Pith. sign in

REVIEW 3 minor 55 references

Local Pauli responses, analyzed jointly as a linear system, identify both coherent and dissipative Lindbladian coefficients in open quantum systems.

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 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-01 01:28 UTC pith:4GG7SOKW

load-bearing objection A technically sound and genuinely new framework for learning Lindbladians from short-time Pauli responses, provided the generator lies exactly in the known candidate dictionary; the paper deserves a serious referee, with attention to the misspecification caveat and the lack of numerics.

arxiv 2607.25795 v1 pith:4GG7SOKW submitted 2026-07-28 quant-ph

Efficient Lindbladian Learning from Constant-Time Pauli Responses

Jiaxing Song , Yukun Zhang , Xiao Yuan , Yusen Wu This is my paper
classification quant-ph MSC 81S2281P1881P45 PACS 03.65.Yz03.67.-a
keywords Lindbladian learningopen quantum systemsPauli response functionsGKSL generatorcoherent-dissipative ambiguitylocal response inversionquantum process tomographyHamiltonian learning
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.

Learning the generator of an open quantum system is harder than Hamiltonian learning because a local response mixes coherent coupling with dissipative noise, and even different dissipative terms can masquerade as each other. This paper shows that if the possible generator terms are known as a Pauli–GKSL dictionary, the mixing is not an obstruction: ordering the candidate terms by support makes the response matrix block-triangular and invertible. Multiplying raw responses by the inverse produces normalized responses whose linear-in-time term is exactly one coefficient each. The paper gives two protocols—Chebyshev–Lobatto endpoint differentiation and single-time projected contraction—that estimate all M coefficients to entrywise accuracy epsilon using O~(M/epsilon^2) experimental shots and comparable classical post-processing. If correct, the result turns local response inversion into a scalable calibration and diagnostic tool for noisy many-body quantum devices.

Core claim

The central claim is that the coherent–dissipative ambiguity in Lindbladian learning is algebraically resolvable from local Pauli response data. For a known nonredundant Pauli–GKSL dictionary, the paper constructs an M by M response matrix G whose entries are the linear-in-time contributions of each candidate generator term to each measured response. Because a term supported inside a region can only contaminate responses whose outside Pauli labels are extended by a common string, ordering dissipative terms by decreasing support makes the dissipative block triangular with nonzero diagonal; Hamiltonian terms form an identity block, so G is invertible. With H = G^{-1}, the normalized responses

What carries the argument

The load-bearing object is the real response matrix G_{j beta} = F_j(L^dagger_beta), built from raw Pauli response functionals F_j over a known local Pauli–GKSL dictionary. Its key structural property is support-ordered block triangularity: after ordering dissipative candidates by decreasing support, a response row only sees candidates whose support is a common extension of its own, making the dissipative block triangular and the Hamiltonian block an identity; hence G is invertible. The inverse H = G^{-1} defines normalized response functionals C_alpha, and the paper proves the normalized-response identity g_alpha(t,theta) = t theta_alpha + O(t^2). The uniform bounded-overlap regime ensures

Load-bearing premise

The true dynamics is exactly e^{t L_theta} for a time-independent Markovian generator drawn from a known, complete, nonredundant Pauli–GKSL dictionary with coefficients in [-1,1] and bounded overlap; if the real generator contains a term outside this dictionary, such as an unmodeled long-range coupling, time dependence, or non-Markovian memory, the response matrix is misspecified and the estimators fail silently.

What would settle it

On a system whose generator includes a known dissipative term supported on more than S_D sites, or a Hamiltonian term not in the dictionary, run the single-time protocol at t = t*. If the claim is false, the projected iteration will converge to a parameter whose response residual stays above the statistical error even as the number of shots grows, and the normalized response g_alpha(t)/t will show a systematic drift larger than O(t) between two different short times.

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

Share X Bluesky LinkedIn Reddit HN

If this is right

  • Both coherent coupling strengths and Markovian noise rates can be estimated simultaneously from the same short-time Pauli response data, instead of fitting one effective Hamiltonian.
  • The protocols work at evolution times t = O(1), avoiding the need for long-time dynamics or full process tomography.
  • The total experimental and classical cost scales as O~(M/epsilon^2) for M candidate coefficients, making the approach viable for many-body systems whose generator has a local, known dictionary.
  • Computing G and H is a one-time preprocessing step for a fixed dictionary, and the inverse can be reused across different systems sharing that dictionary.
  • An optional projection onto the physical set of GKSL parameters yields a valid Lindbladian estimate at the cost of a factor 2 in accuracy.

