Beyond Lindblad Dynamics: Rigorous Guarantees for Thermal and Ground State Preservation under System Bath Interactions

Ke Wang; Zhiyan Ding

REVIEW 2 major objections 5 minor 6 cited by

System–bath quantum state preparation works even at constant coupling strength, not just in the weak-coupling limit.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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System–bath interaction state preparation is shown to approximately preserve thermal and ground states at constant (or larger) coupling, with mixing time O(Γ^{-2}) and total runtime eO(n^7/ε^2) for several models.

T0 review reviewed 2026-08-03 challenge →

load-bearing objection A serious but unfinished paper: the new Dyson-cancellation technique is plausible, but the key lemma is a formal analytic continuation with no proof of convergence. the 2 major comments →

arxiv 2512.03457 v3 pith:AR7WZLVV submitted 2025-12-03 quant-ph

Beyond Lindblad Dynamics: Rigorous Guarantees for Thermal and Ground State Preservation under System Bath Interactions

Ke Wang , Zhiyan Ding This is my paper

classification quant-ph

keywords quantum thermal state preparationsystem-bath interactionbeyond Lindblad limitconstant couplingfixed point preservationmixing timeKMS detailed balanceDyson series

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

Quantum thermal and ground states can be prepared by repeatedly letting a system interact with a small bath, then discarding the bath. Standard theory only guarantees this in the weak-coupling limit, where the coupling must shrink as the desired accuracy grows, making the procedure slow. This paper proves that even when the coupling strength is held constant, the discrete quantum channel still has a fixed point arbitrarily close to the target state, and that mixing to that state happens in a number of iterations that scales with the inverse square of the coupling strength. If true, the result removes the need for vanishing coupling, substantially improving the efficiency and physical realism of these state-preparation algorithms.

Core claim

The paper establishes that the quantum channel ΦΓ induced by a system–bath interaction with coupling strength Γ = Θ(1) (independent of accuracy ε) approximately fixes the target thermal state σβ and, under a spectral gap assumption, the target ground state. Concretely, the fixed-point error bound is ∥ρfix(ΦΓ) − σβ∥1 = O(βΓ²/(σ²) τmix + exponentially small truncation terms), which can be made arbitrarily small by increasing the filter width σ. It further proves a mixing-time bound tmix = eO(λgap⁻¹ log(1/ε)) for the rescaled mixing time, leading to total simulation time Ttotal = eO(n⁷/ε²) for several Hamiltonian families (high-temperature spin, weakly interacting fermions/spins at all temperat

What carries the argument

The central object is the discrete quantum channel ΦΓ defined by the system–bath interaction, expanded as a Dyson series. The key mechanism is a time-domain framework (Lemma D.3 and Lemma D.4) that uses the Baker–Campbell–Hausdorff expansion to express σβ eG† σβ⁻¹ as an imaginary-time shift in the Gaussian filter integrals. This shift produces cancellations between even and odd higher-order terms, allowing an all-orders proof of state preservation. For mixing, the channel is decomposed into a second-order KMS detailed-balance Lindbladian with a positive spectral gap plus a higher-order remainder; a perturbation theory shows the remainder (bounded by O(α²σ)) does not close the gap when α²σ is

Load-bearing premise

The central proof (Lemma D.4) assumes that the Baker–Campbell–Hausdorff expansion can be resummed into an imaginary-time shift of the integration variable inside the Gaussian filter, with boundary terms vanishing and the infinite series converging—an analytic continuation that is not rigorously justified.

What would settle it

A direct numerical check of the fixed-point error for a specific Hamiltonian (e.g., TFIM with L=8) at constant Γ = Θ(1) and increasing σ: if the trace distance ∥ρfix(ΦΓ) − σβ∥1 fails to decrease below a small threshold for σ growing as predicted, or if the spectral gap of ΦΓ closes as σ grows, the central claim is falsified.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

Constant coupling Γ = Θ(1) is sufficient for accurate thermal and ground state preparation, eliminating the need for polynomially small couplings and thus reducing iteration counts by a factor of σ/ε.
The mixing time of the system–bath discrete channel scales as the inverse square of the coupling strength, matching the scaling found in Lindblad dynamics but now outside the weak-coupling regime.
The end-to-end total Hamiltonian simulation time for thermal state preparation improves to Ttotal = eO(n⁷/ε²), saving a factor of n³/ε² compared to prior Lindblad-limit analyses.
The general perturbation framework can be combined with existing spectral-gap bounds for KMS detailed-balance Lindbladians, yielding mixing-time guarantees for a broad class of physical models (spin, fermionic, 1D chains).
Numerical experiments on transverse-field Ising, Hubbard, and ANNNI models confirm the predicted α⁻² mixing scaling and show robustness even in a strong-coupling regime beyond the theoretical scope.

Where Pith is reading between the lines

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

If the fixed-point result extends to the strong-coupling regime Γ = Θ(σ) observed numerically (where the spectral gap still scales as α² up to α/√σ ≈ 0.5), the total simulation time could be further reduced, possibly to near-linear in n.
The time-domain cancellation mechanism might apply to other discrete open-system channels beyond the specific system–bath setting, such as repeated-interaction or collision models, where higher-order Dyson terms are currently ignored.
The n⁷/ε² bound is likely not tight; the analysis could be sharpened by exploiting the quasi-local structure of the higher-order terms (M2) to relax the α²σ = O(λgap) condition to α²σ = O(1), which would remove extra factors of n.
A testable prediction: for a system where KMS Lindbladians mix rapidly, the discrete channel's rescaled mixing time should remain bounded as the filter width σ increases at constant coupling, which can be directly checked by computing the spectral gap of ΦΓ for larger system sizes.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

A serious but unfinished paper: the new Dyson-cancellation technique is plausible, but the key lemma is a formal analytic continuation with no proof of convergence.

read the letter

Ke Wang and Ding have written a paper that is worth reading but not yet fully credible. The main result — that the discrete system-bath channel approximately preserves the thermal or ground state when Γ is independent of the accuracy ε — would be a real advance over the weak-coupling analyses in [17,19,22,46]. The time-domain detailed-balance transformation and the all-orders Dyson cancellation are genuinely new and the numerical experiments are consistent. The paper is clearly the work of people who know the area.

