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REVIEW 3 major objections 4 minor 92 references

The paper introduces an operator length — the averaged rightmost non-identity site in the Pauli expansion — and shows that in the many-body localized regime it grows logarithmically in time for arbitrarily weak interactions, while in Anders

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2026-08-03 09:47 UTC pith:LXLA7NJR

load-bearing objection Smart, useful MPO diagnostics for operator spread; the Δ=1 log law looks solid, but the 'arbitrarily weak interactions' claim is extrapolated beyond the data. the 3 major comments →

arxiv 2601.12446 v3 pith:LXLA7NJR submitted 2026-01-18 quant-ph cond-mat.dis-nncond-mat.str-el

Operator delocalization in disordered spin chains via exact MPO marginals

Jonnathan Pineda , Mario Collura , Gianluca Passarelli , Procolo Lucignano , Davide Rossini , Angelo Russomanno This is my paper
classification quant-ph cond-mat.dis-nncond-mat.str-el
keywords operator lengthoperator massmany-body localizationAnderson localizationmatrix product operatorsPauli basisoperator entanglement entropyclassical shadows
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 paper is trying to establish that operator delocalization in a disordered spin chain can be read off from two simple Pauli-basis marginals — operator mass and the newly introduced operator length — and that these quantities sharply separate Anderson localization from many-body localization. The central numerical claim is that in the interacting disordered XXZ chain at strong disorder, the operator length grows as h(t) ~ a ln t + b over many time decades, while in the non-interacting case it rapidly saturates; the same logarithmic growth is found for operator mass and operator entanglement entropy, and it persists for arbitrarily weak interactions. The paper further claims that this growth is quantitatively captured by an effective l-bit model and that both marginals can be computed exactly and efficiently from a matrix-product-operator representation, avoiding stochastic sampling. A sympathetic reader would care because this offers an exact, experimentally accessible diagnostic that distinguishes the two localized regimes without needing full operator tomography.

Core claim

On the paper's own terms: for the disordered XXZ chain, after starting from a single-site sigma-z operator, the average operator length h(t) exhibits a clear logarithmic increase, h(t) ~ a ln t + b, over several time decades for W greater than about 5 and Delta = 1, while for Delta = 0 it rapidly saturates. Even arbitrarily weak interactions produce unbounded logarithmic operator spreading, and the behavior is quantitatively captured by an effective l-bit model. The mechanism is dephasing between exponentially localized integrals of motion: an effective coupling J_{1r} ~ e^{-r/xi} induces dephasing on timescales t_r ~ e^{r/xi}, so inverting gives r(t) ~ xi ln t. The paper also claims the nov

What carries the argument

The central object is the Pauli-string expansion of the time-evolved operator. Each string is assigned a mass (number of non-identity Pauli matrices) and a length (position of the rightmost non-identity site); averaging these against the squared Hilbert-Schmidt amplitudes defines m(t) and h(t). In the MPS representation these become expectation values of simple diagonal operators: h(t) is obtained from the Renyi-2 entropies of the operator state, and m(t) from a factorized mass superoperator. The log growth is explained by the l-bit Hamiltonian with exponentially decaying couplings J_{jl} = W_{jl} e^{-kappa |j-l|}, whose Heisenberg solution gives h(t) ~ (1/kappa) ln t until finite-size satur

Load-bearing premise

The load-bearing premise is that the MPS/MPO time evolution is converged enough that the observed logarithmic growth of h(t) at L = 20 and t < about 500 is physical, not an artifact of bond-dimension truncation; the paper asserts the bond dimension grows only linearly in time but provides no explicit convergence tests against larger bond dimensions or exact data.

What would settle it

Run the same MPS time evolution at L = 20, W = 6.5, Delta = 1 with bond dimension chi and 2 chi, and compare h(t) at t = 500; if the logarithmic slope changes by more than the reported error bars, the logarithmic law is a truncation artifact rather than a property of the exact dynamics.

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

If this is right

  • If the central claim is correct, the operator length is a sharp dynamical probe: it grows logarithmically in the MBL regime and saturates in the Anderson-localized regime, cleanly separating the two phases.
  • Because the marginals are computed exactly from the MPO, the full probability distributions of operator mass and length are available without stochastic sampling; this removes a sampling bottleneck for these diagnostics.
  • The logarithmic growth of operator entanglement implies the MPO bond dimension grows only linearly in time in the MBL regime, so long-time simulations remain polynomially efficient and are reliable at the accessible sizes.
  • The proposed experimental protocol — Choi-state preparation, controlled forward/backward evolution, and Bell-pair classical shadows — gives a shot-efficient route to measuring the operator marginals on current quantum platforms.
  • The time-averaged operator length and mass, rescaled by ln L, collapse onto a disorder-dependent curve, providing a clean numerical signature of the MBL logarithmic light cone.

