Pith. sign in

REVIEW 2 major objections 3 minor 9 cited by

A 'boundary scrambling' circuit makes the full dynamics of all higher-order OTOCs exactly solvable in the thermodynamic limit, its free cumulants matching full eigenstate-thermalization predictions.

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 ↗

A deterministic Floquet circuit with a dual-unitary bulk exactly reproduces random-circuit higher-order OTOC dynamics, yielding analytic free cumulants that match full eigenstate thermalization hypothesis predictions.

T0 review reviewed 2026-08-04 challenge →

load-bearing objection A new exactly solvable boundary scrambling model with correct 2-OTOC dynamics; the higher-order stability proof needs one more honest step before the strongest claims are taken at face value. the 2 major comments →

arxiv 2509.08060 v1 pith:INCN2IPM submitted 2025-09-09 quant-ph cond-mat.stat-mechmath-phmath.MPnlin.CDnlin.SI

Free Cumulants and Full Eigenstate Thermalization from Boundary Scrambling

Felix Fritzsch , Gabriel O. Alves , Michael A. Rampp , Pieter W. Claeys This is my paper
classification quant-ph cond-mat.stat-mechmath-phmath.MPnlin.CDnlin.SI
keywords boundary scramblingfree cumulantsout-of-time-order correlatorsfull eigenstate thermalizationdual-unitary circuitsinfluence matrixfree probabilitynoncrossing permutations
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

This paper introduces 'boundary scrambling': a one-dimensional Floquet circuit whose bulk is a zig-zag chain of dual-unitary gates (unitary in both space and time), with the leftmost site coupled through one freely chosen gate. The authors claim that in the thermodynamic limit the full dynamics of all higher-order out-of-time-order correlators (OTOCs) of boundary observables becomes exactly solvable, decomposing into free cumulants — connected correlation functions from free probability. These dynamical free cumulants coincide exactly with the eigenstate-based predictions of the 'full' eigenstate thermalization hypothesis (ETH) ansatz, and they also reproduce the known solvable random-circuit model, so no randomness or averaging is needed. All OTOCs decay at the same rate, twice as fast as the time-ordered two-point function, with the first analytical predictions for frequency-resolved, higher-order correlations between eigenstate matrix elements. The exact results remain stable when the bulk is perturbed away from dual-unitarity, a property the authors prove through a projection identity for a higher-order Markovian influence matrix. If correct, this is the first structured, experimentally realizable setting in which eigenstate statistics and dynamical scrambling are two views of the same free-cumulant structure.

Core claim

In the thermodynamic limit, the boundary scrambling circuit reproduces, without randomness, exactly the higher-order influence matrix found earlier in a solvable random-circuit model. Consequently every k-OTOC of the boundary observable is exactly computable and decomposes into free cumulants. For the 2-OTOC with a, b eigenoperators of the quantum channel at eigenvalue λ, C2(t) = λ^(2t) tr(abab)/d + t λ^(2(t−1))[(■|M□◦|•) − λ²(■|•)/d], splitting into free cumulants k2(t) ∝ λ^t and k4(t) ∝ λ^(2t) plus a linear-in-time Jordan correction. These dynamical free cumulants coincide with the eigenstate-based full-ETH sums over pairwise distinct quasienergy eigenstates, verified by exact diagonalizat

What carries the argument

The carrying object is the higher-order influence matrix |I): the state obtained by contracting the folded bulk circuit in space and projecting the right boundary onto the eigenvalue-1 eigenspace of the spatial transfer matrix T, spanned by multichains of noncrossing permutations (for the 2-OTOC, domain walls between the identity and swap permutations). It acts as a Markovian bath for the boundary site, and at order k its bond dimension equals the k-th Catalan number. The load-bearing identity is PTP|I) = |I): for generic unitary gates the transfer matrix projected onto the multichain space keeps |I) as a unit eigenvector, giving stability away from dual-unitarity. Its proof runs through a l

Load-bearing premise

The load-bearing premise is a combinatorial lemma (App. A): sums of Möbius functions over equivalence classes of noncrossing permutations vanish unless singletons propagate. The paper demonstrates it with a single worked example (ℓ = 6) said to 'illustrate the general idea', with no fully general algebraic proof written out, so the projection identity PTP|I) = |I) and the claimed stability away from dual-unitarity rest on it. It also assumes, following earlier work, that the

What would settle it

Check the lemma in an unshown case: fix ℓ = 8 with a singleton, sum the Möbius-function terms over all σ1 in the same equivalence class below a fixed σ2, and verify the sum is exactly zero unless the singleton is in σ2; any nonzero residue falsifies Eq. (A15) and with it the projection identity behind the stability claim. Alternatively, simulate the boundary scrambler with bulk Trotter step τ = π/5 (non-dual-unitary) and compare the measured 2-OTOC against Eq. (20): the paper predicts deviations that stay small and grow smoothly with time, so a deviation of order one at short times would falsi