The soft spot is exactly the one flagged by the stress test: Lemma D.4. The proof moves a formal BCH expansion into a Taylor shift inside a Gaussian integral without justifying the interchange of the infinite series and the integral, and without controlling boundary terms. In the finite-dimensional setting the ingredients for a rigorous argument are present — the Gaussian decay dominates the exponential growth of the Heisenberg-picture operators — but the paper does not supply them, and the stated bounds have no ||H|| dependence at all. That is a red flag: any rigorous version of the analytic continuation will almost certainly produce factors like exp(β||H||) in the error, or require σ relative to ||H||. So the theorem statements as written are not supported.

There is also a mismatch between the advertising and the actual end-to-end result. The abstract and intro emphasize Γ=Θ(1), but Corollary F.3 chooses α = Θ(√(ε/n³)) and σ = eΘ(n²/ε), which makes Γ = α√σ = Θ(n^{-1/2}). That is still ε-independent, and it explains the n⁷/ε² total runtime, but it is not the constant coupling regime the introduction promises. The fixed-point theorem itself does allow Γ=Θ(1) for fixed n, and that part may be correct, but the complexity improvement is really about removing ε-dependence, not about operating at constant coupling.

The mixing-time analysis also leans on [22, Theorem 8] and [46] in places that make the circularity burden real: Theorem D.1's second half is derived by combining the new fixed-point estimate with a mixing-time theorem from [22], and the mixing-time proof in Appendix F uses Lemma D.4 again. So the two halves are interconnected, and the load-bearing lemma is not established.

Who is this for? Researchers working on dissipative state preparation who want to know whether the Lindblad limit is fundamental. The paper deserves peer review — an editor should send it out rather than desk-reject. A referee should ask for a rigorous treatment of Lemma D.4 and a correction of the Γ claim. My own verdict is skeptical but conditional: if the analytic continuation can be made rigorous, the paper's main ideas will likely stand.

Referee Report

2 major / 5 minor

Summary. This paper analyzes the discrete quantum channel ΦΓ induced by a system–bath interaction with coupling strength Γ and Gaussian filter fσ. The main claims are: (1) even when Γ=Θ(1), the fixed point of ΦΓ is arbitrarily close to the target thermal state σβ (Theorem D.1) or ground state |ψ0⟩ (Theorem E.1), provided σ is chosen sufficiently large; (2) a perturbative spectral-gap analysis gives a rescaled mixing-time bound tmix = eO(λgap^{-1} log(1/ε)) and an end-to-end total simulation time Ttotal = eO(n^7/ε^2) for several model families (Theorem F.2, Corollary F.3). The proof expands ΦΓ to all orders in the Dyson series, isolates cancellations among even/odd higher-order terms, and introduces a time-domain identity (Lemma D.4) for the detailed-balance transformation σβ^{-1} eG† σβ. Numerical experiments on TFIM, Hubbard, and ANNNI models support the qualitative predictions.

Significance. If the proofs are correct, the result is substantial: it removes the weak-coupling (Lindblad-limit) restriction in system–bath preparation algorithms and improves the known end-to-end complexity compared with [22,46]. The paper's ambition to control all orders of the Dyson expansion rather than truncating at the Lindblad term is appropriate, and the time-domain detailed-balance framework would be a useful tool if made fully rigorous. The numerical evidence is a strength and goes beyond the proven regime. However, the central fixed-point preservation lemma (Lemma D.4) is asserted through an unproved analytic-continuation step, and the statement of Theorem D.1 is formally self-referential. These issues are load-bearing for the main claims and currently prevent acceptance.

major comments (2)

[Appendix D.2, Eqs. (D31)–(D35), Lemma D.4] The proof of Lemma D.4 expands σβ eG σβ^{-1} via the Baker–Campbell–Hausdorff formula into an infinite commutator series and resums it into the imaginary-time shift sk → sk − iβ. The Hamiltonian H is unbounded, so the commutator series is not known to converge in operator norm; the interchange of the infinite sum with the integral requires a dominated-convergence argument that is not supplied; and the integration-by-parts boundary terms at sk → ±∞ are asserted to vanish without proof. The Gaussian factor formally decays along the shifted contour, but no analytic-continuation or contour-deformation argument is given for the system-operator products AS(∑s_p)···, which are not entire operator-valued functions in the relevant sense for unbounded H. Since Term 1.1, Term 1.2, and the final O(βΓ² log σ/σ²) estimate all rely on Eq. (D14), and since Lemma F.4 in the mixing-time proof uses Lemma D
[Theorem D.1 and Appendix F] The statement of Theorem D.1 chooses σ = eΘ(β tmix,Φ(ε/4)/ε), where tmix,Φ is the rescaled mixing time of the very channel ΦΓ being analyzed. The text says that the second part, Eq. (D2), follows by combining the first part with [22, Theorem 8], but as written the statement is circular: one cannot choose a parameter as a function of the quantity one is trying to bound. The end-to-end complexity result in Theorem III.3 depends on a non-circular version of this argument. The paper should restate Theorem D.1 using an a priori upper bound on tmix,Φ (which Theorem F.2 is intended to supply) and then derive Eq. (D2), rather than placing tmix,Φ in the hypothesis. A brief note explaining the logical ordering of Theorems D.1 and F.2 would resolve this.

minor comments (5)