Where Pith is reading between the lines

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

  • Going beyond the paper: if the logarithmic growth is truly unbounded for arbitrarily weak interactions, then in the thermodynamic limit the operator length should eventually saturate only at the system boundary; comparing the observed log slope at different L can test whether the MBL regime survives at asymptotically large sizes or is cut off by rare-region or avalanche effects.
  • Going beyond the paper: the exact marginals could be repurposed as a benchmark for nonstabilizerness estimators, since both are defined on the same Pauli distribution; discrepancies between exact marginals and sampled magic quantities would identify where sampling overhead enters.
  • Going beyond the paper: the same MPO-marginal technique transfers directly to monitored circuits, where the operator length distribution could track measurement-induced entanglement transitions, a direction the paper itself gestures toward.
  • Going beyond the paper: a direct experimental test on a 12-20 qubit device — measuring h(t) via Bell-pair shadows in both Delta = 0 and Delta = 1 disorder — would either confirm or falsify the logarithmic separation at finite times.

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

3 major / 4 minor

Summary. The paper introduces a new Pauli-basis diagnostic, the operator length h(t), complementary to the operator mass m(t), and shows how both can be computed exactly within an MPS/MPO representation of the time-evolved operator, avoiding stochastic sampling. The method is applied to the disordered XXZ chain. For disorder strengths W≳5, the interaction strength Δ=1 gives a clear logarithmic growth of h(t), m(t), and the operator entanglement entropy over several decades in exact diagonalization at L=12 up to t=10^5, while Δ=0 saturates quickly. The authors claim that even arbitrarily weak interactions produce unbounded logarithmic operator spreading and that this is quantitatively captured by a phenomenological ℓ-bit model. An experimental protocol based on Choi-state preparation and Bell-pair classical shadows is also proposed.

Significance. If the conclusions hold, the operator length is a useful, experimentally accessible probe that sharply distinguishes Anderson localization from MBL, complementing the OTOC and operator entanglement. The exact MPO-computation part is an important technical contribution: Eqs. (20)–(21) and the mass generating function in Eqs. (26)–(27) give deterministic, polynomial-cost access to marginals that would otherwise require sampling. The ED data for Δ=1 at strong disorder are clean and support the central logarithmic-growth observation. However, the two most striking claims—arbitrary-weak-interaction relevance and quantitative agreement with the ℓ-bit model—are not established by the presented evidence, and the tensor-network convergence is not documented. These issues are fixable and do not invalidate the core methodology.

major comments (3)
  1. [Sec. 4.2, Fig. 3] The abstract and Sec. 4.2 claim that operator spreading occurs for arbitrarily weak interactions, but the smallest simulated interaction is Δ=0.2. For Δ=0.2, h(t) rises by only about 0.2 over three time decades (Fig. 3b), and the curves have no disorder-averaging error bars and no MPO truncation-error estimate. In that window a slow transient or a truncation-induced drift can masquerade as a logarithmic law. Please add ED data at smaller Δ (e.g., Δ=0.02, 0.05, 0.1) for L=12 to longer times, and include bootstrap or jackknife error bars. Without that, the singular-at-Δ=0 claim is not supported.
  2. [Appendix A, Sec. 4.2] The text says the ℓ-bit model 'quantitatively reproduces' or 'quantitatively captures' the XXZ behavior, but no quantitative comparison is made. The model contains a free coupling-decay parameter κ and its own disorder amplitude W; Appendix A reports slopes for κ=1,0.5,0.32 but does not fix κ from the XXZ localization length and does not compare the slopes to the a(W) extracted from Fig. 1. Please fit the XXZ h(t) curves to a ln t + b and compare the slopes with the ℓ-bit prediction at the same effective κ, or tone the claim down to qualitative agreement.
  3. [Sec. 4.2, Sec. 3] The assertion that in MBL the MPO bond dimension grows only linearly in time, 'enabling faithful simulations', is not accompanied by convergence tests. No comparison is shown between different bond dimensions, different Trotter time steps, or between ED and MPS at L=12 over the MPS time window. Since the Δ=0.2 logarithmic trend covers a very small variation of h(t), truncation could affect the extracted slope. Please add truncation-error plots and an ED/MPS cross-check on h(t) and m(t).
minor comments (4)
  1. [General] Typos and formatting: 'folowing' (Sec. 4.1), 'nonlcal' (end of Sec. 2), 'su bstituting' (Appendix A), 'Choirepresentation' (Sec. 3), and 'emergentℓ-bits' spacing. Please proofread.
  2. [Eq. (37)] The notation h(L,W) and m(L,W) for time averages conflicts with the operator length h(t) and mass m(t); it would be clearer to use something like ⟨h⟩_L(W) or a separate symbol.
  3. [Fig. 1] The figure shows three disorder strengths but no error bars; reporting the standard error over the 48 realizations would help assess the significance of the slope and is standard practice.
  4. [Sec. 5] The experimental section could state the number of shots needed to estimate P(l) and P(m) to a given accuracy; the discussion of statistical efficiency is qualitative.

