arxiv: 2509.25331 · v2 · submitted 2025-09-29 · 🪐 quant-ph · cond-mat.stat-mech· cond-mat.str-el· hep-th

Krylov Winding and Emergent Coherence in Operator Growth Dynamics

Rishik Perugu , Bryce Kobrin , Michael O. Flynn , Thomas Scaffidi This is my paper

Pith reviewed 2026-05-18 12:35 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mechcond-mat.str-elhep-th

keywords quantum chaosoperator growthKrylov basissize windingLyapunov exponentSYK modelmany-body dynamics

0 comments

The pith

In quantum chaotic systems the operator wavefunction acquires a phase that winds linearly with the Krylov index, producing size winding when a low-rank basis mapping and growth-bound saturation are present.

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

The paper shows that Krylov winding emerges generically in chaotic quantum systems as a direct consequence of the universal operator growth bound. This winding supplies the mechanism that turns into size winding once operators of equal size align in phase through a low-rank mapping between the Krylov and size bases and once the Lyapunov exponent saturates the bound λ_L = 2α. When saturation fails the phase instead grows superlinearly with size. A reader cares because the result supplies a microscopic, non-holographic account of how scrambled operators can still carry a coherent phase that increases with complexity.

Core claim

Krylov winding is a generic feature of quantum chaotic systems and is a direct consequence of the universal operator growth bound hypothesis. It gives rise to size winding under two additional conditions: a low-rank mapping between the Krylov and size bases, which ensures phase alignment among operators of the same size, and saturation of the chaos-operator growth bound λ_L ≤ 2α. For systems which do not saturate this bound, with h = λ_L / 2α < 1, the winding with Pauli size ℓ becomes superlinear, behaving as ℓ^{1/h}. These results are illustrated in the SYK model and its variants and in a disordered k-local spin model.

What carries the argument

Krylov winding: the linear increase of the phase of the operator wavefunction with the Krylov index, which follows from the universal operator growth bound and converts into size winding under a low-rank Krylov-to-size mapping plus bound saturation.

If this is right

Size winding appears automatically once the low-rank mapping holds and the growth bound saturates.
When saturation fails the phase grows as size to the power 1/h with h = λ_L / 2α.
The same winding mechanism operates in both the SYK family and disordered k-local spin chains.
Coherent phases in scrambled operators can arise without holography once the growth bound is respected.

Where Pith is reading between the lines

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

The same winding structure could appear in other bases that are approximately low-rank related to the size basis, offering a route to detect chaos through phase measurements.
Open-system extensions might show how dissipation modifies the superlinear regime when the bound is not saturated.
Quantum simulators could test the predicted crossover from linear to superlinear winding by tuning the effective Lyapunov exponent.

Load-bearing premise

The low-rank mapping between the Krylov and size bases that produces phase alignment among all operators of the same size.

What would settle it

A measurement in a system known to saturate λ_L = 2α that finds the operator phase growing nonlinearly with size, or a chaotic system lacking the low-rank mapping that nevertheless shows size winding.

Figures

Figures reproduced from arXiv: 2509.25331 by Bryce Kobrin, Michael O. Flynn, Rishik Perugu, Thomas Scaffidi.

**Figure 2.** Figure 2: FIG. 2. Fourier transform [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Peak momentum [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

read the original abstract

The operator wavefunction provides a fine-grained description of quantum chaos and of the irreversible growth of simple operators into increasingly complex ones. Remarkably, at finite temperature this wavefunction can acquire a phase that increases linearly with the operator's size, a phenomenon called \emph{size winding}. Although size winding occurs naturally in a holographic setting, the emergence of a coherent phase in a scrambled operator remains mysterious from the standpoint of a thermalizing quantum many-body system. In this work, we elucidate this phenomenon by introducing the related concept of \textit{Krylov winding}, whereby the operator wavefunction acquires a phase which winds linearly with the Krylov index. We show that Krylov winding is a generic feature of quantum chaotic systems and is a direct consequence of the universal operator growth bound hypothesis. It gives rise to size winding under two additional conditions: (i) a low-rank mapping between the Krylov and size bases, which ensures phase alignment among operators of the same size, and (ii) the saturation of the ``chaos-operator growth'' bound $\lambda_L \leq 2 \alpha$ (with $\lambda_L$ the Lyapunov exponent and $\alpha$ the growth rate), which ensures a linear phase dependence on size. For systems which do not saturate this bound, with $h = \lambda_L / 2\alpha <1$, the winding with Pauli size $\ell$ becomes \emph{superlinear}, behaving as $\ell^{1/h}$. We illustrate these results with two classes of microscopic models: the Sachdev-Ye-Kitaev (SYK) model and its variants, and a disordered $k$-local spin model.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Krylov winding follows from the growth bound and yields superlinear size winding when h<1, but the low-rank mapping to size basis is an added condition rather than a derived consequence.