[Lemma D.2, line after Eq. (D8)] The proof of the T→∞ approximation uses ∥γ∥L1 = 1, but this assumption is not stated in Theorem D.1, which only assumes decay at infinity and γ′, eγ′ ∈ L1. If an L1 bound on γ is needed, it should be stated explicitly; otherwise the constant in Eq. (D5) should carry ∥γ∥L1.
[Eq. (E32)] The text refers to 'Theorem D.2' but the intended reference appears to be Lemma D.2. Please correct the cross-reference.
[Theorem F.1 and Eq. (C6)] Theorem F.1 states γ(ω) = g(ω) with the Gaussian choice in Eq. (F5), while Eq. (C6) defines γ(ω) = [g(ω)+g(−ω)]/(1+e^{βω}). The relation between these two definitions should be clarified, since the numerical and analytical sections seem to use the same symbol γ in different ways.
[Appendix D.2, Eq. (D33)] The sign in the exponent after the change of variables appears to be −i(β/σ)(s1+2s2+⋯+ksk), while the final expression in Eq. (D35) uses the same structure; a short verification of the conjugation step would help the reader.
[Appendix D.3, Eq. (D39)] The splitting into regions |⋯| ≥ δ and |⋯| < δ is not written consistently: the text defines δ = O(1/σ) but the definition of δ is deferred until after the bound is used. Please state the choice of δ before Eq. (D39).

Circularity Check

0 steps flagged

No significant circularity: fixed-point and mixing-time estimates are independently established; the apparent σ-vs-tmix self-reference is closed by Theorem F.2.

full rationale

The central chain is not circular. Theorem D.1's first part bounds ∥ΦΓσβ−σβ∥ directly from the Dyson expansion (Theorem C.2), Lemma D.3, and Lemma D.4, and the RHS (D1) contains no τmix. The conditional second part introduces tmix,Φ only through the standard perturbation transfer quoted from [22, Theorem 8]; this is a relation, not an input. The potential self-reference is removed downstream: Theorem F.2 proves an independent bound tmix,Φ = eO(λ_gap^{-1} log(...)) from the KMS spectral gap λ_gap, requiring only α²σ=O(λ_gap) and σ=eΩ(β/λ_gap), and it does not invoke the fixed-point error it is meant to support. For the model families, λ_gap=Ω(1/n) comes from external results [29,32–35,47] (no author overlap), so the coupling and mixing claims do not reduce to a fit. Self-citations to [22] supply the algorithm and a general CPTP lemma; those theorems are not the target result and are not used as the sole justification of the central bound. The main genuine weakness is Lemma D.4's BCH imaginary-time resummation, which is not rigorously justified for unbounded H, and footnote 53 admits an unproven expectation for fermionic gap bounds; these are correctness/completeness gaps, not circularity. Numerical experiments test the independently stated α^{-2} scaling rather than renamed inputs.

Axiom & Free-Parameter Ledger

6 free parameters · 5 axioms · 0 invented entities

The central proof mixes new derivation with imported prior results. The main imported items are the KMS-Lindbladian approximation (Theorem F.1) and model-specific spectral-gap/monotonicity bounds; both are external to this paper. The proof also assumes formal convergence of the Dyson/BCH/imaginary-time machinery in Lemma D.4, which is the least-supported step. No new physical entities are introduced; the only free choices are algorithmic/proof parameters Γ, α, σ, T, N, η.

free parameters (6)

Γ (coupling strength) = Θ(1) in fixed-point theorems; eΘ(n^{-1/2}) in Corollary F.3
User-specified coupling parameter, not fitted to data. The advertised 'constant coupling' is only true at fixed system size; the end-to-end complexity analysis uses Γ shrinking with n.
α = Γ/√σ (rescaled coupling) = eΘ(√(ε/n³)) in Corollary F.3
Chosen to satisfy the perturbation condition α²σ = O(λgap); this is a parameter introduced by the proof, not a measured constant.
σ (Gaussian filter width) = eΘ(n²/ε) in Corollary F.3; eΘ(β tmix/ε) in Theorem D.1
User-specified filter width controlling fixed-point accuracy and interaction-time cost.
T (interaction time per iteration) = eΘ(σ)
User-specified interval length; chosen so that the T→∞ truncation error e^{-T²/4σ²} is negligible.
N (Dyson truncation order in ground-state proof) = Θ(log(1/(αϵ)))
Proof parameter chosen so the high-order Dyson tail is below α²ϵ/3.
η (spectral-grid spacing for H in ground-state proof) = Θ(σ^{-2}ϵ)
Ad hoc discretization parameter introduced to make the Bohr-frequency set finite; the final bound depends polynomially on η^{-1}.

axioms (5)

domain assumption The Dyson expansion (C10) of ΦΓ converges absolutely for Γ=O(1) and Gaussian f, permitting term-by-term expectation/trace and resummation.
Used throughout Appendix D/E to write ΦΓ as an infinite series; only tail bounds are sketched, not a complete convergence theorem.
ad hoc to paper The Baker–Campbell–Hausdorff expansion σβ eG σβ^{-1} = Σ (-β)^m/m! ad_H^m(eG) can be resummed into an imaginary-time shift s_k → s_k - iβ with no boundary terms.
Eqs. (D31)–(D35) in Lemma D.4; this is the formal core of the fixed-point preservation and is not rigorously justified.
domain assumption The chosen g(ω) in Eq. (F5) makes the effective γ(ω) satisfy normalization, KMS balance, and ||γ'||_{L1}+||eγ'||_{L1}=O(1).
Required by Theorem D.1 and Theorem F.1; the paper states conditions but does not verify them explicitly for the chosen g.
domain assumption The approximation M1 ≈ -i[Hσ,Lamb,·] + Lσ,KMS with error O(σ e^{-T²/4σ²} + β/σ) holds (Theorem F.1).
Imported from [22, Theorem 27] and [46, Appendix B]; load-bearing for the mixing-time and total-complexity results and not proved in this paper.
domain assumption The KMS Lindbladian Lσ,KMS has spectral gap λgap=Ω(1/n) for the listed model families, and the gap is monotone non-decreasing in σ as proved in [46].
Needed to convert the perturbation theorem into the eO(n^7/ε^2) runtime; relies on [29,32–35,47,46] and one bullet misattributes a spin-case bound to an expectation footnote.