Circularity Check

0 steps flagged

No significant circularity: the central operator-length result is measured directly in ED/MPS, and the ℓ-bit model is an explicitly phenomenological explanatory model rather than a fitted predictor.

full rationale

The paper's main numerical claim—logarithmic growth of the operator length/mass and operator entanglement in the disordered interacting XXZ chain—is obtained by direct exact-diagonalization (L=12, t up to 10^5) and by MPS time evolution, neither of which assumes the logarithmic law. The ℓ-bit model in Appendix A is presented as an effective model whose exponential couplings J_jl = W_jl e^{-κ|j-l|} are imported from the MBL literature (e.g., Imbrie's analytical results and established ℓ-bit papers), not fitted to the XXZ slopes. No parameter is tuned to reproduce the XXZ h(t) data; the comparison is qualitative ('the same logarithmic behavior'). The self-citations that do appear are for an elementary Pauli-basis fact (footnote [57]) and for previously established ℓ-bit approximations ([26,93,94]); none of these is the load-bearing input of the central derivation. The paper does not claim to derive the logarithmic light cone from first principles, so the fact that the ℓ-bit model encodes exponential interactions and then yields logarithmic spreading is an explanatory consistency check, not a circular reduction of the numerical result to its assumptions.

Axiom & Free-Parameter Ledger

2 free parameters · 4 axioms · 0 invented entities

No new physical entities are postulated; “operator length” is a diagnostic, not a new force or conserved quantity. The main external input is the exponential ℓ-bit coupling, which is a domain assumption with prior-literature support but is not independently derived here. The numeric claims rest additionally on MPS convergence and on the representativeness of finite-time disorder averages.

free parameters (2)
  • κ (ℓ-bit coupling decay rate) = 1, 0.5, 0.32 (chosen in Fig. A1; not mapped to XXZ disorder W)
    Controls the exponential decay J_jl = W_jl e^{-κ|j-l|} in Eq. (A.1); the resulting logarithmic slope scales approximately as 1/κ. No independent determination or mapping from the XXZ parameters is given, so the claimed quantitative agreement is not established.
  • W (amplitude of ℓ-bit random couplings) = 1 (Fig. A1)
    Sets the overall coupling scale in the ℓ-bit model; it affects the time shift but not the logarithmic slope, and is chosen for convenience.
axioms (4)
  • domain assumption The ℓ-bit Hamiltonian Eq. (A.1) with exponentially decaying interactions is a valid effective description of the MBL regime, and n≥3 terms plus onsite fields can be neglected.
    Used in Appendix A to derive the logarithmic growth; imported from Refs. [23, 73–75, 92, 96, 74, 97], some co-authored by the present authors. The exponential coupling form is effectively the target log-light-cone mechanism.
  • domain assumption MPS/MPO truncation with a bond dimension growing linearly in time is faithful for the simulated times and system sizes.
    Invoked in Sec. 4.2 to justify L=20 results to t=500; no convergence checks or truncation-error bounds are reported.
  • domain assumption Disorder-averaged finite-time data (N_r=48–200; t≤10^5 for L=12, t≤500 for MPS) are representative of the asymptotic MBL regime.
    Supports the claim of robust logarithmic growth; the ultimate stability of MBL in the thermodynamic limit is still debated (Refs. [31–36]).
  • standard math Pauli-basis orthonormality, the Choi–Jamiołkowski mapping, and MPS canonical-form identities.
    Standard background used throughout Sec. 3; unproblematic but relied upon for the exact-marginal formulas.

pith-pipeline@v1.3.0-alltime-deepseek · 22484 in / 16265 out tokens · 182436 ms · 2026-08-03T09:47:31.244049+00:00 · methodology