Where Pith is reading between the lines

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

  • The fixed-point residual of the single-time protocol could be turned into a model-misspecification test: after fitting, one can check whether the residual shrinks with increasing shot count; if it does not, an unmodeled generator term is likely present.
  • The support-ordered triangularization relies only on the notion of a common operator extension, so a similar response-inversion construction may transfer to other algebraically closed operator bases, such as fermionic or qudit Pauli-like bases, whenever such an ordering exists.
  • The near-linear cost suggests a practical calibration primitive: on a noisy device with a known local noise model, one could periodically re-estimate all generator coefficients using only constant-depth state preparation and local measurements.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 3 minor

Summary. The paper studies learning the generator of a time-independent Lindbladian evolution on N qubits, assuming a known, complete, nonredundant Pauli–GKSL dictionary with bounded overlap. It defines raw Pauli response functionals whose linear-in-time part is a known response matrix G mapping generator coefficients to response slopes, and proves that support ordering makes the dissipative block triangular, so G is invertible. Under bounded-overlap assumptions, H=G^{-1} has O(1) row norm and row sparsity. The normalized responses g_alpha(t,theta)=sum_j H_alpha j F_j(e^{tL^dagger_theta}-I) then satisfy g_alpha(t,theta)=t theta_alpha+O(t^2). Two estimators are proposed: Chebyshev–Lobatto interpolation at r=O(log(1/epsilon)) short times, and a single-time projected contraction at t_*=Theta(Lambda^{-2}) with geometric convergence. Both estimate all M coefficients to entrywise accuracy epsilon using O~(M/epsilon^2) experimental shots and O~(M/epsilon^2) total classical post-processing.

Significance. If it holds, this is a significant advance: it extends the local-response paradigm from Hamiltonian learning to Markovian open systems by explicitly resolving the coherent-dissipative and common-extension ambiguities, and it does so with concrete algorithms and explicit complexity bounds. The paper's central chain is sound: Lemma 2's response formula, Proposition 1's block-triangular invertibility, Theorem 7's uniform inverse bounds, and the finite-time error analyses of Theorems 8 and 10 are internally consistent. The main caveat is the Problem 1 assumption of a known complete dictionary; a true generator with an unmodeled term, time dependence, or non-Markovianity would make G misspecified and silently bias the estimates. The paper explicitly declares structure learning out of scope in Appendix A, so this is a scoping limitation rather than an internal inconsistency.

minor comments (3)
  1. [Appendix C, Lemma 3 (Eq. C4)] The stated identity F^W_{S,R}(Phi)=F^W_{R,S}(Phi) is not correct for a general Hermiticity-preserving map; the proof applies trace cyclicity incorrectly. The correct relation is F^W_{S,R}(Phi)=overline{F^W_{R,S}(Phi)}. Fortunately, every use in Section C survives under this correction: the combinations F^H_C, F^diag_A, F^Re_AB, F^Im_AB remain real on Hermiticity-preserving maps. Please fix the lemma and its proof.
  2. [Problem 1 and Appendix A] The framework assumes a known, complete, nonredundant dictionary. If the true generator contains a term outside the dictionary, the estimator is biased and no diagnostic is provided. Since the paper explicitly excludes structure discovery, this is not a blocking issue, but a one-sentence warning near Eq. (1) would help readers avoid overgeneralizing the claims to structure-learning settings.
  3. [General notation] The paper uses O~ informally in several places without a single definition. Given that the stated sample and post-processing bounds mix polylogarithmic factors from r, n_it, and the union bound, a brief formal definition of O~ in Section IV or the appendices would improve precision.