reviewed 2026-08-03 · how reviews work

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Cite this review

Pith. "Pith review of Beyond Lindblad Dynamics: Rigorous Guarantees for Thermal and Ground State Preservation under System Bath Interactions." pith.science (2026). https://pith.science/paper/AR7WZLVV

@misc{pith2026251203457,
  author       = {Pith},
  title        = {Pith review of: Beyond Lindblad Dynamics: Rigorous Guarantees for Thermal and Ground State Preservation under System Bath Interactions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AR7WZLVV}},
  note         = {Machine review of arXiv:2512.03457}
}

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read the original abstract

We establish new theoretical results demonstrating the efficiency and robustness of system-bath interaction algorithms for quantum thermal and ground state preparation. We rigorously show that, even when the coupling strength is chosen independently of the desired accuracy, the induced quantum channel admits the target thermal and ground states as approximate fixed points with arbitrarily high precision. This contrasts with prior analyses, which typically rely on the leading order Lindbladian approximation and require the coupling strength to decrease polynomially with the target error tolerance. Our proof introduces new techniques for controlling all orders of the Dyson expansion and for analyzing the associated multidimensional operator Fourier transforms. Building on these new estimation, we demonstrate an improved end-to-end complexity analysis of thermal and ground state preparation for the system-bath interaction algorithms. These bounds substantially improve upon prior results, and numerical simulations further confirm the robustness of the system-bath interaction framework across both weak and strong coupling regimes.

Figures

Figures reproduced from arXiv: 2512.03457 by Ke Wang, Zhiyan Ding.

**Figure 1.** Figure 1: Analysis Flowchart Theorem III.1 (Informal: thermal state). For any inverse temperature β > 0, we can choose the coupling strength Γ = Θ(1) and T = Ω( e σ) such that ∥ρfix(ΦΓ) − σβ∥1 = O Γ 2β σ2 τmix . Theorem III.2 (Informal: ground state). Assume H has a spectral gap ∆. Given precision ϵ > 0, we can choose Γ = Θ(1) , T = Ω( e σ), and σ = Ω(∆ e −1ploylog(τmix/ϵ)) such that ∥ρfix(ΦΓ) − σβ∥1 ≤ ϵ. Here, … view at source ↗

**Figure 2.** Figure 2: (a) Fidelity of thermal state preparation for TFIM with [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: Thermal state preparation for TFIM with L = 4 sites in the regime Γ = Θ(1). In (a)–(c), we use σ = 2, 4, 8, choose the coupling parameter α such that α √ σ = 1.0, 0.5, 0.25, 0.10, 0.05, 0.01 and set the interaction time T = 5σ; the frequency ω is sampled uniformly from the interval [0, 5]. (a)The evolution of infidelity, i.e., 1 − F, versus the iteration steps. (b) The fix point infidelity between the targ… view at source ↗

**Figure 4.** Figure 4: Thermal and groud state preparation for TFIM with 4 sites in the regime Γ = Θ( [PITH_FULL_IMAGE:figures/full_fig_p037_4.png] view at source ↗

**Figure 5.** Figure 5: Thermal state preparation of Hubbard with [PITH_FULL_IMAGE:figures/full_fig_p037_5.png] view at source ↗

**Figure 6.** Figure 6: Ground/Thermal state preparation of TFIM with [PITH_FULL_IMAGE:figures/full_fig_p038_6.png] view at source ↗

**Figure 7.** Figure 7: Ground state preparation of the Hubbard model with [PITH_FULL_IMAGE:figures/full_fig_p038_7.png] view at source ↗

**Figure 8.** Figure 8: Ground state preparation in the regime Γ = Θ( [PITH_FULL_IMAGE:figures/full_fig_p039_8.png] view at source ↗

**Figure 9.** Figure 9: State preparation for larger systems in the strong coupling regime Γ = Θ( [PITH_FULL_IMAGE:figures/full_fig_p039_9.png] view at source ↗

discussion (0)

Forward citations

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Reference graph

Works this paper leans on

27 extracted references · 4 linked inside Pith · cited by 4 Pith papers

[1]

Dyson series expansion of ΦΓ In this section, we introduce the Dyson series expansion of Φ Γ in Eq. (1). We use the subindex {−1, 1} to relabel the system and bath operator as S1 = AS, S −1 = A† S, B 1 = BE, B −1 = B† E. Define the system evolution operator US(t) = exp( −itH). The Dyson series expansion of Φ Γ is summarized in the following theorem: 10 Th...
[2]

We show that it admits the same structural form as eG† k,AS after a β-dependent shift of the integration variable

Proof of Lemma D.4 The main goal of this section is to prove an explicit expression of σ−1 β eG† k,AS σβ. We show that it admits the same structural form as eG† k,AS after a β-dependent shift of the integration variable. In addition, for notation simplicity, we assume AS is hermitian. The extension to the non-hermitian AS is straightforward. We first calc...
[3]

Bounds of T erm 1.1 and T erm 1.2 In this section, we bound Term 1.1 and Term 1.2 separately so that the bound of Term 1 can be obtained by Term1 ≤ X n≥1 (2Γ)2n ((2π)1/4)2n ∥AS∥2n 2nX k=0 even (Term1.1 + Term1.2) . (D36) 19 We notice that Term 1.1 can be upper bounded by the inequality | sin(x)| ≤ |x| as, Term 1.1 = 2 exp (2n − k)β2 4σ2 Z −∞<s1≤s2≤···≤s2n...
[4]