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read the original abstract

We investigate operator delocalization in disordered one-dimensional spin chains by introducing -- besides the already known operator mass -- a complementary measure of operator complexity: the operator length. Like the operator nonstabilizerness, both these quantities are defined from the expansion of time-evolved operators in the Pauli basis. They characterize, respectively, the number of sites on which an operator acts nontrivially and the spatial extent of its support. We show that both the operator mass and length can be computed efficiently and exactly within a matrix-product-state (MPS) framework, providing direct access to their full probability distributions, without resorting to stochastic sampling. Applying this approach to the disordered XXZ spin-1/2 chain, we find sharply distinct behaviors in non-interacting and interacting regimes. In the Anderson-localized case, operator mass, length, and operator entanglement entropy rapidly saturate, signaling the absence of scrambling. By contrast, in the many-body localized (MBL) regime, for arbitrarily weak interactions, all quantities exhibit a robust logarithmic growth in time, consistent with the known logarithmic light cone of quantum-correlation propagation in MBL. We demonstrate that this behavior is quantitatively captured by an effective $\ell$-bit model and persists across system sizes accessible via tensor-network simulations.

Figures

Figures reproduced from arXiv: 2601.12446 by Angelo Russomanno, Davide Rossini, Gianluca Passarelli, Jonnathan Pineda, Mario Collura, Procolo Lucignano.

Figure 1
Figure 1. Figure 1: The average operator length h(t) as a function of the time t, for the model of Eq. (34), for a system with L = 12 sites, as obtained by means of ED computations up to t = 105 . Here we fix ∆ = 1, while the various data sets are for different values of W (see legend). Averages are performed over Nr = 48 disorder realizations. Notice the logarithmic scale on the horizontal axis and the logarithmic growth in … view at source ↗
Figure 2
Figure 2. Figure 2: MPS dynamics of the integrated operator entanglement (a), the operator length h(t) (b), and the operator mass m(t) (c), for L = 12 and L = 20, up to t = 500. Notice the logarithmic increase occurring only for the interacting case ∆ = 1 (dashed lines are guide for the eyes). Averages have been taken over Nr = 200 disorder realizations [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Same as in [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Time-averaged operator length h(L, W) and mass m(L, W), obtained by integrating h(t, W) and m(t, W) over the time window [0, L]. Panel (a): dependence on the system size L, for different fixed disorder strengths. Panel (b): dependence on the disorder W, for fixed system size. Panel (c): rescaled data h(L, W)/ ln L and m(L, W)/ ln L, showing the expected collapse in the many-body localized regime, consisten… view at source ↗

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Reference graph

Works this paper leans on

92 extracted references · 22 canonical work pages

  1. [1]

    Calabrese P 2020SciPost Phys. Lect. Notes20 URLhttps://scipost.org/10. 21468/SciPostPhysLectNotes.20

  2. [2]

    Phys.14988–990 URLhttp://dx.doi.org/10.1038/ s41567-018-0295-5

    Swingle B 2018Nat. Phys.14988–990 URLhttp://dx.doi.org/10.1038/ s41567-018-0295-5

  3. [3]

    Larkin A and Ovchinnikov Y N 1969Sov. Phys. JETP281200

  4. [4]

    High Energ

    Shenker S H and Stanford D 2014J. High Energ. Phys.201467 URLhttp: //dx.doi.org/10.1007/JHEP03(2014)067

    doi:10.1007/jhep03(2014)067 2014

  5. [5]

    Phys.16199–204 URLhttp://dx.doi.org/10

    Xu S and Swingle B 2019Nat. Phys.16199–204 URLhttp://dx.doi.org/10. 1038/s41567-019-0712-4

  6. [6]

    High Energ

    Gu Y, Qi X L and Stanford D 2017J. High Energ. Phys.2017ISSN 1029-8479 URLhttp://dx.doi.org/10.1007/JHEP05(2017)125

    work page doi:10.1007/jhep05(2017)125 2017

  7. [7]

    Song X Y, Jian C M and Balents L 2017Phys. Rev. Lett.119ISSN 1079-7114 URLhttp://dx.doi.org/10.1103/PhysRevLett.119.216601 REFERENCES27

    work page doi:10.1103/physrevlett.119.216601

  8. [8]

    Ben-Zion D and McGreevy J 2018Phys. Rev. B97ISSN 2469-9969 URL http://dx.doi.org/10.1103/PhysRevB.97.155117

    work page doi:10.1103/physrevb.97.155117

  9. [9]

    Chowdhury D and Swingle B 2017Phys. Rev. D96(6) 065005 URLhttps: //link.aps.org/doi/10.1103/PhysRevD.96.065005

    work page doi:10.1103/physrevd.96.065005

  10. [10]

