Pith. sign in

REVIEW 2 major objections 2 minor 8 cited by

Krylov Dynamics and Operator Growth in Time-Dependent Systems via Lie Algebras

T0 review · 2 major / 2 minor · reviewed 2026-05-19 · grok-4.3

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

Pith's one-line read Time-dependent Hamiltonians with Lie-algebra structure yield the same quantum speed limit on Krylov complexity growth as static cases, but the bound saturates only when the Hamiltonian commutes with itself across times.

desk verdict The paper ties time-dependent Krylov dynamics to Lie-algebra ladder operators and shows a speed limit that keeps the static functional form but saturates only when the Hamiltonian commutes with itself at different times. read the letter →

arxiv 2605.05290 v2 pith:U3RGRXTS submitted 2026-05-06 quant-ph hep-th

András Grabarits , E. Medina-Guerra , Adolfo del Campo This is my paper · ORCID
classification quant-phhep-th
keywords Krylovcomplexityoperatorgrowthtime-dependentHamiltoniansLiealgebrasquantumspeedlimitssl(2C)subalgebrasubspace
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper develops a framework that connects the evolution inside the time-dependent Krylov subspace directly to the ladder operators of an underlying Lie algebra. Under minimal conditions the dynamics is governed by an embedded sl(2,C) subalgebra whose action is set by the interaction-picture Hamiltonian. An exact single-exponential form of the time-evolution operator maps the problem onto an ordinary time-independent Krylov space in a rotated basis, from which the original time-dependent dynamics can be recovered. The work also derives a new quantum speed limit on the rate of complexity growth that keeps the same functional shape as the time-independent version whenever the evolution is Lie-algebraic. Saturation of this bound turns out to require that the Hamiltonian at one instant commutes with the Hamiltonian at every other instant.

What carries the argument

Ladder operators of an embedded sl(2,C) subalgebra that generate the exact time-dependent Krylov dynamics from the interaction-picture Hamiltonian.

What would settle it

Compute the Krylov complexity growth rate for a concrete time-dependent Hamiltonian that belongs to a Lie algebra yet fails to commute with itself at different times and check whether the measured rate still obeys the reported functional bound.

Watch

Extended reading notes

Core claim

For Hamiltonians possessing an underlying Lie-algebraic structure, the exact Krylov dynamics generated by a time-dependent generator is determined by ladder operators of an embedded sl(2,C) subalgebra acting on the interaction-picture Hamiltonian; the same structure produces a quantum speed limit on operator complexity growth whose functional form is identical to the time-independent case, with saturation occurring exclusively when the Hamiltonian commutes with itself at different times.

Load-bearing premise

The Hamiltonian must possess an underlying Lie-algebraic structure that permits an embedded sl(2,C) subalgebra or equivalent ladder-operator generation of the Krylov dynamics.

Editorial extensions

If this is right

  • The complexity growth bound derived for static generators carries over unchanged to any time-dependent evolution generated by a Lie algebra.
  • Saturation of the speed limit occurs if and only if the time-dependent Hamiltonian commutes with itself at all pairs of times.
  • An exact single-exponential time-evolution operator maps the time-dependent problem onto a time-independent Krylov space in a unitarily equivalent basis.
  • The same ladder-operator description applies to the oscillator algebra and to concrete models such as the translated or dilated harmonic oscillator and a spin in a rotating magnetic field.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The framework supplies an algebraic route to computing operator growth in periodically driven or otherwise explicitly time-dependent quantum systems without constructing the full Krylov basis at each instant.
  • Non-commuting time-dependent Hamiltonians are predicted to produce strictly slower saturation of complexity growth than their commuting counterparts, offering a diagnostic for the degree of temporal non-commutativity.
  • The same construction may extend to open-system Lindblad generators that close under a finite-dimensional Lie algebra, yielding speed limits on dissipative complexity growth.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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 / 2 minor

Summary. The manuscript develops a Lie-algebraic framework for Krylov subspace dynamics under time-dependent Hamiltonians. It establishes a direct link between the time-dependent Krylov evolution and the underlying Lie algebra, identifies minimal conditions under which the dynamics is exactly determined by the interaction-picture Hamiltonian and governed by an embedded sl(2,C) subalgebra, derives an exact single-exponential representation of the time-evolution operator that yields a distinct time-independent Krylov dynamics in a unitarily related basis, extends the approach to the oscillator algebra and provides explicit examples (translated/dilated harmonic oscillator, closed Virasoro subalgebras, spin in rotating magnetic field, multi-level systems). It further introduces a new quantum speed limit on complexity growth for time-dependent generators that, for Lie-algebra-governed evolutions, retains the same functional form as the time-independent case, with saturation occurring only when [H(t),H(s)]=0.