read the letter

The paper's main contribution is showing that Krylov winding arises as a direct consequence of the universal operator growth bound in chaotic systems. Under the right conditions this produces the linear size winding seen in holography, and when the bound is not saturated it gives a concrete superlinear scaling with Pauli size that scales as ℓ to the power 1/h. That scaling prediction and the framing around Krylov index are new relative to the cited literature and give a first-principles route from growth bounds to phase coherence at finite temperature. The SYK examples and the disordered k-local spin model supply useful checks that keep the discussion grounded rather than purely formal. The argument is internally consistent and engages the existing operator growth and scrambling literature without obvious contradictions. The soft spot is the low-rank mapping between Krylov and size bases. The abstract presents this as an independent requirement alongside bound saturation, without deriving that the overlap matrix stays low-rank from the growth bound or from generic chaotic dynamics. If the mapping is only approximate or model-dependent, the step from Krylov winding to linear size dependence does not follow generically. The paper does not appear to close this gap with a general proof, so the claim that Krylov winding is sufficient for size winding in broad classes of systems rests on how restrictive that extra condition turns out to be. Readers already working on many-body chaos, SYK variants, or holographic models of scrambling will get the most out of the conceptual link and the scaling result. The work is coherent enough on its own terms to deserve referee time so that experts can verify the derivations and test how generic the mapping actually is.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces Krylov winding, a linear phase accumulation with Krylov index in the operator wavefunction, as a generic feature of quantum chaotic systems that follows directly from the universal operator growth bound hypothesis. It further claims that, under two additional conditions—a low-rank mapping between the Krylov and size bases ensuring phase alignment for same-size operators, and saturation of the chaos bound λ_L ≤ 2α—this produces size winding with linear phase dependence on Pauli size ℓ; when the bound is not saturated (h = λ_L / 2α < 1), the dependence becomes superlinear as ℓ^{1/h}. Results are illustrated in the SYK model and a disordered k-local spin model.

Significance. If the claims hold, the work provides a valuable microscopic mechanism linking Krylov complexity to emergent size winding and coherence in thermalizing systems, extending beyond holographic settings. Explicit use of the universal growth bound, the derivation of superlinear scaling when h < 1, and concrete illustrations in SYK and spin models are positive features that support broader applicability in quantum chaos studies.

major comments (2)

[Abstract] Abstract, paragraph on conditions for size winding: the low-rank mapping between Krylov and size bases (condition (i)) is presented as an independent additional requirement needed for phase alignment and linear size dependence, yet it is not derived from the universal operator growth bound hypothesis nor shown to be enforced by chaotic dynamics. This assumption is load-bearing for the central claim that Krylov winding generically produces size winding.
[Model illustrations] Section on model illustrations: explicit checks confirming that the low-rank overlap matrix holds and that λ_L saturates 2α (or quantifying deviations via h) are not provided for the SYK and disordered spin examples, leaving the applicability of the size-winding conclusions dependent on unverified assumptions in the concrete cases.

minor comments (1)

[Notation] The notation for the superlinear exponent 1/h and its relation to the phase dependence on ℓ should be tied to an explicit equation to improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. The comments help clarify the scope of our claims regarding Krylov winding and its relation to size winding. We respond to each major comment below and indicate where revisions will be made.

read point-by-point responses

Referee: [Abstract] Abstract, paragraph on conditions for size winding: the low-rank mapping between Krylov and size bases (condition (i)) is presented as an independent additional requirement needed for phase alignment and linear size dependence, yet it is not derived from the universal operator growth bound hypothesis nor shown to be enforced by chaotic dynamics. This assumption is load-bearing for the central claim that Krylov winding generically produces size winding.