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

If this is right

  • Every k-OTOC between a boundary observable and the bath decays at the same rate, twice as fast as the time-ordered two-point function, with closed-form prefactors and time scales given by Eqs. (20)–(21).
  • Dynamical free cumulants and eigenstate-based full-ETH free cumulants coincide exactly, giving the first structured model where the two definitions provably match; exact diagonalization confirms this for the 2-OTOC.
  • Frequency-resolved free cumulants take explicit Lorentzian-type forms (Eq. 22), yielding the first analytical predictions for higher-order correlations between eigenstate matrix elements — correlations that vanish under the standard i.i.d.-Gaussian ETH assumption.
  • A(t) and B become asymptotically free in the double limit L→∞ then t→∞, so all higher-order OTOCs vanish — an operational signature of maximal scrambling.
  • The exact OTOC dynamics is stable under small deviations from dual-unitarity: the influence matrix remains a unit eigenvector of the projected transfer matrix, with corrections suppressed by the spectral gap of T and confirmed numerically.

Where Pith is reading between the lines

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

  • Because the paper proves the projection identity PTP|I) = |I) for generic unitary gates, not only near dual-unitarity, the residual error is the leakage into non-multichain states; deriving a quantitative bound on that leakage from the spectral gap of T — a step the authors do not take — would turn their perturbative stability statement into a rigorous estimate.
  • If the Lorentzian functional forms of k2(ω) and k4(ω) are the universal line shapes of full-ETH cumulants, as the paper suspects, then measuring these two quantities in other ergodic Floquet systems (for example kicked chains) would test whether boundary scrambling belongs to the same universality class as generic quantum chaos.
  • The model cleanly separates the scrambling layer (the dual-unitary bulk) from the observable (the free boundary gate U, which tunes the decay rate λ), so on current digital quantum processors the boundary scrambler could serve as a deterministic, low-depth stand-in for random circuits in freeness-based scrambling benchmarks — a direction the authors mention but do not develop into a concrete proto
Share X Bluesky LinkedIn Reddit HN

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

2 major / 3 minor

Summary. The paper introduces 'boundary scrambling,' a Floquet circuit in which a boundary qudit interacts with a bulk of dual-unitary gates in a zig-zag layout. It claims that, in the thermodynamic limit, the bulk can be compressed into a Markovian influence matrix built from noncrossing permutations, so that all k-OTOCs have exact dynamics governed by free cumulants. For the 2-OTOC it derives explicit formulas C2(t) (Eq. 20), k2(t), k4(t) (Eq. 21), and frequency-resolved versions (Eq. 22), confirmed by finite-size numerics and exact diagonalization using full-ETH expressions. The paper further claims that this structure is stable under perturbations away from dual-unitarity, via the projected transfer-matrix identity PT P|I)=|I) (Eq. 25).

Significance. Conditional on the proof being completed, this is a significant advance: it provides a deterministic, non-random model with exact multi-point OTOC dynamics, unifying full-ETH free cumulants and random-circuit influence matrices, and giving explicit frequency-domain predictions for eigenstate matrix-element correlations. The 2-OTOC derivation is a genuine strength: Eq. (20) is derived from an explicit Jordan-block calculation, and Fig. 2 verifies both time- and frequency-domain formulas against exact diagonalization. The paper also honestly flags the non-Hermitian subtlety that prevents an exact eigenvector conclusion in App. A. However, the advertised higher-order and stability results currently rest on an unfinished combinatorial proof and on imported statements from Ref. [40].

major comments (2)
  1. [Appendix A, Eqs. (A15) and (A12)] The theorem (A11) is not established as written. Lemma (A15) as printed has no dependence on nu_1 or the gate contraction, and is false in that form: for k=2, sigma_2=cycle and n=1, the sum over sigma_1 in {identity, cycle} equals d^{-2}-d^{-3} != 0 although n is not a singleton in sigma_2. The proof inserts loop prefactors from Eq. (A17) that are absent from the lemma statement. Eq. (A12) likewise has a nu-independent LHS and nu-dependent RHS. Since Eq. (A14) and Eq. (25) depend on this theorem, the stability claim is unsupported as written. Please restate the identities with all contraction prefactors and prove that every equivalence class reduces to the single-cycle split leading to (A22).
  2. [Section V and Appendix A (use of Ref. [40])] The k>2 free-cumulant dynamics is imported from Ref. [40]. Section V states that the influence matrix 'returns the random-circuit influence matrix at all orders k' and that asymptotic k-OTOC dynamics follows from leading eigenmodes 'argued in Ref. [40]'. Ref. [40] is a preprint by two of the present authors, and the relevant statements are not reproduced or proved here. The paper's headline claim about 'full dynamics of higher-order OTOCs' is therefore conditional on an external text. The authors should either include the necessary theorem/proof for the generator eigenmodes or explicitly mark these statements as assumptions from Ref. [40].
minor comments (3)
  1. [Appendix D, text after Fig. 3] The phrase 'small peak at tau approx 5' should read 't approx 5'; tau denotes the Trotter step and is not a time variable.
  2. [Section V, first paragraph] The statement that 'all higher-order OTOCs decay with the same rate, twice as fast as the time-ordered two-point correlation function' should be qualified as a late-time/asymptotic statement; the exact 2-OTOC in Eq. (20) contains both lambda^{2t} and t lambda^{2(t-1)} contributions, and the free cumulant decomposition in Eq. (21) has an intermediate regime.
  3. [Appendix D, Fig. 3] The inset labeled 'var ctilde_2(t)/C_2(t)' is described as a 'relative variance'; please define this quantity explicitly, since the text elsewhere refers to 'fluctuations' over circuit realizations.