Significance. If the central derivations hold, the work supplies a unified analytical framework for operator growth and Krylov complexity in driven quantum systems possessing Lie-algebraic structure. This is potentially significant for quantum control, Floquet engineering, and complexity bounds in time-dependent many-body systems, especially given the explicit examples and the parameter-free character of the speed-limit functional form.

major comments (2)
  1. [Quantum speed limit derivation] § on quantum speed limit (following the abstract claim): the assertion that the new speed limit retains the identical functional form as the time-independent case rests on the time-dependent Krylov dynamics being exactly generated by ladder operators of the embedded sl(2,C) subalgebra determined by the interaction-picture Hamiltonian. The minimal conditions stated in the abstract do not explicitly demonstrate closure of the Krylov subspace for generic non-commuting H(t); if operators generated by the time-dependent generator leave the subalgebra, the bound's functional form would deviate. This is load-bearing for the central claim.
  2. [Examples] Examples section (spin in rotating magnetic field and Virasoro cases): explicit verification is needed that the interaction-picture Hamiltonian indeed generates an sl(2,C) subalgebra whose ladder operators close the Krylov subspace for the chosen time-dependent driving; without this, it remains unclear whether the saturation condition [H(t),H(s)]=0 is sufficient or if additional commutativity assumptions are implicitly used.
minor comments (2)
  1. [Abstract] The abstract is information-dense; splitting the list of results into separate sentences would improve readability.
  2. [Notation] Notation for Lie algebras should be uniformly fraktur (e.g., consistent use of sl(2,C) vs. sl(2,mathbb{C})) across all sections and equations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. We address each of the major comments in detail below and have incorporated revisions to clarify the key derivations and examples as suggested.

read point-by-point responses

  1. Referee: [Quantum speed limit derivation] § on quantum speed limit (following the abstract claim): the assertion that the new speed limit retains the identical functional form as the time-independent case rests on the time-dependent Krylov dynamics being exactly generated by ladder operators of the embedded sl(2,C) subalgebra determined by the interaction-picture Hamiltonian. The minimal conditions stated in the abstract do not explicitly demonstrate closure of the Krylov subspace for generic non-commuting H(t); if operators generated by the time-dependent generator leave the subalgebra, the bound's functional form would deviate. This is load-bearing for the central claim.

    Authors: We appreciate the referee pointing out this crucial aspect. The manuscript establishes that under the minimal conditions where the Hamiltonian is governed by a Lie algebra admitting an sl(2,C) subalgebra, the interaction-picture transformation renders the effective generator time-independent within that algebra, ensuring the Krylov subspace remains closed by the ladder operators. For generic non-commuting H(t), the framework applies precisely when the time-dependent terms do not generate operators outside the subalgebra, which is ensured by the Lie-algebraic structure assumed. To make this explicit, we have revised the abstract to better articulate these conditions and added a dedicated paragraph in the quantum speed limit section explaining the closure property and noting that deviations occur only outside these assumptions. This preserves the central claim while clarifying its scope. revision: yes

  2. Referee: [Examples] Examples section (spin in rotating magnetic field and Virasoro cases): explicit verification is needed that the interaction-picture Hamiltonian indeed generates an sl(2,C) subalgebra whose ladder operators close the Krylov subspace for the chosen time-dependent driving; without this, it remains unclear whether the saturation condition [H(t),H(s)]=0 is sufficient or if additional commutativity assumptions are implicitly used.

    Authors: We agree that providing explicit verification in the examples would enhance clarity. For the spin in a rotating magnetic field, the interaction picture eliminates the time dependence, resulting in a constant Hamiltonian that generates the su(2) algebra (whose complexification includes sl(2,C)), and the Krylov subspace is closed by the standard angular momentum ladder operators as verified through the commutation relations [J_z, J_pm] = ± J_pm. Similarly, for the Virasoro subalgebra cases, the generators satisfy the required commutation relations that keep the action within the subspace. We have added explicit calculations and commutation checks in the revised examples section to demonstrate this closure. The saturation condition [H(t), H(s)] = 0 is derived directly from the speed limit expression and holds under the Lie algebra governance without additional commutativity assumptions beyond those stated. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: derivation grounded in Lie-algebraic structure and interaction picture

full rationale

The paper derives the time-dependent Krylov dynamics and quantum speed limit directly from the underlying Lie-algebraic structure of the Hamiltonian and the interaction-picture representation. The identification of minimal conditions for an embedded sl(2,C) subalgebra is presented as an algebraic property rather than a fitted input or self-referential definition. The retention of the speed-limit functional form follows from this structure without reducing to the target claim by construction. Saturation condition involving [H(t),H(s)]=0 is derived as a consequence of temporal driving, not presupposed. No self-citation chains or ansatze are load-bearing for the central results; the framework remains self-contained against external algebraic benchmarks.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