    Patel A A, Chowdhury D, Sachdev S and Swingle B 2017Phys. Rev. X7(3) 031047 URLhttps://link.aps.org/doi/10.1103/PhysRevX.7.031047

    work page doi:10.1103/physrevx.7.031047

  11. [11]

    Nahum A, Ruhman J, Vijay S and Haah J 2017Phys. Rev. X7(3) 031016 URL https://link.aps.org/doi/10.1103/PhysRevX.7.031016

    doi:10.1103/physrevx.7.031016

  12. [13]

    Khemani V, Vishwanath A and Huse D A 2018Phys. Rev. X8(3) 031057 URL https://link.aps.org/doi/10.1103/PhysRevX.8.031057

    doi:10.1103/physrevx.8.031057

  13. [14]

    Rakovszky T, Pollmann F and von Keyserlingk C W 2018Phys. Rev. X8(3) 031058 URLhttps://link.aps.org/doi/10.1103/PhysRevX.8.031058

    doi:10.1103/physrevx.8.031058

  14. [15]

    Alba V and Calabrese P 2017Proc. Natl. Acad. Sci. U.S.A.1147947 URL https://doi.org/10.1073/pnas.1703516114

    doi:10.1073/pnas.1703516114

  15. [16]

    21468/SciPostPhys.4.3.017

    Alba V and Calabrese P 2018SciPost Phys.4017 URLhttps://doi.org/10. 21468/SciPostPhys.4.3.017

  16. [17]

    Swingle B and Chowdhury D 2017Phys. Rev. B95(6) 060201 URLhttps: //link.aps.org/doi/10.1103/PhysRevB.95.060201

    doi:10.1103/physrevb.95.060201

  17. [18]

    Bull.62707–711 ISSN 2095-9273 URLhttps://www.sciencedirect.com/science/article/ pii/S2095927317301925

    Fan R, Zhang P, Shen H and Zhai H 2017Sci. Bull.62707–711 ISSN 2095-9273 URLhttps://www.sciencedirect.com/science/article/ pii/S2095927317301925

    2095

  18. [19]

    Chen Y 2016 Universal logarithmic scrambling in many body localization (Preprint1608.02765) URLhttps://arxiv.org/abs/1608.02765

    Pith/arXiv arXiv 2016

  19. [20]

    Phys.529ISSN 1521-3889 URL http://dx.doi.org/10.1002/andp.201600318

    Huang Y, Zhang Y and Chen X 2016Ann. Phys.529ISSN 1521-3889 URL http://dx.doi.org/10.1002/andp.201600318

    work page doi:10.1002/andp.201600318

  20. [21]

    ˇZnidariˇ c M, Prosen T c v and Prelovˇ sek P 2008Phys. Rev. B77(6) 064426 URL https://link.aps.org/doi/10.1103/PhysRevB.77.064426

    work page doi:10.1103/physrevb.77.064426

  21. [22]

    Bardarson J H, Pollmann F and Moore J E 2012Phys. Rev. Lett.109(1) 017202 URLhttps://link.aps.org/doi/10.1103/PhysRevLett.109.017202

    work page doi:10.1103/physrevlett.109.017202

  22. [23]

    Serbyn M, Papi´ c Z and Abanin D A 2013Phys. Rev. Lett.110(26) 260601 URL https://link.aps.org/doi/10.1103/PhysRevLett.110.260601

    work page doi:10.1103/physrevlett.110.260601

  23. [24]

    De Chiara G, Montangero S, Calabrese P and Fazio R 2006J. Stat. Mech.2006 P03001 URLhttps://doi.org/10.1088/1742-5468/2006/03/P03001

    work page doi:10.1088/1742-5468/2006/03/p03001 2006

  24. [25]

    Deng D L, Li X, Pixley J H, Wu Y L and Das Sarma S 2017Phys. Rev. B95 ISSN 2469-9969 URLhttp://dx.doi.org/10.1103/PhysRevB.95.024202

    work page doi:10.1103/physrevb.95.024202

  25. [27]

    Kim I H, Chandran A and Abanin D A 2014 Local integrals of motion and the logarithmic lightcone in many-body localized systems (Preprint1412.3073) URL https://arxiv.org/abs/1412.3073 REFERENCES28

    Pith/arXiv arXiv 2014

  26. [28]

    Elgart A and Klein A 2024Commun. Math. Phys.405239 URLhttps: //doi.org/10.1007/s00220-024-05127-y

    doi:10.1007/s00220-024-05127-y

  27. [29]