Circularity Check

1 steps flagged

2-OTOC derivation is self-contained and benchmarked, but the advertised all-order free-cumulant dynamics is imported from the authors' own Ref. [40].

specific steps
  1. self citation load bearing [Section V (Higher-order OTOCs and stability), first two paragraphs]
    "Crucially, the influence matrix of the bulk returns the random-circuit influence matrix at all orders k, such that the results of Ref. [40] can be directly applied, again circumventing the need for randomness and averaging. ... The generator G of the dynamical semigroup can be similarly extended, and the asymptotic k-OTOC dynamics follows from the leading eigenmodes of this generator, argued in Ref. [40] to correspond to free cumulants."

    For k>2 the paper does not independently derive the central spectral statement that leading eigenmodes of the extended generator G yield free cumulants; it transfers this statement from Ref. [40], prior work by two of the present authors. The advertised all-order prediction therefore reduces, at this step, to accepting the same-authors' earlier result. Ref. [40] is not machine-checked, code-reproduced, or independently replicated in this manuscript, so it functions as a load-bearing self-citation rather than as an external proof. The k=2 content of the paper is not affected, which is why the overall circularity score is moderate rather than maximal.

full rationale

The explicit 2-OTOC results (Eq. 20), the free-cumulant decomposition (Eq. 21), and the frequency-domain forms (Eq. 22) are derived in this manuscript from the folded-circuit transfer matrix and the Jordan decomposition of G (App. B), then verified against exact-diagonalization full-ETH predictions (Eq. 3) and finite-size numerics (Fig. 2). No fitted parameter is relabeled as a prediction. The identification of the influence matrix with the projected boundary (Eq. A13) is supported by a proof attempt in App. A; the general proof of the key Lemma (A15) is, however, only illustrated on a single ℓ=6 example ('This example illustrates the general idea'), so the perturbative-stability corollary (Eq. A14) carries a rigor gap—this is an incomplete proof, not circularity. The genuine circularity-sensitive point is the claimed extension to arbitrary k-OTOCs: Section V explicitly invokes Ref. [40] (same authors) for both the random-circuit influence matrix at all orders and the correspondence of leading eigenmodes to free cumulants. Since that prior work is not independently verified here, a load-bearing part of the all-order claim rests on self-citation. The central 2-OTOC claim remains independently content and externally benchmarked, so a moderate score of 4 is appropriate rather than a higher forced-by-construction score.

Axiom & Free-Parameter Ledger

1 free parameters · 5 axioms · 0 invented entities

The derivation is mostly self-contained for the 2-OTOC, with the standard machinery of dual-unitarity and influence matrices. The main imported assumptions are the full-ETH free-cumulant decomposition, the multichain eigenbasis of the transfer matrix, and the dominance of a single channel eigenvalue λ. The general-order extension relies on the authors' prior random-circuit paper (Ref. [40]).

free parameters (1)
  • λ (eigenvalue of the quantum channel M of the boundary gate U) = ≈ 0.9 in the numerics
    The boundary gate U is chosen (Hermitian, randomly generated) so that the channel M has a single nontrivial eigenvalue λ ≈ 0.9. The analytical formulas hold for general λ; this is a model parameter, not fitted to the target data.
axioms (5)
  • domain assumption Full ETH free-cumulant decomposition of k-OTOCs: C2 = k4 + 2 k2^2 and Eq. (3) for k2, k4 in terms of eigenstate matrix elements
    Taken from Refs. [21,23,26,27]; used in Section III to define the quantities compared to the dynamical calculation.
  • standard math At the dual-unitary point the eigenvalue-1 eigenspace of the spatial transfer matrix T is spanned by the noncrossing-permutation multichain states
    Established in Refs. [37,54-56] and used in Section III.A and App. A as the basis for projecting the right boundary onto the influence matrix.
  • domain assumption Bulk gates are dual-unitary (unitary in both time and space)
    Model definition, Section II; enables the graphical identities (5) that compress the bulk into the influence matrix.
  • domain assumption The thermodynamic limit L→∞ can be taken by projecting T onto its leading eigenspace; corrections away from dual-unitarity are suppressed by the gap of T
    Used in Sections III.A and V; the non-Hermitian subtlety is disclosed in App. A, so the stability is approximate rather than exact.
  • domain assumption Generic observables are dominated at late times by the leading eigenmode of the quantum channel M with eigenvalue λ
    Used in Section III.B to derive C2(t) = λ^(2t)...; for observables overlapping multiple eigenmodes, prefactors change but the form is argued to persist.