Assuming Γ2 = α2σ = O(1), we have ∥M2,∞∥2↔2 = O α2σ , (F9) ∥σ−1/4 β M2,∞[σ1/4 β · σ1/4 β ]σ−1/4 β ∥2↔2 = O α2σ , (F10) Proof of Lemma F.4

Proof of Theorem F.2 Recall (F2), we define M2,∞ = 1 α2 EAS  X n≥2 Γ2n(−1)n 2nX k=0 (−1)k Z γ(ω)G† 2n−k,AS ,∞(ω)ρGk,AS ,∞(ω)dω   with Gk,AS ,∞(ω) = Z −∞<t1≤···≤tk<∞ AS(t1)A† S(t2) · · ·e−iω Pk p=1(−1)ptp fσ(t1) · · ·fσ(tk)dt1 · · ·dtk We first provide an upper bound for the norm of M2,∞: Lemma F.4. Assuming Γ2 = α2σ = O(1), we have ∥M2,∞∥2↔2 = O α2σ ,...
[16]

We define eGk,H,AS (ω) = eGk,AS (ω) = Z −∞<t1≤···≤tk<∞ AS(t1)A† S(t2) · · ·e−iω Pk p=1(−1)ptp f (t1) · · ·f (tk)dt1 · · ·dtk

Proof of Lemma D.3 Notice that the expression of eGk,AS depends on the system Hamiltonian H and the coupling operator AS. We define eGk,H,AS (ω) = eGk,AS (ω) = Z −∞<t1≤···≤tk<∞ AS(t1)A† S(t2) · · ·e−iω Pk p=1(−1)ptp f (t1) · · ·f (tk)dt1 · · ·dtk . Then, eG† k,H,AS (ω) = Z −∞<t1≤···≤tk<∞ · · ·AS(t2)A† S(t1)eiω Pk p=1(−1)ptp f (t1) · · ·f (tk)dt1 · · ·dtk ...
[19]

|es1| | √ 2n − kes1 − √ ket1| des1det1 Z T (−∞<s2≤···≤s2n−k<∞, −∞<t2≤···≤tk<∞) e− P2n−k p=2 es2 p−Pk p=2 et2 p des2 · · ·des2n−kdet2 · · ·detk + √ 2n − k 1 2σ Z |γ′(ω)|dω Z |√2n−kes1− √ ket1|∈(δ,∞) e−(es2 1+et2
[20]

1 | √ 2n − kes1 − √ ket1| des1det1 Z T (−∞<s2≤···≤s2n−k<∞, −∞<t2≤···≤tk<∞) e− P2n−k p=2 es2 p−Pk p=2 et2 p 2n−kX p=2 es2 p !1/2 des2 · · ·des2n−kdet2 · · ·detk . (D44) Next, we perform the additional change of variable ( x = √ 2n − kes1 − √ ket1 y = √ kes1 + √ 2n − ket1 , (D45) which gives I ≤ √ 2n − k 1 2σ ∥γ′∥L1 1 2n Z |x|∈(δ,∞) e− x2 +y2 2n | √ 2n − kx...
[21]

ΦΓρ = ρ + α2M1[ρ] + α2M2[ρ]

Approximate ΦΓ using KMS-Lindbladian dynamics Following Theorem C.2, we define M1[ρ] = −EAS 2X k=0 (−1)kσ Z γ(ω)G† 2−k,AS (ω)ρGk,AS (ω)dω ! , (F1) M2[ρ] = 1 α2 EAS ∞X n=2 Γ2n(−1)n 2nX k=0 (−1)k Z γ(ω)G† 2n−k,AS (ω)ρGk,AS (ω)dω ! , (F2) Here M1 corresponds to the Lindbladian derived in [22], as shown in Theorem F.1, and M2 collects all higher-order correct...
[22]

Perturbation result for the spectral gap Define the rescaled mixing time tmix,Φ(ϵ) as in Eq. (C3). According to Theorem D.1, to achieve ϵ-accuracy in trace distance, the total Hamiltonian simulation time is Ttotal := τmix,Φ(ϵ/4) × 2T = eO tmix,Φ(ϵ/4) σ α2 = eO β2t3 mix,Φ(ϵ/4) (α2σ)ϵ2 ! , where we use σ = eΘ(βtmix,Φ(ϵ/4)/ϵ) from Theorem D.1 in the last equ...
[23]

T otal Runtime Bounds Beyond the Lindblad Limit Combining Theorem F.2 with the exising literature of spectral gap estimation for KMS generators [29, 32–35, 47], we can derive explicit total runtime bounds for the system-bath interaction algorithm beyond the Lindblad limit for various physical models [52]. We summarize the models and present our results be...
[25]

Transverse field Ising model (TFIM) Recall the transverse field Ising model (TFIM) in (3): H = −J L−1X i=1 ZiZi+1 − g LX i=1 Xi , where we set J = 1 , g= 1 .2

Thermal state preparation a. Transverse field Ising model (TFIM) Recall the transverse field Ising model (TFIM) in (3): H = −J L−1X i=1 ZiZi+1 − g LX i=1 Xi , where we set J = 1 , g= 1 .2. We now provide more detailed numerical results for thermal state preparation with different choices of parameters. • The thermal state setting with L = 4 in the regime ...
[26]

(d) Spectral gap of Φ Γ with different α, σ, β = ∞, and the same values of α

(c) Infidelity between target ground state ( β = ∞) and the stationary state of Φ Γ for varying σ and α = 2, 1, 0.5, 0.2, 0.1, 0.02. (d) Spectral gap of Φ Γ with different α, σ, β = ∞, and the same values of α. b. 1-D Hubbard model Consider the 1-D Hubbard model defined on L = 2, 4 spinful sites with open boundary conditions in (4) H = −t L−1X j=1 X σ∈{↑,...
[27]