Authors: We agree that the low-rank mapping is presented as an additional condition rather than a direct consequence of the universal operator growth bound. The manuscript derives Krylov winding (linear phase accumulation with Krylov index) strictly from the bound hypothesis, while size winding requires the two extra conditions to ensure phase alignment across same-size operators and linear (or superlinear) dependence on Pauli size. We do not claim that chaotic dynamics universally enforces the low-rank property; it is a physically motivated assumption that facilitates the mapping in the systems of interest. In the revised manuscript, we will update the abstract and relevant sections to emphasize this separation more explicitly and discuss the motivation for the low-rank condition without implying it follows automatically from chaos. revision: partial
Referee: [Model illustrations] Section on model illustrations: explicit checks confirming that the low-rank overlap matrix holds and that λ_L saturates 2α (or quantifying deviations via h) are not provided for the SYK and disordered spin examples, leaving the applicability of the size-winding conclusions dependent on unverified assumptions in the concrete cases.

Authors: We concur that adding explicit verifications would strengthen the link between the general framework and the model results. For the SYK model, the chaos bound saturation (h=1) is known from prior literature, but we will include a direct computation or reference to the overlap matrix rank. For the disordered k-local spin model, we will add numerical or analytical estimates of the Krylov-size overlap matrix and the value of h to confirm or quantify deviations from the assumptions. These additions will be placed in the main text or an appendix of the revised version. revision: yes

Circularity Check

0 steps flagged

No significant circularity; Krylov winding follows from external growth bound with explicit additional conditions stated

full rationale

The paper states that Krylov winding is a direct consequence of the external universal operator growth bound hypothesis. Size winding is derived only after introducing two additional conditions (low-rank mapping and saturation of λ_L ≤ 2α) that are presented as independent requirements rather than derived internally. The parameter h = λ_L / 2α is defined from these quantities to describe superlinear behavior when the bound is not saturated, but this is a straightforward re-expression, not a fit renamed as prediction. No self-citation forms a load-bearing chain, no ansatz is smuggled, and no quantity is defined in terms of itself. The derivation chain remains self-contained against the stated external hypothesis and explicit assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the universal operator growth bound hypothesis as an external domain assumption and on the existence of a low-rank Krylov-size mapping whose justification is not supplied in the abstract. No free parameters are fitted inside the derivation; h is defined from λ_L and α rather than adjusted to data. No new particles or forces are postulated.

axioms (1)

domain assumption universal operator growth bound hypothesis
Invoked as the origin of generic Krylov winding (abstract).

pith-pipeline@v0.9.0 · 5847 in / 1350 out tokens · 31916 ms · 2026-05-18T12:35:51.809812+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel; dAlembert_cosh_solution_aczel echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

Krylov wavefunction φ_n(t_β) = sqrt[...] tanh[α(t_β)]^n / cosh[α(t_β)]^{2Δ} with phase θ_K(t) = Arg[tanh(α t_β)] and exponential decay exp(−2 n e^{-2α t} cos(α β/2))
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean alpha_pin_under_high_calibration; J_uniquely_calibrated_via_higher_derivative refines

?

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

Operator growth hypothesis b_n ∼ α n; chaos-operator growth bound λ_L ≤ 2α with h = λ_L / 2α controlling linear vs superlinear winding ℓ^{1/h}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

81 extracted references · 81 canonical work pages · 3 internal anchors

[1]

Hayden and J

P. Hayden and J. Preskill, Black holes as mirrors: quan- tum information in random subsystems, Journal of High Energy Physics2007, 120–120 (2007)

work page 2007
[2]

Sekino and L

Y. Sekino and L. Susskind, Fast scramblers, Journal of High Energy Physics2008, 065–065 (2008)

work page 2008
[3]

Maldacena, S

J. Maldacena, S. H. Shenker, and D. Stanford, A bound on chaos, Journal of High Energy Physics2016, 10.1007/jhep08(2016)106 (2016)

work page doi:10.1007/jhep08(2016)106 2016
[4]

S. H. Shenker and D. Stanford, Black holes and the butterfly effect, Journal of High Energy Physics2014, 10.1007/jhep03(2014)067 (2014)

work page doi:10.1007/jhep03(2014)067 2014
[5]

S. H. Shenker and D. Stanford, Stringy effects in scram- bling (2015), arXiv:1412.6087 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2015
[6]

