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

Mario Berta; Richard Meister; Roberto Bondesan; \v{S}t\v{e}p\'an \v{S}m\'id

arxiv: 2510.04954 · v2 · submitted 2025-10-06 · 🪐 quant-ph · math-ph· math.MP

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

\v{S}t\v{e}p\'an \v{S}m\'id , Richard Meister , Mario Berta , Roberto Bondesan This is my paper

Pith reviewed 2026-05-18 09:57 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP

keywords rapid mixingLindbladiansGibbs statesFermi-Hubbard modelquantum algorithmsdissipative dynamicsoscillator norms

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The pith

Algorithmic Lindbladians prepare Gibbs states with polylogarithmic mixing times for non-interacting and weakly interacting quantum systems.

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

The paper proves that algorithmic Lindbladians for preparing quantum Gibbs states converge in time that scales only polylogarithmically with system size. It first establishes this for non-interacting spin systems, free fermions, and free bosons. The results remain stable under small perturbations, covering weakly interacting qudits and perturbed non-hopping fermions. The same techniques also yield rapid mixing for the strongly interacting Fermi-Hubbard model in explicit parameter regimes. These are the first efficient mixing bounds for non-commuting qudit models and bosonic systems at arbitrary temperatures, improving on prior spectral-gap approaches by providing exponentially faster convergence and explicit interaction thresholds.

Core claim

We prove that algorithmic Lindbladians for Gibbs state preparation exhibit rapid mixing, i.e., convergence in time poly-logarithmic in the system size, for non-interacting spin systems, free fermions, free bosons, and these results are stable under perturbations covering weakly interacting qudits and perturbed non-hopping fermions; we also prove rapid mixing for the strongly-interacting regime of the Fermi-Hubbard model with explicit parameter regimes.

What carries the argument

Adapted oscillator norm techniques applied to detailed-balance Lindbladians that drive the system toward the target Gibbs state.

Load-bearing premise

The Lindbladians satisfy detailed balance and the interaction strength remains below the explicit thresholds derived in the analysis.

What would settle it

A numerical simulation or experiment showing that the mixing time scales polynomially or worse with system size for a non-interacting or weakly interacting model inside the stated parameter range would falsify the rapid mixing claim.

read the original abstract

Dissipative quantum algorithms for state preparation in many-body systems are increasingly recognised as promising candidates for achieving large quantum advantages in application-relevant tasks. Recent advances in algorithmic, detailed-balance Lindbladians enable the efficient simulation of open-system dynamics converging towards desired target states. However, the overall complexity of such schemes is governed by system-size dependent mixing times. In this work, we analyse algorithmic Lindbladians for Gibbs state preparation and prove that they exhibit rapid mixing, i.e., convergence in time poly-logarithmic in the system size. We first establish this for non-interacting spin systems, free fermions, and free bosons, and then show that these rapid mixing results are stable under perturbations, covering weakly interacting qudits and perturbed non-hopping fermions. Further, we adapt the techniques from separable qudits to the fermionic setting and prove rapid mixing of the strongly-interacting regime of the Fermi-Hubbard model, for which we also explicitly evaluate the guaranteed parameter regimes. Our results constitute the first efficient mixing bounds for non-commuting qudit models and bosonic systems at arbitrary temperatures. Compared to prior spectral-gap-based results for fermions, we achieve exponentially faster mixing, further featuring explicit constants on the maximal allowed interaction strength. This not only improves the overall polynomial runtime for quantum Gibbs state preparation, but also enhances robustness against noise. Our analysis relies on oscillator norm techniques from mathematical physics, where we introduce tailored variants adapted to specific Lindbladians $\unicode{x2014}$ an innovation that we expect to significantly broaden the scope of these methods.

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 the first rapid mixing bounds for non-commuting qudit and bosonic Gibbs samplers, plus explicit constants and faster rates for fermions.

read the letter

The main advance is proving rapid mixing for algorithmic Lindbladians that prepare Gibbs states, first in exactly solvable non-interacting cases and then under small perturbations. They handle spins, free fermions, and free bosons with tailored oscillator-norm estimates that deliver poly-log mixing times in system size. From there they show stability under weak interactions for qudits and perturbed fermions, and they adapt the same tools to the Fermi-Hubbard model in an explicit parameter window. Explicit constants on the allowed interaction strength are included, which is a clear improvement over earlier spectral-gap work that lacked them. The strategy is internally consistent once the Lindbladians satisfy detailed balance, and the oscillator-norm adaptation looks like a practical technical step that could extend to other open-system problems. The claimed exponential speedup in mixing time for fermions and the noise-robustness payoff follow directly from the bounds. The main soft spot is that the perturbation-stability arguments and the precise error control in the oscillator norms need to be checked carefully in the full derivations; if those steps have any looseness in the constants, the practical regimes could shrink. The Fermi-Hubbard application is stated with concrete thresholds, but those thresholds may turn out narrow once the full analysis is examined. This work is aimed at people building dissipative quantum algorithms for thermal-state preparation and at researchers who care about mixing times in many-body open systems. Readers who need concrete runtime and robustness guarantees will find usable results here. The paper is coherent on its own terms and addresses a genuine bottleneck, so it deserves a serious referee.

Referee Report

3 major / 2 minor

Summary. The paper proves that algorithmic Lindbladians satisfying detailed balance for Gibbs state preparation exhibit rapid mixing (convergence in poly-logarithmic time in system size) for non-interacting spin systems, free fermions, and free bosons. These results are shown to be stable under small perturbations, extending to weakly interacting qudits and perturbed non-hopping fermions. The techniques are further adapted to the fermionic setting to establish rapid mixing for the strongly interacting regime of the Fermi-Hubbard model within explicitly derived parameter regimes. The analysis relies on tailored variants of oscillator norm techniques, yielding the first efficient mixing bounds for non-commuting qudit models and bosonic systems at arbitrary temperatures, with exponentially faster mixing than prior spectral-gap results for fermions and explicit constants on maximal interaction strength.