Circularity Check

0 steps flagged

No circularity found: the central recovery claim is a self-contained coordinate-construction and finite-time analysis under stated assumptions.

full rationale

The derivation chain is not circular. The raw responses F_j are defined independently (Eq. (2)/(C1)) from the candidate dictionary, and the response matrix G_jβ = F_j(L†_β) is a known dictionary-dependent matrix (Eq. (5)/(C25)). The normalized response functionals C_α = Σ_j H_αj F_j are then defined with H = G^{-1} (Eq. (6)/(C27)), so the identity C_α(L†_β) = δ_αβ is a direct consequence of matrix inversion, not an independent empirical input. Corollary 1 then derives g_α(t,θ) = t θ_α + O(t^2) by Taylor expansion of e^{tL†_θ} and use of HG = I (Eq. (C36)). This is a coordinate normalization, not a fitted prediction: the quantity being estimated is the coefficient θ_α itself, and the normalized response is engineered so that its linear slope equals that coefficient. The sample-complexity statements (Theorems 1–2, Propositions 2, 6, 7) rest on bounds on |Z|, Hoeffding's inequality, and the Taylor bounds of Appendix E, all of which are proved from the stated locality assumptions without importing the conclusion. The invertibility and locality of H are proved in Proposition 1 and Theorem 7 from support ordering and bounded overlap; these arguments do not assume the coefficients being estimated. The Chebyshev-Lobatto and contraction error analyses (Theorems 8, 10 and Corollaries 5, 8) are independent analytic bounds. There is no load-bearing self-citation, no uniqueness theorem imported from the authors' prior work, and no fitted input renamed as a prediction. The paper explicitly scopes itself to a known nonredundant Pauli–GKSL dictionary (Problem 1, Appendix A), which is a stated assumption, not a circular conclusion. One minor technical slip exists but is not circular: Lemma 3 states F_{S,R} = F_{R,S} for Hermiticity-preserving maps, where the correct identity is F_{S,R} = conj(F_{R,S}); the real-valued response combinations used later remain valid under this corrected relation, so the central argument is unaffected.

Axiom & Free-Parameter Ledger

5 free parameters · 7 axioms · 0 invented entities

No parameter is fitted to any data; the learning guarantees are derived analytically. The quantities listed as free parameters are design-time constants - explicit formulas in known locality scales and target accuracy epsilon - included for completeness under the rubric's 'chosen by hand' clause; none is informed by the target coefficients. Axioms 1-4 and 7 are the modeling and experimental premises that define the problem and the regime; if any fails, the guarantees do not apply. No new physical entities are postulated; the normalized response coordinates are derived constructs fully determined by the known dictionary.

free parameters (5)
  • Chebyshev time scale tau* = 1/(2*Lambda) = 1/(2*Lambda)
    Chosen so Lambda*tau* = 1/2 < 1, making the Taylor tail geometric (Cor. 5). Analytically determined by the known locality scale; not fitted to data.
  • Single-time contraction time t* = 1/(4*Lambda*(1+6*L_C*Lambda)) = 1/(4*Lambda*(1+6*L_C*Lambda))
    Hand-chosen so the Jacobian bound (Cor. 7, Eq. (G22)) yields contraction factor q <= 1/2 for all truncation degrees; an analytic design choice, not fitted.
  • Interpolation degree r* = min{n >= 2 : 2^n >= 12 L_C Lambda n^2 / epsilon}
    Prescribed by Eq. (F18) from target accuracy epsilon; deterministic, not a fit.
  • Contraction factor q* = 1/2 = 1/2
    Chosen in Cor. 8 (Eq. (G25)); any q in (0,1) works; affects only constants.
  • Taylor truncation degree m (single-time method) = O(log(1/epsilon)) per Eq. (G30)
    Set by the tail-error allocation; deterministic function of epsilon and the known bounds.
axioms (7)
  • domain assumption Exact time-homogeneous Markovian GKSL dynamics: measured evolution is exactly e^{t L_theta} with L_theta = sum_alpha theta_alpha L^+_alpha
    Eq. (1) and Problem 1. Non-Markovian or time-dependent deviations are outside the model; no robustness bound is provided.
  • domain assumption A complete, nonredundant, known Pauli-GKSL candidate dictionary A
    Problem 1, App. B. If the true generator contains terms outside A, G is misspecified and the estimates are biased without bound.
  • domain assumption Bounded coefficients theta in Theta_GKSL subset of [-1,1]^M
    Eq. (1), Eq. (B13); the projection Omega = [-1,1]^M in Algorithm 2 relies on it.
  • domain assumption Uniform bounded-overlap regime: S_D = O(1), Lambda = O(1), with Lambda = 2(d+1) or s*kappa_0
    Eq. (E12). Needed for L_C = O(1) and row sparsity r_0 = O(1) (Thm. 7), and hence for the sample and classical scalings; if Lambda grows with system size, all constants and conditioning degrade.
  • standard math Pauli support propagation: disjoint supports imply L^+_a(Z) = 0; supp(L^+_a(Z)) subset of supp(Z) union supp(L^+_a)
    Eq. (E3), basic Pauli composition; used throughout App. E for Taylor bounds.
  • standard math Standard probability and analysis tools: Hoeffding's inequality, Banach fixed-point theorem, Taylor expansion, Pauli orthogonality
    Used in Props. 2/6/7 and Thm. 9; no special or unproved machinery beyond these.
  • domain assumption Experimental access: product Pauli eigenstate preparation on the N system qubits without ancillas, tunable short-time evolution, Pauli measurements
    App. D protocol; if such preparations are not available, the unbiased estimators do not apply.