Hosur, X.-L

P. Hosur, X.-L. Qi, D. A. Roberts, and B. Yoshida, Chaos in quantum channels, Journal of High Energy Physics 2016, 10.1007/jhep02(2016)004 (2016)

work page doi:10.1007/jhep02(2016)004 2016
[7]

D. A. Roberts, D. Stanford, and L. Susskind, Local- ized shocks, Journal of High Energy Physics2015, 10.1007/jhep03(2015)051 (2015)

work page doi:10.1007/jhep03(2015)051 2015
[8]

Nahum, S

A. Nahum, S. Vijay, and J. Haah, Operator spreading in random unitary circuits, Physical Review X8, 021014 (2018). 6

work page 2018
[9]

D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi, and E. Altman, A universal operator growth hypothesis, Phys. Rev. X9, 041017 (2019)

work page 2019
[10]

C. W. von Keyserlingk, T. Rakovszky, F. Pollmann, and S. L. Sondhi, Operator hydrodynamics, otocs, and en- tanglement growth in systems without conservation laws, Physical Review X8, 021013 (2018)

work page 2018
[11]

Swingle, Unscrambling the physics of out-of-time- order correlators, Nature Physics14, 988 (2018)

B. Swingle, Unscrambling the physics of out-of-time- order correlators, Nature Physics14, 988 (2018)

work page 2018
[12]

Khemani, A

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

work page 2018
[13]

Xu and B

S. Xu and B. Swingle, Scrambling dynamics and out-of- time-ordered correlators in quantum many-body systems, PRX Quantum5, 010201 (2024)

work page 2024
[14]

J. Kim, E. Altman, and X. Cao, Dirac fast scramblers, Phys. Rev. B103, L081113 (2021)

work page 2021
[15]

D. A. Roberts and B. Yoshida, Chaos and complex- ity by design, Journal of High Energy Physics2017, 10.1007/jhep04(2017)121 (2017)

work page doi:10.1007/jhep04(2017)121 2017
[16]

Von Keyserlingk, F

C. Von Keyserlingk, F. Pollmann, and T. Rakovszky, Op- erator backflow and the classical simulation of quantum transport, Physical Review B105, 245101 (2022)

work page 2022
[17]

X. Mi, P. Roushan, C. Quintana, S. Mandra, J. Mar- shall, C. Neill, F. Arute, K. Arya, J. Atalaya, R. Bab- bush,et al., Information scrambling in quantum circuits, Science374, 1479 (2021)

work page 2021
[18]

D. A. Abanin, R. Acharya, L. Aghababaie-Beni, G. Aigeldinger, A. Ajoy, R. Alcaraz, I. Aleiner, T. I. Andersen, M. Ansmann, F. Arute,et al., Constructive interference at the edge of quantum ergodic dynamics, arXiv preprint arXiv:2506.10191 (2025)

work page arXiv 2025
[19]

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 Physics13, 781 (2017)

work page 2017
[20]

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, Physical Review X7, 031011 (2017)

work page 2017
[21]

Qi and A

X.-L. Qi and A. Streicher, Quantum epidemiology: op- erator growth, thermal effects, and syk, Journal of High Energy Physics2019, 10.1007/jhep08(2019)012 (2019)

work page doi:10.1007/jhep08(2019)012 2019
[22]

Here the inner product is defined as (A|B) = Tr[A †B]

work page
[23]

A. R. Brown, H. Gharibyan, S. Leichenauer, H. W. Lin, S. Nezami, G. Salton, L. Susskind, B. Swingle, and M. Walter, Quantum gravity in the lab. i. teleporta- tion by size and traversable wormholes, PRX quantum 4, 010320 (2023)

work page 2023
[24]

Nezami, H

S. Nezami, H. W. Lin, A. R. Brown, H. Gharibyan, S. Leichenauer, G. Salton, L. Susskind, B. Swingle, and M. Walter, Quantum gravity in the lab. ii. teleporta- tion by size and traversable wormholes, PRX quantum 4, 010321 (2023)

work page 2023
[25]

P. Gao, D. L. Jafferis, and A. C. Wall, Traversable worm- holes via a double trace deformation, Journal of High Energy Physics2017, 1 (2017)

work page 2017
[26]

Maldacena, D

J. Maldacena, D. Stanford, and Z. Yang, Diving into traversable wormholes, Fortschritte der Physik65, 1700034 (2017)

work page 2017
[27]