For those large scale numerical experiments, including TFIM-8, Hubbard-4, we simulate the state vector instead of density operator to reduce the computational cost

Ground state preparation We also consider the ground state preparation of TFIM, Hubbard and ANNNI model. For those large scale numerical experiments, including TFIM-8, Hubbard-4, we simulate the state vector instead of density operator to reduce the computational cost. Thus, we report the evolution of energy along a single trajectory. Unless stated otherw...
[42]

Quantum algorithm for the simulation of open-system dynamics and thermalization

Hong-Yi Su and Ying Li. Quantum algorithm for the simulation of open-system dynamics and thermalization. Phys. Rev. A , 101:012328, Jan 2020

2020
[43]

X. Mi, A. A. Michailidis, S. Shabani, K. C. Miao, P. V. Klimov, J. Lloyd, E. Rosenberg, R. Acharya, I. Aleiner, T. I. Andersen, M. Ansmann, F. Arute, K. Arya, A. As- faw, J. Atalaya, J. C. Bardin, A. Bengtsson, G. Bortoli, A. Bourassa, J. Bovaird, L. Brill, M. Broughton, B. B. Buckley, D. A. Buell, T. Burger, B. Burkett, N. Bush- nell, Z. Chen, B. Chiaro,...

2024
[44]

Michailidis, Xiao Mi, Vadim Smelyanskiy, and Dmitry A

Jerome Lloyd, Alexios A. Michailidis, Xiao Mi, Vadim Smelyanskiy, and Dmitry A. Abanin. Quasiparticle cool- ing algorithms for quantum many-body state prepara- tion. PRX Quantum , 6:010361, March 2025

2025
[45]

Thermal state preparation by re- peated interactions at and beyond the Lindblad limit

Carlos Ramon-Escandell, Alessandro Prositto, and Dvira Segal. Thermal state preparation by re- peated interactions at and beyond the Lindblad limit. arXiv:2506.12166, 2025

arXiv 2025
[46]

Alhambra, Daniel Stilck Fran¸ ca, and Cambyse Rouz´ e

Samuel Slezak, Matteo Scandi, ´Alvaro M. Alhambra, Daniel Stilck Fran¸ ca, and Cambyse Rouz´ e. Polynomial- time thermalization and gibbs sampling from system- bath couplings. arXiv/2601.16154, 2026

arXiv 2026
[47]

Rapid mixing of quantum gibbs samplers for weakly-interacting quantum systems

ˇStˇ ep´ anˇSm ´ ıd, Richard Meister, Mario Berta, and Roberto Bondesan. Rapid mixing of quantum gibbs samplers for weakly-interacting quantum systems. arXiv/2510.04954, 2025

Pith/arXiv arXiv 2025
[48]

The annni model-theoretical analysis and experimental application

Walter Selke. The annni model-theoretical analysis and experimental application. Physics Reports, 170(4):213– 264, 1988

1988
[49]

Ignacio Cirac, and Bar- bara Kraus

Daniel Molpeceres, Sirui Lu, J. Ignacio Cirac, and Bar- bara Kraus. Quantum algorithms for cooling: a simple case study. arXiv:2503.24330, 2025

Pith/arXiv arXiv 2025
[50]

Chi-Fang Chen and Fernando G. S. L. Brand˜ ao. Fast thermalization from the eigenstate thermalization hy- pothesis. arXiv:2112.07646, 2023

Pith/arXiv arXiv 2023
[51]

Efficient quantum Gibbs sampling with local circuits

Dominik Hahn, Ryan Sweke, Abhinav Deshpande, and Oles Shtanko. Efficient quantum Gibbs sampling with local circuits. arXiv:2506.04321, 2025

Pith/arXiv arXiv 2025
[52]

However, rapid mixing directly implies a lower bound for the spectral gap

Some of the result here prove the rapid mixing prop- erty for KMS generators using oscillator norm technique instead of spectral gap. However, rapid mixing directly implies a lower bound for the spectral gap. Thus, we can directly apply their result in our case
[53]

Although it has not been explicitly proved in the liter- ature, we expect similar results should also hold for lo- cal fermionic Hamiltonians using similar techniques as in [29, 32–34]
[54]

Quantum Gibbs samplers: The commuting case

Michael J Kastoryano and Fernando GSL Brandao. Quantum Gibbs samplers: The commuting case. Com- mun. Math. Phys. , 344(3):915–957, 2016

2016
[55]

Generators of detailed balance quantum Markov semigroups

Franco Fagnola and Veronica Umanit` a. Generators of detailed balance quantum Markov semigroups. Infin. Di- mens. Anal. Quantum Probab. Relat. Top. , 10(03):335– 363, 2007. Appendix A: Organization of the appendix The appendix is organized as follows. • Appendix B: A review of related works on dissipative state preparation algorithms, with a focus on syst...

2007

This paper was first reviewed by deepseek-v4-flash on August 3, 2026.

[1] [1]

Dyson series expansion of ΦΓ In this section, we introduce the Dyson series expansion of Φ Γ in Eq. (1). We use the subindex {−1, 1} to relabel the system and bath operator as S1 = AS, S −1 = A† S, B 1 = BE, B −1 = B† E. Define the system evolution operator US(t) = exp( −itH). The Dyson series expansion of Φ Γ is summarized in the following theorem: 10 Th...

[2] [2]

We show that it admits the same structural form as eG† k,AS after a β-dependent shift of the integration variable

Proof of Lemma D.4 The main goal of this section is to prove an explicit expression of σ−1 β eG† k,AS σβ. We show that it admits the same structural form as eG† k,AS after a β-dependent shift of the integration variable. In addition, for notation simplicity, we assume AS is hermitian. The extension to the non-hermitian AS is straightforward. We first calc...