pith-pipeline@v1.3.0-alltime-deepseek · 27764 in / 29194 out tokens · 255335 ms · 2026-08-01T01:28:22.175497+00:00 · methodology

0 comments
Cite this review

Pith. "Pith review of Efficient Lindbladian Learning from Constant-Time Pauli Responses." pith.science (2026). https://pith.science/paper/4GG7SOKW

@misc{pith2026260725795,
  author       = {Pith},
  title        = {Pith review of: Efficient Lindbladian Learning from Constant-Time Pauli Responses},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4GG7SOKW}},
  note         = {Machine review of arXiv:2607.25795}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

Learning the generator of an open many-body system is more challenging than Hamiltonian learning: local responses, which can directly reveal coherent interaction terms in closed-system dynamics, may also contain dissipative contributions in open-system dynamics. In this paper, we address this challenge by developing an efficient Lindbladian learning framework for a known local candidate generator dictionary with bounded dissipative support and either bounded dual-interaction-graph degree or bounded unweighted local strength. The framework resolves the coherent-dissipative ambiguity by treating local Pauli responses as a linear system over both types of generator terms. Inverting this response system separates their contributions and makes the individual Lindbladian coefficients accessible from local response data in a fixed short-time window. Within this framework, we develop two efficient learning algorithms: Chebyshev--Lobatto response interpolation, which uses logarithmically many short evolution times and has a post-mean cost linear in $M$, with the stated dependence on $\epsilon$, and Single-time projected response contraction, which uses a single fixed evolution time and globally inverts a truncated response function. Both procedures estimate $M$ candidate coefficients to entrywise accuracy $\epsilon$ using $\widetilde{\mathcal{O}}(M/\epsilon^2)$ sample and classical post-processing complexity. Our theoretical results establish local response inversion as a scalable paradigm for learning, calibrating, and diagnosing complex quantum systems from experimentally accessible short-time data.

Figures

Figures reproduced from arXiv: 2607.25795 by Jiaxing Song, Xiao Yuan, Yukun Zhang, Yusen Wu.

Figure 1
Figure 1. Figure 1: FIG. 1. Overview of the Lindbladian learning framework. (a) Randomized product-Pauli experiments estimate finite-time [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

55 extracted references · 7 linked inside Pith

  1. [1]

    computeGandH=G −1 once from the known candidate dictionary

  2. [2]

    estimate the raw response means and applyHto form the normalized finite-time responsesg α(t,θ)

  3. [3]

    recoverθusing either Chebyshev–Lobatto endpoint differentiation or single-time projected response in- version. III. EXPERIMENT AL RESPONSE ESTIMA TION Each real raw response admits a bounded unbiased estimator obtained from a randomized product-Pauli preparation, short-time evolution, and Pauli measure- ment. Concretely, sampleP∈ P W uniformly and writeRP...