reviewed 2026-08-04 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Free Cumulants and Full Eigenstate Thermalization from Boundary Scrambling." pith.science (2026). https://pith.science/paper/INCN2IPM

@misc{pith2026250908060,
  author       = {Pith},
  title        = {Pith review of: Free Cumulants and Full Eigenstate Thermalization from Boundary Scrambling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/INCN2IPM}},
  note         = {Machine review of arXiv:2509.08060}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

Out-of-time-order correlation functions (OTOCs) and their higher-order generalizations present important probes of quantum information dynamics and scrambling. We introduce a solvable many-body quantum model, which we term boundary scrambling, for which the full dynamics of higher-order OTOCs is analytically tractable. These dynamics support a decomposition into free cumulants and unify recent extensions of the eigenstate thermalization hypothesis with predictions from random quantum circuit models. We obtain exact expressions for (higher-order) correlations between matrix elements and show these to be stable away from the solvable point. The solvability is enabled by the identification of a higher-order Markovian influence matrix, capturing the effect of the full system on a local subsystem. These results provide insight into the emergence of random-matrix behavior from structured Floquet dynamics and show how techniques from free probability can be applied in the construction of exactly-solvable many-body models.

Figures

Figures reproduced from arXiv: 2509.08060 by Felix Fritzsch, Gabriel O. Alves, Michael A. Rampp, Pieter W. Claeys.

Figure 1
Figure 1. Figure 1: FIG. 1. Tensor network representation of (a) the boundary [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Dynamics of the 2-OTOC and its decompo [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Comparison of numerical realizations [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Numerical analysis for typicality of a finite-size realization in the boundary scrambler. (a) Numerical 2-OTOC (square [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗

discussion (0)

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

Forward citations

Cited by 9 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Mimicry of chaos and $k$-design in higher order OTOCs of Luttinger liquids

    cond-mat.str-el 2026-06 unverdicted novelty 7.0

    Higher-order OTOCs in Luttinger liquids map exactly onto the partition function of a non-Hermitian Harper model and become parametrically small up to seventh order for moderate interactions, mimicking k-design behavior.

  2. Scar Full Eigenstate Thermalization Hypothesis

    quant-ph 2026-05 unverdicted novelty 7.0

    The authors introduce the scar full ETH to capture correlations among matrix elements involving scar states, establish its scaling and factorization via typicality, reorganize multi-time correlations with thermal and ...

  3. Page Curve for Local-Operator Entanglement from Free Probability

    quant-ph 2026-05 unverdicted novelty 7.0

    LOE for Haar random dynamics asymptotically matches the Page curve for traceless operators and is independent of the initial operator at leading order.

  4. Timescales for Deep and Full Thermalization

    quant-ph 2026-04 unverdicted novelty 6.0

    In a chaotic quantum system, higher-order correlations reach thermal equilibrium faster than state design moments, both relaxing exponentially.

  5. Quantum many-body operator cascade as a route to chaos

    cond-mat.stat-mech 2026-04 unverdicted novelty 6.0

    Local operators in quantum chaotic systems cascade toward non-local fractal structures whose dimension is tied by unitarity to the decay rate of local correlations, demonstrated exactly in dual-unitary circuits and nu...

  6. Scaling of free cumulants in closed system-bath setups

    cond-mat.stat-mech 2025-11 unverdicted novelty 6.0

    Microcanonical free cumulants of central-system observables scale universally with system-bath interaction strength in closed setups, connecting to thermalization dynamics.

  7. Quantum speed limit for the OTOC from an open systems perspective

    quant-ph 2025-09 unverdicted novelty 6.0

    Derives a quantum speed limit for the OTOC decay rate by mapping scrambling to open-system decoherence bounded by system-environment coupling strength and environmental correlation functions.

  8. Refinements of the Eigenstate Thermalization Hypothesis under Local Rotational Invariance via Free Probability

    cond-mat.stat-mech 2025-11 conditional novelty 5.0

    Under local rotational invariance, the leading factorization of ETH matrix-element correlations is refined by local free cumulants attached to neighboring non-crossing partitions, confirmed numerically in a Floquet sp...

  9. Lecture Notes on Replica Tensor Networks for Random Quantum Circuits

    quant-ph 2026-05 unverdicted novelty 2.0

    Lecture notes and accompanying library teach replica tensor network methods to compute circuit-averaged observables in random quantum circuits by mapping them to classical statistical mechanics models.

