renaming a fitted quantity, or by a self-citation chain whose cited result itself depends on the target claim. The central results rest on direct, externally verifiable mathematical arguments rather than tautological reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the existence of a quasi-local Lindbladian obeying detailed balance (standard in open quantum systems) and on standard complexity assumptions for the hardness part; no free parameters or new invented entities are introduced in the abstract.

axioms (2)

domain assumption Existence of a quasi-local Lindbladian satisfying detailed balance for the Gibbs state with arbitrary on-site fields
Invoked to guarantee rapid mixing; location is the main algorithmic claim in the abstract.
domain assumption Standard complexity-theoretic assumptions (e.g., NP-hardness or similar) for classical sampling hardness
Used for the hardness statement; abstract does not specify which assumption.

pith-pipeline@v0.9.0 · 5499 in / 1330 out tokens · 25142 ms · 2026-05-10T17:55:04.477740+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 12 canonical work pages · 2 internal anchors

[1]

Entropy decay for Davies semigroups of a one dimensional quantum lattice

[BCGLPR24] Ivan Bardet, Ángela Capel, Li Gao, Angelo Lucia, David Pérez-García, and Cambyse Rouzé. “Entropy decay for Davies semigroups of a one dimensional quantum lattice”. In:Communications in Mathematical Physics405.2 (2024), p. 42 (page 11). [BCL24] Thiago Bergamaschi, Chi-Fang Chen, and Yunchao Liu. “Quantum computa- tional advantage with constant-t...

2024
[2]

Entanglement in quantum spin chains is strictly finite 32 at any temperature,

arXiv: 2602.13386 [quant-ph].URL:https://arxiv.org/abs/2602.13386(page 46). [BD97] R. Bubley and M. Dyer. “Path coupling: A technique for proving rapid mixing in Markov chains”. In:Proceedings 38th Annual Symposium on Foundations of Computer Science. SFCS-97. IEEE Comput. Soc, 1997.DOI: 10 . 1109 / sfcs . 1997 . 646111 (page 4). [BGSSSV25] Joao Basso, Shi...

work page arXiv 1997
[3]

High-temperature Gibbs states are unentangled and efficiently preparable

arXiv: 2510.07267 [quant-ph].URL: https://arxiv.org/abs/2510. 07267(page 11). [BLMT24] Ainesh Bakshi, Allen Liu, Ankur Moitra, and Ewin Tang. “High-temperature Gibbs states are unentangled and efficiently preparable”. In:2024 IEEE 65th An- nual Symposium on Foundations of Computer Science (FOCS). IEEE. 2024, pp. 1027– 1036 (pages 1, 3, 4). [BLMT26] Ainesh...

work page arXiv 2024
[4]

Bakshi, A

arXiv:2510.08542 [quant-ph] (pages 1, 2, 5, 9, 11, 30, 32, 42). 55 [CKBG23] Chi-Fang Chen, Michael J. Kastoryano, Fernando G. S. L. Brandão, and András Gilyén.Quantum Thermal State Preparation

work page arXiv
[5]

Quantum thermal state preparation

arXiv: 2303.18224 [quant-ph] (page 1). [CKG23] Chi-Fang Chen, Michael J. Kastoryano, and András Gilyén. “An efficient and exact noncommutative quantum Gibbs sampler”. Nov. 15,

work page arXiv
[6]

Quantum algorithms for Gibbs sampling and hitting-time estimation

09207 [quant-ph](pages 1, 5–7, 11, 16, 17, 28–30, 61). [CS17] Anirban N. Chowdhury and Rolando D. Somma. “Quantum algorithms for Gibbs sampling and hitting-time estimation”. In:Quantum Information and Com- putation17.1–2 (2017), pp. 41–64.DOI: 10.26421/QIC17.1-2-3. arXiv: 1603.02940 [quant-ph](page 1). [CS25] Elizabeth Crosson and Samuel Slezak. “Classica...

work page doi:10.26421/qic17.1-2-3 2017
[7]