  4. [4]

    Arute, K

    F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. Brandao, D. A. Buell,et al., Nature574, 505 (2019)

    2019

  5. [5]

    Morvan, B

    A. Morvan, B. Villalonga, X. Mi, S. Mandra, A. Bengts- son, P. Klimov, Z. Chen, S. Hong, C. Erickson, I. Droz- dov,et al., Nature634, 328 (2024)

    2024

  6. [6]

    Zhong, H

    H.-S. Zhong, H. Wang, Y.-H. Deng, M.-C. Chen, L.-C. Peng, Y.-H. Luo, J. Qin, D. Wu, X. Ding, Y. Hu,et al., Science370, 1460 (2020)

    2020

  7. [7]

    Smith, A

    J. Smith, A. Lee, P. Richerme, B. Neyenhuis, P. W. Hess, P. Hauke, M. Heyl, D. A. Huse, and C. Monroe, Nature Physics12, 907 (2016)

    2016

  8. [8]

    S. J. Evered, D. Bluvstein, M. Kalinowski, S. Ebadi, T. Manovitz, H. Zhou, S. H. Li, A. A. Geim, T. T. Wang, N. Maskara,et al., Nature622, 268 (2023)

    2023

  9. [9]

    Bravyi, A

    S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, Nature627, 778 (2024)

    2024

  10. [10]

    Acharya, L

    R. Acharya, L. Aghababaie-Beni, I. Aleiner, T. I. An- dersen, M. Ansmann, F. Arute, K. Arya, A. Asfaw, N. Astrakhantsev, J. Atalaya,et al., arXiv preprint arXiv:2408.13687 (2024)

    Pith/arXiv arXiv 2024

  11. [11]

    Y. Kim, C. J. Wood, T. J. Yoder, S. T. Merkel, J. M. Gambetta, K. Temme, and A. Kandala, Nature Physics 19, 752 (2023)

    2023

  12. [12]

    T. E. O’Brien, G. Anselmetti, F. Gkritsis, V. Elfving, S. Polla, W. J. Huggins, O. Oumarou, K. Kechedzhi, D. Abanin, R. Acharya,et al., Nature Physics19, 1787 (2023)

    2023

  13. [13]

    Y. Kim, A. Eddins, S. Anand, K. X. Wei, E. Van Den Berg, S. Rosenblatt, H. Nayfeh, Y. Wu, M. Zale- tel, K. Temme, and A. Kandala, Nature618, 500 (2023)

    2023

  14. [14]

    Banaszek, M

    K. Banaszek, M. Cramer, and D. Gross, New Journal of Physics15, 125020 (2013)

    2013

  15. [15]

    Blume-Kohout, New Journal of Physics12, 043034 (2010)

    R. Blume-Kohout, New Journal of Physics12, 043034 (2010)

    2010

  16. [16]

    Eisert, D

    J. Eisert, D. Hangleiter, N. Walk, I. Roth, D. Markham, R. Parekh, U. Chabaud, and E. Kashefi, Nature Reviews Physics2, 382 (2020)

    2020

  17. [17]

    Gross, Y.-K

    D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eis- ert, Physical Review Letters105, 150401 (2010)

    2010

  18. [18]

    Hradil, Physical Review A55, R1561 (1997)

    Z. Hradil, Physical Review A55, R1561 (1997)

    1997

  19. [19]

    Mauro D’Ariano, M

    G. Mauro D’Ariano, M. G. Paris, and M. F. Sacchi, in Advances in Imaging and Electron Physics, Vol. 128 (El- sevier, 2003) pp. 205–308

    2003

  20. [20]

    I. L. Chuang and M. A. Nielsen, Journal of Modern Op- tics44, 2455 (1997)

    1997

  21. [21]

    G. M. D’Ariano and P. Lo Presti, Physical Review Letters 86, 4195 (2001)

    2001

  22. [22]

    Mohseni, A

    M. Mohseni, A. T. Rezakhani, and D. A. Lidar, Physical Review A77, 032322 (2008)

    2008

  23. [23]

    Gebhart, R

    V. Gebhart, R. Santagati, A. A. Gentile, E. M. Gauger, D. Craig, N. Ares, L. Banchi, F. Marquardt, L. Pezz` e, and C. Bonato, Nature Reviews Physics 10.1038/s42254- 022-00552-1 (2023)

    doi:10.1038/s42254- 2023

  24. [24]

    J. Haah, R. Kothari, and E. Tang, Nature Physics20, 1027 (2024)

    2024

  25. [25]

    Huang, Y

    H.-Y. Huang, Y. Tong, D. Fang, and Y. Su, Physical Review Letters130, 200403 (2023)

    2023

  26. [26]