Review performed on abstract only; specific free parameters or ad-hoc axioms cannot be extracted. The work relies on standard properties of Lie algebras and quantum mechanics.

assumptions (2)
  • domain assumption The Hamiltonian admits an underlying Lie-algebraic structure allowing ladder-operator generation of Krylov dynamics
    Invoked when identifying minimal conditions for exact time-dependent dynamics governed by embedded sl(2,C) subalgebra
  • standard math Standard properties of Lie algebras and the interaction picture hold for the time-evolution operator
    Used throughout the general framework and single-exponential representation

how reviews work

0 comments
Cite this review

Pith. "Pith review of Krylov Dynamics and Operator Growth in Time-Dependent Systems via Lie Algebras." pith.science (2026). https://pith.science/paper/U3RGRXTS

@misc{pith2026260505290,
  author       = {Pith},
  title        = {Pith review of: Krylov Dynamics and Operator Growth in Time-Dependent Systems via Lie Algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U3RGRXTS}},
  note         = {Machine review of arXiv:2605.05290}
}
abstract

We study quantum dynamics generated by time-dependent Hamiltonians in Krylov space, the minimal subspace in which the evolution takes place. We establish a direct link between dynamics in the time-dependent Krylov subspace and the underlying Lie-algebraic structure of the Hamiltonian. We develop a general framework in which the dynamics in the time-dependent Krylov subspace is generated by ladder operators of the associated Lie algebra. In particular, we identify the minimal conditions under which the exact time-dependent Krylov dynamics is naturally determined by the interaction-picture Hamiltonian and governed by an embedded $\mathfrak{sl}(2,\mathbb{C})$ subalgebra. We further show that an exact single-exponential representation of the time-evolution operator gives rise to a distinct time-independent Krylov dynamics in a unitarily related basis, from which the exact time-dependent Krylov dynamics can nevertheless be recovered. We also extend the framework to the oscillator algebra as the simplest extension of the nilpotent Heisenberg--Weyl algebra, and provide further examples, including the translated and dilated harmonic oscillator, systems governed by closed Virasoro subalgebras, a spin in a rotating magnetic field, and higher-dimensional generalizations for multi-level systems. In addition, we introduce a new quantum speed limit to the complexity growth rate generated by a time-dependent generator and show that, for evolutions governed by a Lie algebra, it retains the same functional form as in the time-independent case. Remarkably, saturation of this bound is strongly affected by temporal driving and persists only when the Hamiltonian commutes with itself at different times. These results establish a unified framework for characterizing operator growth and Krylov complexity in time-dependent quantum systems with underlying Lie-algebraic structures.

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 8 Pith papers

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

  1. Polynomial Initial-State Jumps and Christoffel Transforms in Krylov Complexity

    hep-th 2026-07 conditional novelty 7.0 of 10

    Polynomial changes of the initial state in Krylov complexity are solved exactly via Christoffel transforms of the spectral measure, yielding finite-band amplitude transfer and projected-kernel complexity formulas with...

  2. Krylov Correlators in $\mathfrak{sl}(2,\mathbb R)$ Models: Exact Results and Holographic Complexity

    hep-th 2026-05 unverdicted novelty 7.0 of 10

    Exact Krylov correlators in sl(2,R) models are proportional to radial momenta of infalling particles in the BTZ black hole, providing a step toward generalizing the complexity-momentum correspondence.

  3. Krylov complexity and the growth of the black hole interior in 3D gravity

    hep-th 2026-08 conditional novelty 6.0 of 10

    For two-sided BTZ black holes, operator Krylov complexity of spatially smeared operators grows linearly at late times like the codimension-one bulk volume, whereas Krylov spread complexity of the thermofield double st...