Efficient decoding for the Hayden-Preskill protocol

B. Yoshida and A. Kitaev, Efficient decoding for the hayden-preskill protocol, arXiv preprint arXiv:1710.03363 (2017)

work page internal anchor Pith review Pith/arXiv arXiv 2017
[28]

K. A. Landsman, C. Figgatt, T. Schuster, N. M. Linke, B. Yoshida, N. Y. Yao, and C. Monroe, Verified quantum information scrambling, Nature567, 61 (2019)

work page 2019
[29]

M. S. Blok, V. V. Ramasesh, T. Schuster, K. O’Brien, J.-M. Kreikebaum, D. Dahlen, A. Morvan, B. Yoshida, N. Y. Yao, and I. Siddiqi, Quantum information scram- bling on a superconducting qutrit processor, Physical Re- view X11, 021010 (2021)

work page 2021
[30]

Schuster, B

T. Schuster, B. Kobrin, P. Gao, I. Cong, E. T. Khabi- boulline, N. M. Linke, M. D. Lukin, C. Monroe, B. Yoshida, and N. Y. Yao, Many-body quantum telepor- tation via operator spreading in the traversable wormhole protocol, Phys. Rev. X12, 031013 (2022)

work page 2022
[31]

F. M. Haehl, A. Streicher, and Y. Zhao, Six-point func- tions and collisions in the black hole interior, Journal of High Energy Physics2021, 1 (2021)

work page 2021
[32]

H. W. Lin, J. Maldacena, and Y. Zhao, Symmetries near the horizon, Journal of High Energy Physics2019(2019)

work page 2019
[33]

T.-G. Zhou, Y. Gu, and P. Zhang, Size winding mech- anism beyond maximal chaos, Journal of High Energy Physics2024, 10.1007/jhep11(2024)044 (2024)

work page doi:10.1007/jhep11(2024)044 2024
[34]

Gao, Commuting syk: a pseudo-holographic model, Journal of High Energy Physics2024, 1 (2024)

P. Gao, Commuting syk: a pseudo-holographic model, Journal of High Energy Physics2024, 1 (2024)

work page 2024
[35]

Jafferis, A

D. Jafferis, A. Zlokapa, J. D. Lykken, D. K. Kolch- meyer, S. I. Davis, N. Lauk, H. Neven, and M. Spiropulu, Traversable wormhole dynamics on a quantum processor, Nature612, 51 (2022)

work page 2022
[36]

Kobrin, T

B. Kobrin, T. Schuster, and N. Y. Yao, Comment on” traversable wormhole dynamics on a quantum proces- sor”, arXiv preprint arXiv:2302.07897 (2023)

work page arXiv 2023
[37]

Sachdev and J

S. Sachdev and J. Ye, Gapless spin-fluid ground state in a random quantum Heisenberg magnet, Phys. Rev. Lett. 70, 3339 (1993)

work page 1993
[38]

Sachdev, Bekenstein-Hawking Entropy and Strange Metals, Phys

S. Sachdev, Bekenstein-Hawking Entropy and Strange Metals, Phys. Rev. X5, 041025 (2015)

work page 2015
[39]

Kitaev, A simple model of quantum holography, http://online.kitp.ucsb.edu/online/entangled15/kitaev/ (2015)

A. Kitaev, A simple model of quantum holography, http://online.kitp.ucsb.edu/online/entangled15/kitaev/ (2015)

work page 2015
[40]

Maldacena and D

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

work page 2016
[41]

Kitaev and S

A. Kitaev and S. J. Suh, The soft mode in the sachdev- ye-kitaev model and its gravity dual, Journal of High Energy Physics2018, 10.1007/jhep05(2018)183 (2018)

work page doi:10.1007/jhep05(2018)183 2018
[42]

Chowdhury, A

D. Chowdhury, A. Georges, O. Parcollet, and S. Sachdev, Sachdev-Ye-Kitaev models and beyond: Window into non-Fermi liquids, Rev. Mod. Phys.94, 035004 (2022)

work page 2022
[43]

Y. Chen, H. Zhai, and P. Zhang, Tunable quantum chaos in the sachdev-ye-kitaev model coupled to a thermal bath, Journal of High Energy Physics2017, 10.1007/jhep07(2017)150 (2017)

work page doi:10.1007/jhep07(2017)150 2017
[44]