    Bakshi, A

    A. Bakshi, A. Liu, A. Moitra, and E. Tang, arXiv:2405.00082 (2024)

    Pith/arXiv arXiv 2024

  27. [27]

    H.-Y. Hu, M. Ma, W. Gong, Q. Ye, Y. Tong, S. T. Flam- mia, and S. F. Yelin 10.48550/arXiv.2502.11900 (2025)

    doi:10.48550/arxiv.2502.11900 2025

  28. [28]

    Cedillo Vayson de Pradenne, J

    C. Cedillo Vayson de Pradenne, J. Cotler, and H.-Y. Huang, arXiv:2606.05690 (2026)

    Pith/arXiv arXiv 2026

  29. [29]

    Gorini, A

    V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Journal of Mathematical Physics17, 821 (1976)

    1976

  30. [30]

    Lindblad, Communications in Mathematical Physics 48, 119 (1976)

    G. Lindblad, Communications in Mathematical Physics 48, 119 (1976)

    1976

  31. [31]

    Breuer and F

    H.-P. Breuer and F. Petruccione,The Theory of Open Quantum Systems(Oxford University Press, Oxford, 2002)

    2002

  32. [32]

    Diehl, A

    S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. P. B¨ uchler, and P. Zoller, Nature Physics4, 878 (2008)

    2008

  33. [33]

    Verstraete, M

    F. Verstraete, M. M. Wolf, and J. I. Cirac, Nature Physics 5, 633 (2009)

    2009

  34. [34]

    D. S. Fran¸ ca, T. M¨ obus, C. Rouz´ e, and A. H. Werner, arXiv:2510.08500 (2025)

    arXiv 2025

  35. [35]

    Breuer, E.-M

    H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, Re- views of Modern Physics88, 021002 (2016)

    2016

  36. [36]

    R. T. Birke, J. B. Severin, M. A. Marciniak, E. Hogedal, A. Nylander, I. Ahmad, A. Osman, J. Bizn´ arov´ a, M. Rommel, A. F. Roudsari, J. Bylander, G. Tancredi, D. S. Fran¸ ca, A. Werner, C. W. Warren, J. Hastrup, S. Krøjer, and M. Kjaergaard, arXiv:2602.14694 (2026)

    arXiv 2026

  37. [37]

    Ivashkov, N

    P. Ivashkov, N. Romanov, W. Gong, A. Gu, H.-Y. Hu, and S. F. Yelin, arXiv:2603.05492 (2026)

    arXiv 2026

  38. [38]

    Heightman, R

    T. Heightman, R. Aseguinolaza Gallo, E. Jiang, J. Saave- dra, A. Ac ´ ın, and M. P lodzie´ n, arXiv:2603.07778 (2026)

    arXiv 2026

  39. [39]

    Romanov, P

    N. Romanov, P. Ivashkov, W. Gong, I. Kannan, A. Gu, H.-Y. Hu, and S. F. Yelin, arXiv:2606.18188 (2026)

    Pith/arXiv arXiv 2026

  40. [40]

    I. Arad, Z. Chen, N. Guo, P. Rebentrost, and Z. Yu, arXiv:2606.20535 (2026)

    Pith/arXiv arXiv 2026

  41. [41]

    M¨ obus, T

    T. M¨ obus, T. Bergamaschi, D. S. Fran¸ ca, and C. Rouz´ e, arXiv:2606.23652 (2026)

    Pith/arXiv arXiv 2026

  42. [42]

    Lewis, E

    L. Lewis, E. Tang, and J. Wright, arXiv:2606.30358 (2026). 7 Appendix A: Comparison with related and concurrent work Recent progress on learning open quantum dynamics has followed several different directions, depending on whether the goal is reconstructing the full dynamical map, identifying the interaction structure, or estimating the parameters of a pr...

    Pith/arXiv arXiv 2026

  43. [43]

    For a Hamiltonian coefficienth C with|supp(C)| ≤S D, use F H C (Φ) := 1 2i F supp(C) C,I (Φ)−F supp(C) I,C (Φ) .(C17)

  44. [44]

    For|supp(C)|> S D, use the direct response F H C :=R C (C18) from Eq. (C8)

  45. [45]

    For a diagonal dissipative coefficientK AA, use F diag A :=F supp(A) A,A .(C19)

  46. [46]

    If only one coordinate is candidate, include only its corresponding real row

    For an ordered off-diagonal pairA < B, set WAB = supp(A)∪supp(B), F BA =F WAB B,A , F AB =F WAB A,B .(C20) Define the real off-diagonal raw responses F Re AB(Φ) := 1 2 (FBA(Φ) +F AB(Φ)),(C21) F Im AB(Φ) :=− i 2 (FBA(Φ)−F AB(Φ)).(C22) If bothk R AB andk I AB are candidate, include both real rows. If only one coordinate is candidate, include only its corres...