Reference graph

Works this paper leans on

81 extracted references · 47 canonical work pages · cited by 9 Pith papers · 1 internal anchor

  1. [1]

    Srednicki, The approach to thermal equilibrium in quantized chaotic systems, J

    M. Srednicki, The approach to thermal equilibrium in quantized chaotic systems, J. Phys. A32, 1163 (1999)

    1999

  2. [2]

    J. M. Deutsch, Eigenstate thermalization hypothesis, Rep. 7 Prog. Phys.81, 082001 (2018)

    work page 2018

  3. [3]

    D’Alessio, Y

    L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Adv. Phys. 65, 239 (2016)

    2016

  4. [4]

    S. H. Shenker and D. Stanford, Black holes and the but- terfly effect, J. High Energ. Phys.2014(3), 67

    work page 2014

  5. [5]

    Maldacena, S

    J. Maldacena, S. H. Shenker, and D. Stanford, A bound on chaos, J. High Energy Phys.2016(8)

    work page 2016

  6. [6]

    Swingle, G

    B. Swingle, G. Bentsen, M. Schleier-Smith, and P. Hayden, Measuring the scrambling of quantum information, Phys. Rev. A94, 040302 (2016)

    2016

  7. [7]

    Swingle, Unscrambling the physics of out-of-time-order correlators, Nat

    B. Swingle, Unscrambling the physics of out-of-time-order correlators, Nat. Phys.14, 988 (2018)

    2018

  8. [8]

    Rakovszky, F

    T. Rakovszky, F. Pollmann, and C. W. von Keyserlingk, Diffusive Hydrodynamics of Out-of-Time-Ordered Corre- lators with Charge Conservation, Phys. Rev. X8, 031058 (2018)

    work page 2018

  9. [9]

    C. W. von Keyserlingk, T. Rakovszky, F. Pollmann, and S. L. Sondhi, Operator Hydrodynamics, OTOCs, and Entanglement Growth in Systems without Conservation Laws, Phys. Rev. X8, 021013 (2018)

    2018

  10. [10]

    Nahum, S

    A. Nahum, S. Vijay, and J. Haah, Operator Spreading in Random Unitary Circuits, Phys. Rev. X8, 021014 (2018)

    2018

  11. [11]

    Khemani, A

    V. Khemani, A. Vishwanath, and D. A. Huse, Operator spreading and the emergence of dissipative hydrodynamics under unitary evolution with conservation laws, Phys. Rev. X8, 031057 (2018)

    2018

  12. [12]

    G¨ arttner, J

    M. G¨ arttner, J. G. Bohnet, A. Safavi-Naini, M. L. Wall, J. J. Bollinger, and A. M. Rey, Measuring out- of-time-order correlations and multiple quantum spectra in a trapped-ion quantum magnet, Nature Phys13, 781 (2017)

    work page 2017

  13. [13]

    J. Li, R. Fan, H. Wang, B. Ye, B. Zeng, H. Zhai, X. Peng, and J. Du, Measuring Out-of-Time-Order Correlators on a Nuclear Magnetic Resonance Quantum Simulator, Phys. Rev. X7, 031011 (2017)

    work page 2017

  14. [14]

    E. J. Meier, J. Ang’ong’a, F. A. An, and B. Gadway, Exploring quantum signatures of chaos on a Floquet syn- thetic lattice, Phys. Rev. A100, 013623 (2019)

    work page 2019

  15. [15]

    K. X. Wei, P. Peng, O. Shtanko, I. Marvian, S. Lloyd, C. Ramanathan, and P. Cappellaro, Emergent Prethermal- ization Signatures in Out-of-Time Ordered Correlations, Phys. Rev. Lett.123, 090605 (2019)

    work page 2019

  16. [16]

    K. A. Landsman, C. Figgatt, T. Schuster, N. M. Linke, B. Yoshida, N. Y. Yao, and C. Monroe, Verified quantum information scrambling, Nat567, 61 (2019)

    work page 2019

  17. [17]

    M. Blok, V. Ramasesh, T. Schuster, K. O’Brien, J. Kreike- baum, D. Dahlen, A. Morvan, B. Yoshida, N. Yao, and I. Siddiqi, Quantum Information Scrambling on a Super- conducting Qutrit Processor, Phys. Rev. X11, 021010 (2021)

    work page 2021

  18. [18]

    X. Mi, P. Roushan, C. Quintana, S. Mandr` a, J. Marshall, et al., Information scrambling in quantum circuits, Sci 374, 1479 (2021)

    work page 2021

  19. [19]

    Braum¨ uller, A

    J. Braum¨ uller, A. H. Karamlou, Y. Yanay, B. Kannan, D. Kim, M. Kjaergaard, A. Melville, B. M. Niedzielski, Y. Sung, A. Veps¨ al¨ ainen, R. Winik, J. L. Yoder, T. P. Orlando, S. Gustavsson, C. Tahan, and W. D. Oliver, Probing quantum information propagation with out-of- time-ordered correlators, Nat. Phys.18, 172 (2022)

    work page 2022

  20. [20]

    D. A. Abanin, R. Acharya, L. Aghababaie-Beni, G. Aigeldinger, A. Ajoy,et al., Constructive inter- ference at the edge of quantum ergodic dynamics, arXiv:2506.10191 (2025)