    Toniolo D and Bose S 2024 Stability of slow Hamiltonian dynamics from Lieb- Robinson bounds (Preprint2405.05958) URLhttps://arxiv.org/abs/2405. 05958

    Pith/arXiv arXiv 2024

  28. [30]

    Abanin D A, Altman E, Bloch I and Serbyn M 2019Rev. Mod. Phys.91(2) 021001 URLhttps://link.aps.org/doi/10.1103/RevModPhys.91.021001

    doi:10.1103/revmodphys.91.021001

  29. [31]

    Ha H, Morningstar A and Huse D A 2023Phys. Rev. Lett.130(25) 250405 URL https://link.aps.org/doi/10.1103/PhysRevLett.130.250405

    work page doi:10.1103/physrevlett.130.250405

  30. [32]

    Colbois J, Alet F and Laflorencie N 2024Phys. Rev. Lett.133(11) 116502 URL https://link.aps.org/doi/10.1103/PhysRevLett.133.116502

    doi:10.1103/physrevlett.133.116502

  31. [33]

    ˇSuntajs J, Bonˇ ca J, Prosen T and Vidmar L 2020Phys. Rev. E102(6) 062144 URLhttps://link.aps.org/doi/10.1103/PhysRevE.102.062144

    doi:10.1103/physreve.102.062144

  32. [34]

    ˇSuntajs J, Bonˇ ca J, Prosen T and Vidmar L 2020Phys. Rev. B102(6) 064207 URLhttps://link.aps.org/doi/10.1103/PhysRevB.102.064207

    doi:10.1103/physrevb.102.064207

  33. [35]

    Sels D and Polkovnikov A 2021Phys. Rev. E104(5) 054105 URLhttps: //link.aps.org/doi/10.1103/PhysRevE.104.054105

    doi:10.1103/physreve.104.054105

  34. [37]

    Leone L, Oliviero S F E and Hamma A 2022Phys. Rev. Lett.128(5) 050402 URL https://link.aps.org/doi/10.1103/PhysRevLett.128.050402

    doi:10.1103/physrevlett.128.050402

  35. [38]

    Oliviero S F E, Leone L and Hamma A 2022Phys. Rev. A106(4) 042426 URL https://link.aps.org/doi/10.1103/PhysRevA.106.042426

    doi:10.1103/physreva.106.042426

  36. [39]

    Gottesman D 1998Phys. Rev. A57(1) 127–137 URLhttps://link.aps.org/ doi/10.1103/PhysRevA.57.127

    doi:10.1103/physreva.57.127

  37. [40]

    Gottesman D 1999 Proceedings of the xxii international collo- quium on group theoretical methods in physics ed Corney S P, Delbourgo R and Jarvis P D (Cambridge, MA: International Press) p 32–43

    1999

  38. [41]

    Schuster T and Yao N Y 2023Phys. Rev. Lett.131(16) 160402 URLhttps: //link.aps.org/doi/10.1103/PhysRevLett.131.160402

    doi:10.1103/physrevlett.131.160402

  39. [42]

    Bandyopadhyay J N and Lakshminarayan A 2005 Entangling power of quantum chaotic evolutions via operator entanglement (Preprintquant-ph/0504052) URL https://arxiv.org/abs/quant-ph/0504052

    arXiv 2005

  40. [43]

    Prosen T c v and Piˇ zorn I 2007Phys. Rev. A76(3) 032316 URLhttps: //link.aps.org/doi/10.1103/PhysRevA.76.032316

    doi:10.1103/physreva.76.032316

  41. [44]

    Piˇ zorn I and Prosen T c v 2009Phys. Rev. B79(18) 184416 URLhttps: //link.aps.org/doi/10.1103/PhysRevB.79.184416

    doi:10.1103/physrevb.79.184416

  42. [45]

    Swingle B, Bentsen G, Schleier-Smith M and Hayden P 2016Phys. Rev. A94(4) 040302 URLhttp://dx.doi.org/10.1103/PhysRevA.94.040302

    doi:10.1103/physreva.94.040302

  43. [46]

    Zhu G, Hafezi M and Grover T 2016Phys. Rev. A94(6) 062329 URLhttps: //link.aps.org/doi/10.1103/PhysRevA.94.062329

    work page doi:10.1103/physreva.94.062329

  44. [47]

    Yao N Y, Grusdt F, Swingle B, Lukin M D, Stamper-Kurn D M, Moore J E and Demler E A 2016 Interferometric approach to probing fast scrambling (Preprint 1607.01801) URLhttps://arxiv.org/abs/1607.01801 REFERENCES29

