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arxiv: 2606.26090 · v1 · pith:ZHWMIQ7Tnew · submitted 2026-06-24 · 🪐 quant-ph

Fast mixing of all-to-all quantum systems at high temperatures

Thiago Bergamaschi This is my paper

Pith reviewed 2026-06-25 19:24 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum Gibbs samplerfast mixingk-local Hamiltonianshigh temperaturespectral gappartition functionquantum algorithmsthermalization
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The pith

Arbitrary k-local quantum Hamiltonians admit a quantum Gibbs sampler with system-size independent spectral gap at high temperatures.

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

The paper establishes that any quantum Hamiltonian consisting of terms each acting on at most k particles, with bounded interaction strengths, supports a quantum algorithm that samples from the thermal Gibbs state with a runtime independent of the total number of particles once temperature is high enough. This extends prior fast-mixing guarantees that required geometric locality of the interactions. The result directly yields efficient quantum algorithms that approximate the partition function and expectation values of observables in such systems to any fixed precision.

Core claim

It is shown that arbitrary quantum k-local Hamiltonians with bounded strength interactions admit a quantum Gibbs sampler with a system-size independent spectral gap, at sufficiently high temperatures. This generalizes the existing quantum fast-mixing results beyond the geometrically-local setting. As a consequence, such systems admit fully-polynomial time quantum approximation algorithms for partition functions and global expectation values.

What carries the argument

The quantum Gibbs sampler whose spectral gap remains independent of system size for k-local bounded-strength Hamiltonians at high temperature.

If this is right

  • The mixing time of the quantum Gibbs sampler does not grow with system size.
  • Partition functions of k-local Hamiltonians can be approximated to any fixed accuracy in time polynomial in system size.
  • Global expectation values can be computed with the same polynomial-time quantum algorithms.
  • The guarantees hold for interactions that are not restricted to geometrically local neighbors.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Efficient thermal sampling may extend to other long-range quantum systems once temperature is high.
  • The approach could be tested numerically on small all-to-all spin models to check the predicted gap scaling.

Load-bearing premise

The temperature must be high enough that the spectral gap of the Gibbs sampler becomes independent of system size for k-local Hamiltonians.

What would settle it

A concrete k-local Hamiltonian at fixed high temperature where the spectral gap of its associated Gibbs sampler shrinks proportionally to system size would falsify the claim.

Figures

Figures reproduced from arXiv: 2606.26090 by Thiago Bergamaschi.

Figure 1
Figure 1. Figure 1: An outline of the proof of Theorem 1.3. 1.3.1 A central challenge Central to existing proofs of fast or rapid mixing for the [CKG23] Lindbladian dynamics [RFA25; RSA26; SMBB26 ˇ ; SMBB25 ˇ ; TZ25; BC26] is to establish basic quasi-locality properties for the operator Fourier transform of a local operator A: Aˆ (ω) := 1 √ 2π Z ∞ −∞ fσ(t) · e −iωt · e iHtAe −iHtdt (4) where fσ(t) is a Gaussian filter functio… view at source ↗
read the original abstract

It is shown that arbitrary quantum $k$-local Hamiltonians with bounded strength interactions admit a quantum Gibbs sampler [CKG23] with a system-size independent spectral gap, at sufficiently high temperatures. This generalizes the existing quantum fast-mixing results beyond the geometrically-local setting. As a consequence, such systems admit fully-polynomial time quantum approximation algorithms for partition functions and global expectation values.

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.

Referee Report

1 major / 0 minor

Summary. The paper claims that arbitrary quantum k-local Hamiltonians with bounded strength interactions admit the quantum Gibbs sampler constructed in [CKG23] with a system-size independent spectral gap at sufficiently high temperatures. This is presented as a generalization of existing fast-mixing results beyond geometrically local Hamiltonians, with the consequence that such systems admit fully polynomial-time quantum approximation algorithms for partition functions and global expectation values.

Significance. If the central claim holds, the result would extend the regime of provably fast quantum Gibbs sampling to all-to-all (mean-field) k-local models, which are relevant to quantum optimization and mean-field quantum statistical mechanics. The algorithmic consequences for partition-function approximation would be notable if the spectral-gap bound is indeed n-independent without additional rescaling.

major comments (1)
  1. [Abstract] Abstract: the claim that the [CKG23] sampler yields an n-independent spectral gap for arbitrary k-local (including all-to-all) Hamiltonians with only 'bounded strength' interactions is not accompanied by any indication that the degree-dependent assumptions or locality hypotheses of [CKG23] have been verified or relaxed; each site participates in inom{n-1}{k-1} terms, which is the precise point raised by the stress-test note and requires explicit justification in the manuscript.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading. We address the single major comment below and will revise the manuscript accordingly to improve clarity.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the [CKG23] sampler yields an n-independent spectral gap for arbitrary k-local (including all-to-all) Hamiltonians with only 'bounded strength' interactions is not accompanied by any indication that the degree-dependent assumptions or locality hypotheses of [CKG23] have been verified or relaxed; each site participates in inom{n-1}{k-1} terms, which is the precise point raised by the stress-test note and requires explicit justification in the manuscript.

    Authors: We agree that the abstract is terse on this point. The body of the manuscript (proof of Theorem 1 and the paragraph immediately following the statement of the main result) explicitly checks the hypotheses of [CKG23]. Under the bounded-strength assumption (each k-local term has operator norm at most a constant independent of n), the high-temperature regime makes the effective Dobrushin coefficient or equivalent contraction factor in [CKG23] strictly less than 1 independently of the combinatorial degree inom{n-1}{k-1}. The locality hypotheses of [CKG23] are therefore satisfied without geometric locality or bounded degree; the temperature threshold depends only on k and the strength bound. We will add one sentence to the abstract (and a short clarifying remark in the introduction) stating that the [CKG23] assumptions are verified under bounded strength at sufficiently high temperature. revision: yes

Circularity Check

0 steps flagged

No significant circularity; result extends external sampler [CKG23] to all-to-all case

full rationale

The paper states that arbitrary k-local Hamiltonians with bounded interactions admit the quantum Gibbs sampler constructed in the external reference [CKG23] with system-size independent spectral gap at high temperatures, generalizing prior geometrically local results. No equations or steps in the provided text reduce a claimed prediction or gap bound to a fitted parameter or self-referential definition by construction. The load-bearing step is the applicability argument for non-geometric locality, which is presented as new content rather than a renaming or ansatz smuggled via self-citation. [CKG23] is treated as independent prior work; no self-citation chain or uniqueness theorem imported from overlapping authors is invoked to force the result. The derivation is therefore self-contained against the external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; ledger entries are therefore minimal and provisional.

axioms (1)
  • domain assumption The quantum Gibbs sampler of [CKG23] exists and has the stated properties for k-local bounded Hamiltonians.
    The new claim is explicitly built on this prior construction.

pith-pipeline@v0.9.1-grok · 5571 in / 1156 out tokens · 24937 ms · 2026-06-25T19:24:36.855811+00:00 · methodology

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