    Pith/arXiv arXiv 2025

  21. [21]

    Foini and J

    L. Foini and J. Kurchan, Eigenstate thermalization hy- pothesis and out of time order correlators, Phys. Rev. E 99, 042139 (2019)

    2019

  22. [22]

    Brenes, S

    M. Brenes, S. Pappalardi, M. T. Mitchison, J. Goold, and A. Silva, Out-of-time-order correlations and the fine structure of eigenstate thermalization, Phys. Rev. E104, 034120 (2021)

    work page 2021

  23. [23]

    Pappalardi, L

    S. Pappalardi, L. Foini, and J. Kurchan, Eigenstate Ther- malization Hypothesis and Free Probability, Phys. Rev. Lett.129, 170603 (2022)

    work page 2022

  24. [24]

    Jindal and P

    S. Jindal and P. Hosur, Generalized Free Cumulants for Quantum Chaotic Systems, arXiv:2401.13829 (2024)

    Pith/arXiv arXiv 2024

  25. [25]

    Vallini and S

    E. Vallini and S. Pappalardi, Long-time Freeness in the Kicked Top, arXiv:2411.12050 (2024)

    Pith/arXiv arXiv 2024

  26. [26]

    Pappalardi, F

    S. Pappalardi, F. Fritzsch, and T. Prosen, Full Eigenstate Thermalization via Free Cumulants in Quantum Lattice Systems, Phys. Rev. Lett.134, 140404 (2025)

    work page 2025

  27. [27]

    M. Fava, J. Kurchan, and S. Pappalardi, Designs via Free Probability, Phys. Rev. X15, 011031 (2025)

    2025

  28. [28]

    G. O. Alves, F. Fritzsch, and P. W. Claeys, Probes of Full Eigenstate Thermalization in Ergodicity-Breaking Quantum Circuits, arXiv:2504.08517 (2025)

    arXiv 2025

  29. [29]

    Fritzsch, T

    F. Fritzsch, T. Prosen, and S. Pappalardi, Microcanonical free cumulants in lattice systems, Phys. Rev. B111, 054303 (2025)

    work page 2025

  30. [30]

    Voiculescu, Free Noncommutative Random Variables, Random Matrices and the II 1 Factors of Free Groups, in Quantum Probability and Related Topics, Vol

    D. Voiculescu, Free Noncommutative Random Variables, Random Matrices and the II 1 Factors of Free Groups, in Quantum Probability and Related Topics, Vol. Volume 6 (World Scientific, 1991) pp. 473–487

    work page 1991

  31. [31]

    Speicher, Free Probability Theory and Random Ma- trices, inAsymptotic Combinatorics with Applications to Mathematical Physics, Vol

    R. Speicher, Free Probability Theory and Random Ma- trices, inAsymptotic Combinatorics with Applications to Mathematical Physics, Vol. 1815 (Springer Berlin Heidel- berg, Berlin, Heidelberg, 2003) pp. 53–73

    work page 2003

  32. [32]

    J. A. Mingo and R. Speicher,Free Probability and Random Matrices, Fields Institute Monographs, Vol. 35 (Springer New York, New York, NY, 2017)

    work page 2017

  33. [33]

    Nica and R

    A. Nica and R. Speicher,Lectures on the combinatorics of free probability, Vol. 13 (Cambridge University Press, 2006)

    work page 2006

  34. [34]

    Three lectures on free probability

    J. Novak and M. LaCroix, Three lectures on free proba- bility, arXiv:1205.2097 (2012)

    work page internal anchor Pith review Pith/arXiv arXiv 2097

  35. [35]

    Speicher, Free Probability Theory, Jahresber

    R. Speicher, Free Probability Theory, Jahresber. Deutsch. Math.-Verein.119, 3 (2017)

    work page 2017

  36. [36]

    Hruza and D

    L. Hruza and D. Bernard, Coherent fluctuations in noisy mesoscopic systems, the open quantum ssep, and free probability, Phys. Rev. X13, 011045 (2023)

    work page 2023

  37. [37]

    H. J. Chen and J. Kudler-Flam, Free independence and the noncrossing partition lattice in dual-unitary quantum circuits, Phys. Rev. B111, 014311 (2025)

    work page 2025

  38. [38]

    Jahnke, P

    V. Jahnke, P. Nandy, K. Pal, H. A. Camargo, and K.-Y. Kim, Free Probability approach to spectral and operator statistics in Rosenzweig-Porter random matrix ensembles, arXiv:2506.04520 (2025)

    arXiv 2025

  39. [39]

    H. A. Camargo, Y. Fu, V. Jahnke, K. Pal, and K.-Y. Kim, Quantum Signatures of Chaos from Free Probability, arXiv:2503.20338 (2025)

    arXiv 2025

  40. [40]