[3] [3]

Bounds of T erm 1.1 and T erm 1.2 In this section, we bound Term 1.1 and Term 1.2 separately so that the bound of Term 1 can be obtained by Term1 ≤ X n≥1 (2Γ)2n ((2π)1/4)2n ∥AS∥2n 2nX k=0 even (Term1.1 + Term1.2) . (D36) 19 We notice that Term 1.1 can be upper bounded by the inequality | sin(x)| ≤ |x| as, Term 1.1 = 2 exp (2n − k)β2 4σ2 Z −∞<s1≤s2≤···≤s2n...

[4] [4]

Assuming Γ2 = α2σ = O(1), we have ∥M2,∞∥2↔2 = O α2σ , (F9) ∥σ−1/4 β M2,∞[σ1/4 β · σ1/4 β ]σ−1/4 β ∥2↔2 = O α2σ , (F10) Proof of Lemma F.4

Proof of Theorem F.2 Recall (F2), we define M2,∞ = 1 α2 EAS  X n≥2 Γ2n(−1)n 2nX k=0 (−1)k Z γ(ω)G† 2n−k,AS ,∞(ω)ρGk,AS ,∞(ω)dω   with Gk,AS ,∞(ω) = Z −∞<t1≤···≤tk<∞ AS(t1)A† S(t2) · · ·e−iω Pk p=1(−1)ptp fσ(t1) · · ·fσ(tk)dt1 · · ·dtk We first provide an upper bound for the norm of M2,∞: Lemma F.4. Assuming Γ2 = α2σ = O(1), we have ∥M2,∞∥2↔2 = O α2σ ,...

[5] [16]

We define eGk,H,AS (ω) = eGk,AS (ω) = Z −∞<t1≤···≤tk<∞ AS(t1)A† S(t2) · · ·e−iω Pk p=1(−1)ptp f (t1) · · ·f (tk)dt1 · · ·dtk

Proof of Lemma D.3 Notice that the expression of eGk,AS depends on the system Hamiltonian H and the coupling operator AS. We define eGk,H,AS (ω) = eGk,AS (ω) = Z −∞<t1≤···≤tk<∞ AS(t1)A† S(t2) · · ·e−iω Pk p=1(−1)ptp f (t1) · · ·f (tk)dt1 · · ·dtk . Then, eG† k,H,AS (ω) = Z −∞<t1≤···≤tk<∞ · · ·AS(t2)A† S(t1)eiω Pk p=1(−1)ptp f (t1) · · ·f (tk)dt1 · · ·dtk ...

[6] [19]

|es1| | √ 2n − kes1 − √ ket1| des1det1 Z T (−∞<s2≤···≤s2n−k<∞, −∞<t2≤···≤tk<∞) e− P2n−k p=2 es2 p−Pk p=2 et2 p des2 · · ·des2n−kdet2 · · ·detk + √ 2n − k 1 2σ Z |γ′(ω)|dω Z |√2n−kes1− √ ket1|∈(δ,∞) e−(es2 1+et2

[7] [20]

1 | √ 2n − kes1 − √ ket1| des1det1 Z T (−∞<s2≤···≤s2n−k<∞, −∞<t2≤···≤tk<∞) e− P2n−k p=2 es2 p−Pk p=2 et2 p 2n−kX p=2 es2 p !1/2 des2 · · ·des2n−kdet2 · · ·detk . (D44) Next, we perform the additional change of variable ( x = √ 2n − kes1 − √ ket1 y = √ kes1 + √ 2n − ket1 , (D45) which gives I ≤ √ 2n − k 1 2σ ∥γ′∥L1 1 2n Z |x|∈(δ,∞) e− x2 +y2 2n | √ 2n − kx...

[8] [21]

ΦΓρ = ρ + α2M1[ρ] + α2M2[ρ]

Approximate ΦΓ using KMS-Lindbladian dynamics Following Theorem C.2, we define M1[ρ] = −EAS 2X k=0 (−1)kσ Z γ(ω)G† 2−k,AS (ω)ρGk,AS (ω)dω ! , (F1) M2[ρ] = 1 α2 EAS ∞X n=2 Γ2n(−1)n 2nX k=0 (−1)k Z γ(ω)G† 2n−k,AS (ω)ρGk,AS (ω)dω ! , (F2) Here M1 corresponds to the Lindbladian derived in [22], as shown in Theorem F.1, and M2 collects all higher-order correct...

[9] [22]

Perturbation result for the spectral gap Define the rescaled mixing time tmix,Φ(ϵ) as in Eq. (C3). According to Theorem D.1, to achieve ϵ-accuracy in trace distance, the total Hamiltonian simulation time is Ttotal := τmix,Φ(ϵ/4) × 2T = eO tmix,Φ(ϵ/4) σ α2 = eO β2t3 mix,Φ(ϵ/4) (α2σ)ϵ2 ! , where we use σ = eΘ(βtmix,Φ(ϵ/4)/ϵ) from Theorem D.1 in the last equ...

[10] [23]

T otal Runtime Bounds Beyond the Lindblad Limit Combining Theorem F.2 with the exising literature of spectral gap estimation for KMS generators [29, 32–35, 47], we can derive explicit total runtime bounds for the system-bath interaction algorithm beyond the Lindblad limit for various physical models [52]. We summarize the models and present our results be...

[11] [25]

Transverse field Ising model (TFIM) Recall the transverse field Ising model (TFIM) in (3): H = −J L−1X i=1 ZiZi+1 − g LX i=1 Xi , where we set J = 1 , g= 1 .2

Thermal state preparation a. Transverse field Ising model (TFIM) Recall the transverse field Ising model (TFIM) in (3): H = −J L−1X i=1 ZiZi+1 − g LX i=1 Xi , where we set J = 1 , g= 1 .2. We now provide more detailed numerical results for thermal state preparation with different choices of parameters. • The thermal state setting with L = 4 in the regime ...