    Pith/arXiv arXiv 2016

  45. [48]

    Yunger Halpern N 2017Phys. Rev. A95(1) 012120 URLhttps://link.aps. org/doi/10.1103/PhysRevA.95.012120

    work page doi:10.1103/physreva.95.012120

  46. [49]

    Yunger Halpern N, Swingle B and Dressel J 2018Phys. Rev. A97(4) 042105 URLhttp://dx.doi.org/10.1103/PhysRevA.97.042105

    doi:10.1103/physreva.97.042105

  47. [50]

    Campisi M and Goold J 2017Phys. Rev. E95(6) 062127 URLhttps://link. aps.org/doi/10.1103/PhysRevE.95.062127

    work page doi:10.1103/physreve.95.062127

  48. [51]

    Phys.13781 URLhttps://link.aps.org/doi/10.1038/nphys4119

    G¨ arttner M, Bohnet J, Safavi-Naini A, Wall M L, Bollinger J J and Reay A M 2017Nat. Phys.13781 URLhttps://link.aps.org/doi/10.1038/nphys4119

    doi:10.1038/nphys4119

  49. [52]

    Li J, Fan R, Wang H, Ye B, Zeng B, Zhai H, Peng X and Du J 2017Phys. Rev. X 7(3) 031011 URLhttps://link.aps.org/doi/10.1103/PhysRevX.7.031011

    doi:10.1103/physrevx.7.031011

  50. [53]

    Choi M D 1975Linear Algebra Appl.10285

  51. [54]

    Jamio lkowski A 1972Rep. Math. Phys.3275

  52. [55]

    Phys.161050 URLhttps: //doi.org/10.1038/s41567-020-0932-7

    Huang H Y, Kueng R and Preskill J 2020Nat. Phys.161050 URLhttps: //doi.org/10.1038/s41567-020-0932-7

    doi:10.1038/s41567-020-0932-7

  53. [56]

    Aaronson S 2018 Shadow tomography of quantum statesProceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing (STOC)pp 325–338

    2018

  54. [57]

    Russomanno A, Passarelli G, Rossini D and Lucignano P 2025Physical Review B112ISSN 2469-9969 URLhttp://dx.doi.org/10.1103/njgn-fksh

    work page doi:10.1103/njgn-fksh

  55. [59]

    Nahum A, Ruhman J, Vijay S and Haah J 2018Phys. Rev. X8(2) 021014 URL https://doi.org/10.1103/PhysRevX.8.021014

    doi:10.1103/physrevx.8.021014

  56. [60]

    Lami G and Collura M 2023Phys. Rev. Lett.131(18) 180401 URLhttps: //link.aps.org/doi/10.1103/PhysRevLett.131.180401

    doi:10.1103/physrevlett.131.180401

  57. [61]

    Haug T and Piroli L 2023Quantum71092 ISSN 2521-327X URLhttps: //doi.org/10.22331/q-2023-08-28-1092

    doi:10.22331/q-2023-08-28-1092 2023

  58. [62]

    Adhikary M, Ranabhat N and Collura M 2025 Quantum complexity across thermal phase transition in the transverse field ising chain with long-range couplings (Preprint2511.13667) URLhttps://arxiv.org/abs/2511.13667

    arXiv 2025

  59. [63]

    Collura M, Lami G, Ranabhat N and Santini A 2024Tensor Network Techniques for Quantum Computation(SISSA Medialab S.r.l.) ISBN 9788898587049 URL https://library.oapen.org/handle/20.500.12657/96026

  60. [64]

    von Keyserlingk C, Rakovszky T, Pollmann F and Sondhi S L 2018Phys. Rev. X8(2) 021013 URLhttps://doi.org/10.1103/PhysRevX.8.021013

    doi:10.1103/physrevx.8.021013

  61. [65]

    Jonay C, Huse D A and Nahum A 2018 Coarse-grained dynamics of operator and state entanglement (Preprint1803.00089) URLhttps://arxiv.org/abs/ 1803.00089

    Pith/arXiv arXiv 2018

  62. [66]

    Bertini B, Kos P and Prosen T 2020SciPost Physics8ISSN 2542-4653 URL http://dx.doi.org/10.21468/SciPostPhys.8.4.067

    work page doi:10.21468/scipostphys.8.4.067

  63. [67]

    Muth D, Unanyan R G and Fleischhauer M 2011Physical Review Letters106 ISSN 1079-7114 URLhttp://dx.doi.org/10.1103/PhysRevLett.106.077202

    work page doi:10.1103/physrevlett.106.077202

  64. [68]