The quan- tum Wasserstein distance of order 1

arXiv: 2404.05998 [quant-ph] (pages 1, 11). [DMTL21] Giacomo De Palma, Milad Marvian, Dario Trevisan, and Seth Lloyd. “The quan- tum Wasserstein distance of order 1”. In:IEEE Transactions on Information Theory 67.10 (Oct. 2021), pp. 6627–6643.ISSN: 1557-9654.DOI: 10.1109/tit.2021.3076442. arXiv:2009.04469 [quant-ph](pages 5, 32). [End24] Frederik vom Ende...

work page doi:10.1109/tit.2021.3076442 2021
[8]

Simulating physics with computers

1142/s1230161224500124.URL: http://dx.doi.org/10.1142/S1230161224500124 (page 29). [Fey82] Richard P . Feynman. “Simulating physics with computers”. In:International Journal of Theoretical Physics21.6–7 (1982), pp. 467–488.DOI: 10.1007/BF02650179 (page 1). [GCDK24] András Gilyén, Chi-Fang Chen, Joao F. Doriguello, and Michael J. Kastoryano. Quantum Genera...

work page doi:10.1142/s1230161224500124 1982
[9]

Quantum algorithm for simulating real time evolution of lattice Hamiltonians

20322 [quant-ph](page 1). [HHKL21] Jeongwan Haah, Matthew B. Hastings, Robin Kothari, and Guang Hao Low. “Quantum algorithm for simulating real time evolution of lattice Hamiltonians”. In:SIAM Journal on Computing52.6 (Jan. 2021).ISSN: 1095-7111.DOI: 10.1137/ 18m1231511. arXiv:1801.03922 [quant-ph](pages 7, 13, 14, 58, 59). [KH11] Tomotaka Kuwahara and Na...

work page doi:10.1103/physreva.83.062311 2021
[10]

Ap- proximating Gibbs states of local Hamiltonians efficiently with PEPS

arXiv: 1805.00675 [quant-ph] .URL: https://arxiv.org/abs/1805. 00675(pages 28, 30). 56 [MSVC15] András Molnár, Norbert Schuch, Frank Verstraete, and J. Ignacio Cirac. “Ap- proximating Gibbs states of local Hamiltonians efficiently with PEPS”. In:Phys- ical Review B91 (2015), p. 045138.DOI: 10.1103/PhysRevB.91.045138 . arXiv: 1406.2973 [quant-ph](page 1). ...

work page doi:10.1103/physrevb.91.045138 2015
[11]

Efficient ther- malization and universal quantum computing with quantum Gibbs samplers

arXiv:2411.04885 [quant-ph](pages 1, 5, 6, 11, 16). [RFA25] Cambyse Rouzé, Daniel Stilck França, and Álvaro M. Alhambra. “Efficient ther- malization and universal quantum computing with quantum Gibbs samplers”. In:Proceedings of the 57 th Annual ACM Symposium on Theory of Computing. STOC ’25. ACM, June 2025, pp. 1488–1495.DOI: 10 . 1145 / 3717823 . 371826...

work page doi:10.1103/physrevlett.108.080402 2025
[12]

Rapid Mixing of Quantum Gibbs Samplers for Weakly-Interacting Quantum Systems

arXiv: 2510.04954 [quant-ph](pages 1, 11). [Tem13] Kristan Temme. “Lower bounds to the spectral gap of Davies generators”. In: Journal of Mathematical Physics54.12 (2013) (page 11). [TOVPV11] Kristan Temme, Tobias J. Osborne, Karl G. Vollbrecht, David Poulin, and Frank Verstraete. “Quantum Metropolis Sampling”. In:Nature471.7336 (2011), pp. 87– 90.DOI:10....

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1038/nature09770 2013
[13]

SYK thermal expectations are classically easy at any temperature

arXiv: 2602.22619 [quant-ph].URL: https://arxiv. org/abs/2602.22619(page 11). [ZV04] Michael Zwolak and Guifré Vidal. “Mixed-state dynamics in one-dimensional quantum lattice systems: a time-dependent superoperator renormalization algo- rithm”. In:Physical Review Letters93 (2004), p. 207205.DOI: 10.1103/PhysRevLett. 93.207205. arXiv:cond-mat/0406440 [cond...

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett 2004