L. Chen, B. Mu, H. Wang, and P. Zhang, Dissecting quantum many-body chaos in the krylov space (2024), arXiv:2404.08207 [quant-ph]

work page arXiv 2024
[45]

Viswanath and G

V. Viswanath and G. Muller,The Recursion Method: Ap- plications to Many-body Dynamics(Springer, New York, 2008)

work page 2008
[46]

T. Xu, T. Scaffidi, and X. Cao, Does scrambling equal chaos?, Physical Review Letters124, 140602 (2020), arXiv:1912.11063 [cond-mat.stat-mech]

work page arXiv 2020
[47]

D. E. Parker, X. Cao, and M. P. Zaletel, Local matrix product operators: Canonical form, compression, and control theory, Phys. Rev. B102, 035147 (2020)

work page 2020
[48]

Cao, A statistical mechanism for operator growth, 7 Journal of Physics A: Mathematical and Theoretical54, 144001 (2021)

X. Cao, A statistical mechanism for operator growth, 7 Journal of Physics A: Mathematical and Theoretical54, 144001 (2021)

work page 2021
[49]

Z. Qi, T. Scaffidi, and X. Cao, Surprises in the deep hilbert space of all-to-all systems: From super- exponential scrambling to slow entanglement growth, Physical Review B108, 054301 (2023), arXiv:2304.11138 [cond-mat.stat-mech]

work page arXiv 2023
[50]

A. De, U. Borla, X. Cao, and S. Gazit, Stochastic sam- pling of operator growth dynamics, Phys. Rev. B110, 155135 (2024)

work page 2024
[51]

Bhattacharjee, X

B. Bhattacharjee, X. Cao, P. Nandy, and T. Pathak, Krylov complexity in saddle-dominated scrambling, Journal of High Energy Physics2022, 174 (2022)

work page 2022
[52]

Nandy, A

P. Nandy, A. S. Matsoukas-Roubeas, P. Mart´ ınez- Azcona, A. Dymarsky, and A. del Campo, Quantum dynamics in krylov space: Methods and applications, Physics Reports1125–1128, 1 (2025)

work page 2025
[53]

Krylov Complexity

E. Rabinovici, A. S´ anchez-Garrido, R. Shir, and J. Son- ner, Krylov complexity, arXiv preprint (2025), submitted on 8 July 2025, arXiv:2507.06286 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2025
[54]

Bhattacharjee, X

B. Bhattacharjee, X. Cao, P. Nandy, and T. Pathak, Op- erator growth in open quantum systems: lessons from the dissipative SYK, arXiv e-prints , arXiv:2212.06180 (2022), arXiv:2212.06180 [quant-ph]

work page arXiv 2022
[55]

Bhattacharjee, S

B. Bhattacharjee, S. Sur, and P. Nandy, Probing quan- tum scars and weak ergodicity-breaking through quan- tum complexity (2022)

work page 2022
[56]

Caputa, N

P. Caputa, N. Gupta, S. S. Haque, S. Liu, J. Murugan, and H. J. R. Van Zyl, Spread complexity and topological transitions in the kitaev chain (2022)

work page 2022
[57]

Caputa and S

P. Caputa and S. Liu, Quantum complexity and topolog- ical phases of matter (2022)

work page 2022
[58]

Rabinovici, A

E. Rabinovici, A. S´ anchez-Garrido, R. Shir, and J. Sonner, Krylov complexity from integrability to chaos, Journal of High Energy Physics2022, 10.1007/jhep07(2022)151 (2022)

work page doi:10.1007/jhep07(2022)151 2022
[59]

F. B. Trigueros and C.-J. Lin, Krylov complexity of many-body localization: Operator localization in Krylov basis, SciPost Phys.13, 037 (2022)

work page 2022
[60]

Caputa, J

P. Caputa, J. M. Magan, and D. Patramanis, Geome- try of krylov complexity, Phys. Rev. Research4, 013041 (2022)

work page 2022
[61]

Dymarsky and M

A. Dymarsky and M. Smolkin, Krylov complexity in con- formal field theory, Phys. Rev. D104, L081702 (2021)

work page 2021
[62]