  47. [47]

    SampleP∈ P W and write RP S=ω(P)Q(P), ω(P)∈ {±1,±i},(D1) whereQ(P) is phase-free and Hermitian

  48. [48]

    IfQ(P)̸=I, sampleσ∈ {±1}uniformly and prepare an ensemble with mean state ρσ,Q(P) = I+σQ(P) D .(D2) Evolve for timet, measurePwith outcomeµ∈ {±1}, and record Z=ω(P) ∗σµ.(D3) IfQ(P) =I, omit the random sign, use a maximally mixed input, and recordZ=ω(P) ∗µ

  49. [49]

    Use the same realization withQ C for a direct response

    For a direct responseR C, replaceQ(P) byQ C, sampleσ∈ {±1}, and prepare ρσ,QC = I+σQ C D ,(D4) Evolve for timet, measureP C with outcomeµ C ∈ {±1}, and record ZC = χC 2 σµC.(D5) WhenQ(P) has multiqubit support, realize its mixed state by sampling product eigenstates of the nonidentity single-qubit Pauli factors, with local eigenvalues uniform subject to p...

  50. [50]

    Two vertices are adjacent when the corresponding supports intersect

    T aylor bounds under bounded dual-graph degree Let the dual interaction graph have one vertex for each candidate dictionary term. Two vertices are adjacent when the corresponding supports intersect. Letdbe the maximum degree and writeN(O) =|{a: supp(O)∩supp(L † a)̸=∅}|. Theorem 5(Mixed Taylor bound from bounded dual-interaction-graph degree).Assume the su...

  51. [51]

    Theorem 6(Repeated-generator bounds under local strength).Assume the support properties in Eq.(E3)

    T aylor bounds under bounded local strength Define the unweighted local dictionary strength and the maximum body size by κ0 := max y∈[N] X a:y∈supp(L † a) L† a ∞→∞ , s:= max a supp(L† a) .(E7) The quantityκ 0 depends only on the prescribed dictionary and not on a coefficient vector. Theorem 6(Repeated-generator bounds under local strength).Assume the supp...

  52. [52]

    The following regime makes this inverse uniformly local

    Uniform bounds on the response inverse The constantL C is determined by the inverse of the candidate response matrix. The following regime makes this inverse uniformly local. We call it the uniform bounded-overlap regime when either locality hypothesis in Eq. (E11) holds and SD =O(1),Λ =O(1),(E12) with constants independent of the system sizeNand dictiona...

  53. [53]

    Each raw-response functional has total scalar coefficient weight at most one

    T aylor bounds for normalized responses Letm≥1 denote the Taylor truncation degree. Each raw-response functional has total scalar coefficient weight at most one. In both locality settings, the scalar matrix elements obey (L† θ)k(O) ∞ ≤Λ kk!,(E32) with the unified scale Λ from Eq. (E11). Combining this bound with∥H α·∥1 ≤L C gives ∂k t gα(0, θ) ≤L CΛkk!.(E...

  54. [54]

    We bound their construction separately from the experimental response cost; the square response family containsMcoefficients andMraw responses

    Classical construction of the truncated response polynomials The single-time method uses coefficient tables for the truncated response map. We bound their construction separately from the experimental response cost; the square response family containsMcoefficients andMraw responses. a. Computational model.All complexity bounds in this section count sparse...

  55. [55]

    Sufficient Jacobian conditions for the finite-time response map Recallg m from Eq. (G1). The required Jacobian condition is sup x∈Ω IM −t −1Dxgm(t, x) ∞→∞ ≤q <1.(G15) The next theorem gives a direct sufficient bound for its left-hand side. 25 Theorem 10(Unified Jacobian remainder bound).Assume either locality hypothesis in Eq.(E11)and letΩ⊆ [−1,1] M be co...