    Alba V, Dubail J and Medenjak M 2019Phys. Rev. Lett.122(25) 250603 URL https://link.aps.org/doi/10.1103/PhysRevLett.122.250603 REFERENCES30

    doi:10.1103/physrevlett.122.250603

  65. [69]

    Essler F H L and Fagotti M 2016Journal of Statistical Mechanics: Theory and Experiment2016064002 ISSN 1742-5468 URLhttp://dx.doi.org/10.1088/ 1742-5468/2016/06/064002

    2016

  66. [70]

    Prosen T c v and ˇZnidariˇ c M 2007Phys. Rev. E75(1) 015202 URLhttps: //link.aps.org/doi/10.1103/PhysRevE.75.015202

    work page doi:10.1103/physreve.75.015202

  67. [71]

    Dubail J 2017Journal of Physics A: Mathematical and Theoretical50234001 URLhttps://doi.org/10.1088/1751-8121/aa6f38

    doi:10.1088/1751-8121/aa6f38

  68. [72]

    Zhou T and Luitz D J 2017Phys. Rev. B95(9) 094206 URLhttps://link. aps.org/doi/10.1103/PhysRevB.95.094206

    doi:10.1103/physrevb.95.094206

  69. [73]

    Huse D A, Nandkishore R and Oganesyan V 2014Phys. Rev. B90(17) 174202 URLhttps://link.aps.org/doi/10.1103/PhysRevB.90.174202

    doi:10.1103/physrevb.90.174202

  70. [74]

    Serbyn M, Papi´ c Z and Abanin D A 2013Phys. Rev. Lett.111(12) 127201 URL https://link.aps.org/doi/10.1103/PhysRevLett.111.127201

    work page doi:10.1103/physrevlett.111.127201

  71. [75]

    Nandkishore R and Huse D A 2015Annu. Rev. Condens. Matter Phys.615–38 ISSN 1947-5462 URLhttp://dx.doi.org/10.1146/ annurev-conmatphys-031214-014726

    1947

  72. [76]

    Mbeng G B, Russomanno A and Santoro G E 2024SciPost Physics Lecture Notes ISSN 2590-1990 URLhttp://dx.doi.org/10.21468/SciPostPhysLectNotes. 82

    doi:10.21468/scipostphyslectnotes 1990

  73. [77]

    Franchini F 2017An Introduction to Integrable Techniques for One-Dimensional Quantum Systems(Springer International Publishing) ISBN 9783319484877 URL http://dx.doi.org/10.1007/978-3-319-48487-7

    doi:10.1007/978-3-319-48487-7

  74. [78]

    Qi X L, Davis E J, Periwal A and Schleier-Smith M 2019 Measuring operator size growth in quantum quench experiments (Preprint1906.00524) URLhttps: //arxiv.org/abs/1906.00524

    Pith/arXiv arXiv 2019

  75. [79]

    Arute Fet al.2019Nature574505 URLhttps://doi.org/10.1038/ s41586-019-1666-5

  76. [80]

    Technol.6025020 URLhttps://doi.org/ 10.1088/2058-9565/abe519

    Jurcevic Pet al.2021Quantum Sci. Technol.6025020 URLhttps://doi.org/ 10.1088/2058-9565/abe519

    doi:10.1088/2058-9565/abe519 2058

  77. [81]

    Monroe Cet al.2021Rev. Mod. Phys.93(2) 025001 URLhttps://link.aps. org/doi/10.1103/RevModPhys.93.025001

    doi:10.1103/revmodphys.93.025001

  78. [82]

    1038/s41586-024-07459-0

    Guo S A, Wu Y K, Ye Jet al.2024Nature630613 URLhttps://doi.org/10. 1038/s41586-024-07459-0

  79. [83]

    1126/science.aah3778) URLhttps://www.science.org/doi/abs/10.1126/ science.aah3778

    Barredo D, de L´ es´ eleuc S, Lienhard V, Lahaye T and Browaeys A 2016 Science3541021–1023 (Preprinthttps://www.science.org/doi/pdf/10. 1126/science.aah3778) URLhttps://www.science.org/doi/abs/10.1126/ science.aah3778

    2016

  80. [84]

    Manetsch H J, Nomura G, Bataille E, Lv X, Leung K H and Endres M 2025 Nature64760 URLhttps://doi.org/10.1038/s41586-025-09641-4

    doi:10.1038/s41586-025-09641-4 2025

Showing first 80 references.