    Fritzsch and P

    F. Fritzsch and P. W. Claeys, Free Probability in a Mini- mal Quantum Circuit Model, arXiv:2506.11197 (2025)

    Pith/arXiv arXiv 2025

  41. [41]

    Dowling, J

    N. Dowling, J. D. Nardis, M. Heinrich, X. Turkeshi, and S. Pappalardi, Free Independence and Unitary Design from Random Matrix Product Unitaries, arXiv:2508.00051 (2025)

    arXiv 2025

  42. [42]

    M. C. Ba˜ nuls, M. B. Hastings, F. Verstraete, and J. I. 8 Cirac, Matrix product states for dynamical simulation of infinite chains, Phys. Rev. Lett.102, 240603 (2009)

    work page 2009

  43. [43]

    Lerose, M

    A. Lerose, M. Sonner, and D. A. Abanin, Influence Matrix Approach to Many-Body Floquet Dynamics, Phys. Rev. X11, 021040 (2021)

    work page 2021

  44. [44]

    Sonner, A

    M. Sonner, A. Lerose, and D. A. Abanin, Influence func- tional of many-body systems: Temporal entanglement and matrix-product state representation, Ann. Phys.435, 168677 (2021)

    work page 2021

  45. [45]

    Bertini, P

    B. Bertini, P. Kos, and T. Prosen, Exact Correlation Functions for Dual-Unitary Lattice Models in 1 + 1 Di- mensions, Phys. Rev. Lett.123, 210601 (2019)

    work page 2019

  46. [46]

    Gopalakrishnan and A

    S. Gopalakrishnan and A. Lamacraft, Unitary circuits of finite depth and infinite width from quantum channels, Phys. Rev. B100, 064309 (2019)

    2019

  47. [47]

    Bertini, P

    B. Bertini, P. W. Claeys, and T. Prosen, Exactly solvable many-body dynamics from space-time duality, arXiv:2505.11489 (2025)

    Pith/arXiv arXiv 2025

  48. [48]

    D. A. Roberts and D. Stanford, Diagnosing chaos using four-point functions in two-dimensional conformal field theory, Phys. Rev. Lett115, 131603 (2015)

    work page 2015

  49. [49]

    R. N. Das, S. Dutta, and A. Maji, Late time dynamics in susy saddle-dominated scrambling through higher-point otoc, Phys. Scr.99, 085246 (2024)

    work page 2024

  50. [50]

    Maldacena and D

    J. Maldacena and D. Stanford, Remarks on the Sachdev- Ye-Kitaev model, Phys. Rev. D94, 106002 (2016)

    2016

  51. [51]

    A. Chan, A. De Luca, and J. T. Chalker, Solution of a minimal model for many-body quantum chaos, Phys. Rev. X8, 041019 (2018)

    2018

  52. [52]

    Yoshimura, S

    T. Yoshimura, S. J. Garratt, and J. T. Chalker, Operator dynamics in Floquet many-body systems, Phys. Rev. B 111, 094316 (2025)

    2025

  53. [53]

    Bertini and L

    B. Bertini and L. Piroli, Scrambling in random unitary circuits: Exact results, Phys. Rev. B102, 064305 (2020)

    2020

  54. [54]

    P. W. Claeys and A. Lamacraft, Maximum velocity quan- tum circuits, Phys. Rev. Res.2, 033032 (2020)

    2020

  55. [55]

    Bertini, P

    B. Bertini, P. Kos, and T. Prosen, Operator Entangle- ment in Local Quantum Circuits I: Chaotic Dual-Unitary Circuits, SciPost Phys.8, 067 (2020)

    2020

  56. [56]

    M. A. Rampp, R. Moessner, and P. W. Claeys, From Dual Unitarity to Generic Quantum Operator Spreading, Phys. Rev. Lett.130, 130402 (2023)

    work page 2023

  57. [57]

    Huang, X

    K. Huang, X. Li, D. A. Huse, and A. Chan, Out-of-time- order correlator, many-body quantum chaos, light-like generators, and singular values, arXiv:2308.16179 (2023)

    Pith/arXiv arXiv 2023

  58. [58]

    Zhou and A

    T. Zhou and A. Nahum, Entanglement membrane in chaotic many-body systems, Phys. Rev. X10, 031066 (2020)

    2020

  59. [59]

    M. P. Fisher, V. Khemani, A. Nahum, and S. Vijay, Ran- dom Quantum Circuits, Annu. Rev. Conden. Ma. P.14, 335–79 (2023)

    work page 2023

  60. [60]

    Jonay and T

    C. Jonay and T. Zhou, Physical theory of two-stage ther- malization, Phys. Rev. B110, L020306 (2024)

    2024

  61. [61]

    Suchsland, B

    P. Suchsland, B. Dou¸ cot, V. Khemani, and R. Moessner, Dynamical correlations and domain wall relocalization in transverse field ising chains, Phys. Rev. B112, 014311 (2025)

    work page 2025

  62. [62]