[12] [26]

(d) Spectral gap of Φ Γ with different α, σ, β = ∞, and the same values of α

(c) Infidelity between target ground state ( β = ∞) and the stationary state of Φ Γ for varying σ and α = 2, 1, 0.5, 0.2, 0.1, 0.02. (d) Spectral gap of Φ Γ with different α, σ, β = ∞, and the same values of α. b. 1-D Hubbard model Consider the 1-D Hubbard model defined on L = 2, 4 spinful sites with open boundary conditions in (4) H = −t L−1X j=1 X σ∈{↑,...

[13] [27]

For those large scale numerical experiments, including TFIM-8, Hubbard-4, we simulate the state vector instead of density operator to reduce the computational cost

Ground state preparation We also consider the ground state preparation of TFIM, Hubbard and ANNNI model. For those large scale numerical experiments, including TFIM-8, Hubbard-4, we simulate the state vector instead of density operator to reduce the computational cost. Thus, we report the evolution of energy along a single trajectory. Unless stated otherw...

[14] [42]

Quantum algorithm for the simulation of open-system dynamics and thermalization

Hong-Yi Su and Ying Li. Quantum algorithm for the simulation of open-system dynamics and thermalization. Phys. Rev. A , 101:012328, Jan 2020

2020

[15] [43]

X. Mi, A. A. Michailidis, S. Shabani, K. C. Miao, P. V. Klimov, J. Lloyd, E. Rosenberg, R. Acharya, I. Aleiner, T. I. Andersen, M. Ansmann, F. Arute, K. Arya, A. As- faw, J. Atalaya, J. C. Bardin, A. Bengtsson, G. Bortoli, A. Bourassa, J. Bovaird, L. Brill, M. Broughton, B. B. Buckley, D. A. Buell, T. Burger, B. Burkett, N. Bush- nell, Z. Chen, B. Chiaro,...

2024

[16] [44]

Michailidis, Xiao Mi, Vadim Smelyanskiy, and Dmitry A

Jerome Lloyd, Alexios A. Michailidis, Xiao Mi, Vadim Smelyanskiy, and Dmitry A. Abanin. Quasiparticle cool- ing algorithms for quantum many-body state prepara- tion. PRX Quantum , 6:010361, March 2025

2025

[17] [45]

Thermal state preparation by re- peated interactions at and beyond the Lindblad limit

Carlos Ramon-Escandell, Alessandro Prositto, and Dvira Segal. Thermal state preparation by re- peated interactions at and beyond the Lindblad limit. arXiv:2506.12166, 2025

arXiv 2025

[18] [46]

Alhambra, Daniel Stilck Fran¸ ca, and Cambyse Rouz´ e

Samuel Slezak, Matteo Scandi, ´Alvaro M. Alhambra, Daniel Stilck Fran¸ ca, and Cambyse Rouz´ e. Polynomial- time thermalization and gibbs sampling from system- bath couplings. arXiv/2601.16154, 2026

arXiv 2026

[19] [47]

Rapid mixing of quantum gibbs samplers for weakly-interacting quantum systems

ˇStˇ ep´ anˇSm ´ ıd, Richard Meister, Mario Berta, and Roberto Bondesan. Rapid mixing of quantum gibbs samplers for weakly-interacting quantum systems. arXiv/2510.04954, 2025

Pith/arXiv arXiv 2025

[20] [48]

The annni model-theoretical analysis and experimental application

Walter Selke. The annni model-theoretical analysis and experimental application. Physics Reports, 170(4):213– 264, 1988

1988

[21] [49]

Ignacio Cirac, and Bar- bara Kraus

Daniel Molpeceres, Sirui Lu, J. Ignacio Cirac, and Bar- bara Kraus. Quantum algorithms for cooling: a simple case study. arXiv:2503.24330, 2025

Pith/arXiv arXiv 2025

[22] [50]

Chi-Fang Chen and Fernando G. S. L. Brand˜ ao. Fast thermalization from the eigenstate thermalization hy- pothesis. arXiv:2112.07646, 2023

Pith/arXiv arXiv 2023

[23] [51]

Efficient quantum Gibbs sampling with local circuits

Dominik Hahn, Ryan Sweke, Abhinav Deshpande, and Oles Shtanko. Efficient quantum Gibbs sampling with local circuits. arXiv:2506.04321, 2025

Pith/arXiv arXiv 2025

[24] [52]

However, rapid mixing directly implies a lower bound for the spectral gap

Some of the result here prove the rapid mixing prop- erty for KMS generators using oscillator norm technique instead of spectral gap. However, rapid mixing directly implies a lower bound for the spectral gap. Thus, we can directly apply their result in our case

[25] [53]

Although it has not been explicitly proved in the liter- ature, we expect similar results should also hold for lo- cal fermionic Hamiltonians using similar techniques as in [29, 32–34]

[26] [54]

Quantum Gibbs samplers: The commuting case

Michael J Kastoryano and Fernando GSL Brandao. Quantum Gibbs samplers: The commuting case. Com- mun. Math. Phys. , 344(3):915–957, 2016

2016

[27] [55]

Generators of detailed balance quantum Markov semigroups

Franco Fagnola and Veronica Umanit` a. Generators of detailed balance quantum Markov semigroups. Infin. Di- mens. Anal. Quantum Probab. Relat. Top. , 10(03):335– 363, 2007. Appendix A: Organization of the appendix The appendix is organized as follows. • Appendix B: A review of related works on dissipative state preparation algorithms, with a focus on syst...

2007

Beyond Lindblad Dynamics: Rigorous Guarantees for Thermal and Ground State Preservation under System Bath Interactions

The reading

Core claim §

What carries the argument

Load-bearing premise §

What would settle it §

If this is right

Where Pith is reading between the lines

discussion (0)

Core claim

Load-bearing premise

What would settle it