Bhattacharjee, P

B. Bhattacharjee, P. Nandy, and T. Pathak, Krylov com- plexity in large-qand double-scaled syk model (2022)

work page 2022
[63]

Baek, Krylov complexity in inverted harmonic oscilla- tor (2022)

S. Baek, Krylov complexity in inverted harmonic oscilla- tor (2022)

work page 2022
[64]

Fan, Universal relation for operator complexity, Phys

Z.-Y. Fan, Universal relation for operator complexity, Phys. Rev. A105, 062210 (2022)

work page 2022
[65]

Guo, Operator growth in su(2) yang-mills theory (2022)

S. Guo, Operator growth in su(2) yang-mills theory (2022)

work page 2022
[66]

Rabinovici, A

E. Rabinovici, A. S´ anchez-Garrido, R. Shir, and J. Son- ner, Operator complexity: a journey to the edge of krylov space, Journal of High Energy Physics2021, 10.1007/jhep06(2021)062 (2021)

work page doi:10.1007/jhep06(2021)062 2021
[67]

C. Liu, H. Tang, and H. Zhai, Krylov complexity in open quantum systems (2022)

work page 2022
[68]

Rabinovici, A

E. Rabinovici, A. S´ anchez-Garrido, R. Shir, and J. Sonner, Krylov localization and suppression of complexity, Journal of High Energy Physics2022, 10.1007/jhep03(2022)211 (2022)

work page doi:10.1007/jhep03(2022)211 2022
[69]

Bhattacharya, P

A. Bhattacharya, P. Nandy, P. P. Nath, and H. Sahu, Operator growth and krylov construction in dissipative open quantum systems (2022)

work page 2022
[70]

Balasubramanian, P

V. Balasubramanian, P. Caputa, J. M. Magan, and Q. Wu, Quantum chaos and the complexity of spread of states, Phys. Rev. D106, 046007 (2022)

work page 2022
[71]

M¨ uck and Y

W. M¨ uck and Y. Yang, Krylov complexity and orthogo- nal polynomials, Nuclear Physics B984, 115948 (2022)

work page 2022
[72]

We note that this interpretation is “dual” to the one in holography, where sizelis regarded as momentum andµ gives the position

work page
[73]

Baldwin and B

C. Baldwin and B. Swingle, Quenched vs annealed: Glassiness from sk to syk, Physical Review X10, 10.1103/physrevx.10.031026 (2020)

work page doi:10.1103/physrevx.10.031026 2020
[74]

This is easily shown using the interlacing theorem since Mnm is a submatrix of the matrix representation of a projector

work page
[75]

Bhattacharjee, X

B. Bhattacharjee, X. Cao, P. Nandy, and T. Pathak, Op- erator growth in open quantum systems: lessons from the dissipative syk, Journal of High Energy Physics2023, 54 (2023)

work page 2023
[76]

Gu and A

Y. Gu and A. Kitaev, On the relation between the mag- nitude and exponent of otocs, Journal of High Energy Physics2019, 10.1007/jhep02(2019)075 (2019)

work page doi:10.1007/jhep02(2019)075 2019
[77]

Y. Gu, A. Kitaev, and P. Zhang, A two-way approach to out-of-time-order correlators, Journal of High Energy Physics2022, 10.1007/jhep03(2022)133 (2022)

work page doi:10.1007/jhep03(2022)133 2022
[78]

Zhang and Y

P. Zhang and Y. Gu, Operator size distribution in large n quantum mechanics of majorana fermions, Journal of High Energy Physics2023, 10.1007/jhep10(2023)018 (2023)

work page doi:10.1007/jhep10(2023)018 2023
[79]

Zhang, Y

P. Zhang, Y. Gu, and A. Kitaev, An obstacle to sub-ads holography for syk-like models, Journal of High Energy Physics2021, 10.1007/jhep03(2021)094 (2021)

work page doi:10.1007/jhep03(2021)094 2021
[80]

Zhang, Information scrambling and entangle- ment dynamics of complex brownian sachdev-ye- kitaev models, Journal of High Energy Physics2023, 10.1007/jhep04(2023)105 (2023)

P. Zhang, Information scrambling and entangle- ment dynamics of complex brownian sachdev-ye- kitaev models, Journal of High Energy Physics2023, 10.1007/jhep04(2023)105 (2023)

work page doi:10.1007/jhep04(2023)105 2023

Showing first 80 references.