    At time t = 0 we hence find a universal valuek 4(t= 0) =−1/4

    Note that for d = 2 and hermitian and traceless a, b normalized as tr(a2)/d = tr(b2)/d = 1, k4(a, b, a, b) = −1/4 does not depend on either a or b. At time t = 0 we hence find a universal valuek 4(t= 0) =−1/4

    work page

  63. [63]

    Fritzsch and T

    F. Fritzsch and T. Prosen, Eigenstate thermalization in dual-unitary quantum circuits: Asymptotics of spectral functions, Phys. Rev. E103, 062133 (2021)

    work page 2021

  64. [64]

    P. Kos, B. Bertini, and T. Prosen, Correlations in Per- turbed Dual-Unitary Circuits: Efficient Path-Integral For- mula, Phys. Rev. X11, 011022 (2021)

    work page 2021

  65. [65]

    Ippoliti and W

    M. Ippoliti and W. W. Ho, Dynamical Purification and the Emergence of Quantum State Designs from the Projected Ensemble, PRX Quantum4, 030322 (2023)

    work page 2023

  66. [66]

    Riddell, C

    J. Riddell, C. von Keyserlingk, T. Prosen, and B. Bertini, Structural stability hypothesis of dual unitary quantum chaos, Phys. Rev. Res.6, 033226 (2024)

    work page 2024

  67. [67]

    Ippoliti and D

    M. Ippoliti and D. M. Long, Infinite temperature at zero energy, arXiv:2509.04410 (2025)

    Pith/arXiv arXiv 2025

  68. [68]

    Prosen, Ruelle resonances in quantum many-body dynamics, J

    T. Prosen, Ruelle resonances in quantum many-body dynamics, J. Phys. A: Math. Gen.35, L737 (2002)

    work page 2002

  69. [69]

    Prosen, Ruelle resonances in kicked quantum spin chain, Physica D187, 244 (2004)

    T. Prosen, Ruelle resonances in kicked quantum spin chain, Physica D187, 244 (2004)

    work page 2004

  70. [70]

    Garc ´ ıa-Mata, M

    I. Garc ´ ıa-Mata, M. Saraceno, R. A. Jalabert, A. J. Roncaglia, and D. A. Wisniacki, Chaos Signatures in the Short and Long Time Behavior of the Out-of-Time Ordered Correlator, Phys. Rev. Lett.121, 210601 (2018)

    work page 2018

  71. [71]

    ˇZnidariˇ c, Momentum-dependent quantum Ruelle- Pollicott resonances in translationally invariant many- body systems, Phys

    M. ˇZnidariˇ c, Momentum-dependent quantum Ruelle- Pollicott resonances in translationally invariant many- body systems, Phys. Rev. E110, 054204 (2024)

    2024

  72. [72]

    Mori, Liouvillian-gap analysis of open quantum many- body systems in the weak dissipation limit, Phys

    T. Mori, Liouvillian-gap analysis of open quantum many- body systems in the weak dissipation limit, Phys. Rev. B 109, 064311 (2024)

    2024

  73. [73]

    Zhang, L

    C. Zhang, L. Nie, and C. von Keyserlingk, Thermalization rates and quantum Ruelle-Pollicott resonances: insights from operator hydrodynamics, arXiv:2409.17251 (2025)

    Pith/arXiv arXiv 2025

  74. [74]

    J. A. Jacoby, D. A. Huse, and S. Gopalakrishnan, Spectral gaps of local quantum channels in the weak-dissipation limit, Phys. Rev. B111, 104303 (2025)

    work page 2025

  75. [75]

    Duh and M

    U. Duh and M. ˇZnidariˇ c, Ruelle-Pollicott resonances of diffusive U(1)-invariant qubit circuits, arXiv:2506.24097 (2025)

    arXiv 2025

  76. [76]

    While Ref. [37] established these eigenstates for a transfer matrix in which the gates did not have the alternating orientation of the spatial transfer matrix of this work, this difference is immaterial for dual-unitary gates

    work page

  77. [77]

    R. P. Stanley,Catalan Numbers(Cambridge University Press, Cambridge, 2015)

    work page 2015

  78. [78]

    Khaneja and S

    N. Khaneja and S. J. Glaser, Cartan decomposition of SU(2n) and control of spin systems, Chem. Phys.267, 11 (2001)

    work page 2001

  79. [79]

    Khaneja, R

    N. Khaneja, R. Brockett, and S. J. Glaser, Time optimal control in spin systems, Phys. Rev. A63, 032308 (2001)

    2001

  80. [80]

    Kraus and J

    B. Kraus and J. I. Cirac, Optimal creation of entanglement using a two-qubit gate, Phys. Rev. A63, 062309 (2001). APPENDICES In Appendix A we prove that the boundary scrambler returns the random-circuit influence matrix at arbitrary orders of the k-OTOC and that it is stable when perturbing away from dual-unitarity. Appendix B illustrates this proof 9 wit...

    work page 2001

Showing first 80 references.

This paper was first reviewed by deepseek-v4-flash on August 4, 2026.