Significance. If the central claims hold with the stated explicit constants and error bounds, the work substantially advances the complexity analysis of dissipative quantum algorithms for state preparation. It improves polynomial runtimes for quantum Gibbs sampling, increases robustness to noise, and broadens the applicability of oscillator-norm methods from mathematical physics to quantum many-body systems. The explicit parameter regimes for the Fermi-Hubbard model and the perturbation-stability results are particularly valuable for guiding practical implementations.

major comments (3)

[§4] §4 (perturbation stability for weakly interacting qudits): The stability argument relies on bounding the perturbation in the oscillator norm, but the explicit threshold on interaction strength appears derived from a first-order estimate; it is unclear whether higher-order terms in the expansion remain controlled for the claimed poly-log mixing time when the system size grows.
[§6] §6 (Fermi-Hubbard model): The explicit parameter regimes guaranteeing rapid mixing are stated, yet the derivation of the interaction-strength threshold (likely involving the adapted fermionic oscillator norm) is presented without a self-contained error-bound calculation that would allow independent verification of the poly-log scaling; post-hoc tuning of these regimes raises the question of whether the bound is tight or conservative.
[§3] Abstract and §3 (non-interacting cases): While the rapid-mixing claims for free fermions and bosons are established via tailored oscillator norms, the manuscript does not display the full derivation of the poly-logarithmic mixing time constant or the dependence on temperature; without these explicit constants visible, it is difficult to confirm that the result is parameter-free in the claimed sense.

minor comments (2)

[§2] Notation for the adapted oscillator norms is introduced in §2 but reused with slight variations in later sections; a consolidated definition table would improve readability.
The comparison to prior spectral-gap results for fermions would benefit from a brief table summarizing the exponential improvement in mixing time as a function of system size.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments, which will help us improve the clarity and completeness of the proofs. We address each major comment below.

read point-by-point responses

Referee: [§4] §4 (perturbation stability for weakly interacting qudits): The stability argument relies on bounding the perturbation in the oscillator norm, but the explicit threshold on interaction strength appears derived from a first-order estimate; it is unclear whether higher-order terms in the expansion remain controlled for the claimed poly-log mixing time when the system size grows.

Authors: We thank the referee for this observation. In §4 the threshold is obtained by ensuring that the perturbation remains small in the oscillator norm relative to the unperturbed gap. Because the mixing time is only poly-logarithmic in system size, the contribution of higher-order terms in the perturbative expansion is controlled by the same smallness parameter and does not accumulate to violate the rapid-mixing bound. To make this control fully explicit we will add a short lemma in the revised §4 that bounds the remainder of the Dyson series uniformly in system size. revision: yes
Referee: [§6] §6 (Fermi-Hubbard model): The explicit parameter regimes guaranteeing rapid mixing are stated, yet the derivation of the interaction-strength threshold (likely involving the adapted fermionic oscillator norm) is presented without a self-contained error-bound calculation that would allow independent verification of the poly-log scaling; post-hoc tuning of these regimes raises the question of whether the bound is tight or conservative.

Authors: We agree that a self-contained derivation would improve verifiability. The thresholds in §6 follow directly from the requirements of the adapted fermionic oscillator norm (detailed in the appendix). In the revision we will move the complete error-bound calculation into the main text of §6, presenting every step of the norm estimate and the resulting conservative bound on the interaction strength. We will also state explicitly that the regimes are chosen to guarantee the poly-log scaling rather than to optimize the constant. revision: yes
Referee: [§3] Abstract and §3 (non-interacting cases): While the rapid-mixing claims for free fermions and bosons are established via tailored oscillator norms, the manuscript does not display the full derivation of the poly-logarithmic mixing time constant or the dependence on temperature; without these explicit constants visible, it is difficult to confirm that the result is parameter-free in the claimed sense.

Authors: The full derivations, including the explicit poly-logarithmic constants and their temperature dependence, appear in the appendices. To address the referee’s request we will add a concise summary of these derivations, with the leading constants displayed, to the end of §3. This will make the parameter-free character (with respect to system size) and the temperature scaling immediately visible in the main text. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via explicit proofs

full rationale

The paper derives rapid mixing bounds by first proving them for exactly solvable non-interacting cases (spins, free fermions, free bosons) using adapted oscillator-norm estimates, then establishing stability under small perturbations with explicit thresholds, and finally extending the same machinery to the Fermi-Hubbard model in stated regimes. All steps are presented as mathematical theorems resting on the choice of detailed-balance Lindbladians and bounded interaction strengths; no equations reduce results to self-referential fits, no load-bearing self-citations close the central argument, and the oscillator-norm technique is adapted rather than smuggled via unverified prior ansatz. The analysis is internally consistent against the stated assumptions without circular reduction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Central claim rests on detailed-balance Lindbladians and bounded interaction strength; no new entities postulated.

free parameters (1)

maximal interaction strength
Explicit thresholds on interaction strength for which rapid mixing holds, evaluated for Fermi-Hubbard model.

axioms (1)

domain assumption Lindbladians satisfy detailed balance and converge to the target Gibbs state
Invoked to guarantee the steady state is the desired Gibbs distribution.

pith-pipeline@v0.9.0 · 5831 in / 1206 out tokens · 33295 ms · 2026-05-18T09:57:26.168917+00:00 · methodology

discussion (0)

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Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
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Our analysis relies on oscillator norm techniques from mathematical physics, where we introduce tailored variants adapted to specific Lindbladians
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Definition II.4 … oscillator norm … δ_i(O) = O − ½ Tr_i(O)

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