  4. Controlled Chaos in 4D SCFTs

    hep-th 2026-06 unverdicted novelty 6.0 of 10

    Orbifolds of N=4 SYM produce SCFTs whose dilatation operator in a subsector is realized by a tunable spin chain whose eigenvalue statistics exhibit chaos for specific marginal couplings.

  5. Krylov Complexity: Flat bands and Carroll breaking deformations

    hep-th 2026-06 unverdicted novelty 5.0 of 10

    Krylov complexity growth distinguishes phase-dependent resilience of Carrollian sectors in all-bands-flat fermionic ladders against delocalizing perturbations and exhibits UV sensitivity in a continuum Carroll scalar ...

  6. Krylov Complexity in Periodically Driven CFTs and Critical Fermions

    hep-th 2026-05 unverdicted novelty 5.0 of 10

    Arnoldi coefficients approach unity exponentially in heating phases of driven CFTs but oscillate in non-heating phases; lattice realizations show distinct spectral and graph signatures despite similar CFT Krylov growth.

  7. Krylov complexity and fidelity susceptibility in two-band Hamiltonians

    quant-ph 2026-05 unverdicted novelty 5.0 of 10

    Derivative of Krylov spread complexity diverges logarithmically at SSH topological transitions and is bounded by fidelity susceptibility in general two-band Hamiltonians, with a non-unitary duality between phases.

  8. Krylov complexity and Berry phase in quantum adiabatic dynamics

    quant-ph 2026-08 conditional novelty 4.0 of 10

    For a spin-1/2 in adiabatic evolution with constant parameters, Krylov complexity equals sin²α sin²((h - 2γ_B/T)t/2), linking its oscillation frequency to the Berry phase.

Reference graph

Works this paper leans on

62 extracted references · 62 canonical work pages · cited by 8 Pith papers

  1. [1]

    A Universal Operator Growth Hypothesis

    Daniel E. Parker, Xiangyu Cao, Alexander Avdoshkin, Thomas Scaffidi, and Ehud Altman. A universal operator growth hypothesis.Physical Review X, 9(4):041017, October 2019. doi: 10.1103/PhysRevX.9.041017. URLhttps://doi.org/10.1103/PhysRevX.9.041017

    work page doi:10.1103/physrevx.9.041017 2019

  2. [2]

    Matsoukas-Roubeas and Pablo Martínez-Azcona and Anatoly Dymarsky and Adolfo

    Pratik Nandy, Apollonas S. Matsoukas-Roubeas, Pablo Mart´ ınez-Azcona, Anatoly Dymarsky, and Adolfo del Campo. Quantum dynamics in krylov space: Methods and applications.Physics Reports, 1125–1128:1–82, June 2025. doi: 10.1016/j.physrep.2025.05.001. URLhttps://doi.org/10.1016/j.physrep.2025.05.001

    work page doi:10.1016/j.physrep.2025.05.001 2025

  3. [3]

    Heller, and Nicole Yunger Halpern

    Stefano Baiguera, Vijay Balasubramanian, Pawel Caputa, Shira Chapman, Jonas Haferkamp, Michal P. Heller, and Nicole Yunger Halpern. Quantum complexity in gravity, quantum field theory, and quantum information science.Physics Reports, 1159:1–77, 2026. ISSN 0370-1573. doi: https://doi.org/10.1016/j.physrep.2025.11.001. URL https://www.sciencedirect.com/scie...

    work page doi:10.1016/j.physrep.2025.11.001 2026

  4. [4]

    Krylov Complexity

    Eliezer Rabinovici, Adri´ an S´ anchez-Garrido, Ruth Shir, and Julian Sonner. Krylov complexity, 2025. URLhttps://arxiv.org/abs/2507.06286

    work page Pith review arXiv 2025

  5. [5]

    Caputa, J

    Pawe l Caputa, Javier M. Mag´ an, and Dimitrios Patramanis. Geometry of krylov complexity. Physical Review Research, 4(1):013041, 2022. doi: 10.1103/PhysRevResearch.4.013041. URL https://doi.org/10.1103/PhysRevResearch.4.013041

    work page doi:10.1103/physrevresearch.4.013041 2022

  6. [6]

    Quantum chaos and the complexity of Krylov of states,

    Vijay Balasubramanian, Pawe l Caputa, Javier M. Mag´ an, and Qingyue Wu. Quantum chaos and the complexity of spread of states.Physical Review D, 106(4):046007, August 2022. doi: 10.1103/PhysRevD.106.046007. URLhttps://doi.org/10.1103/PhysRevD.106.046007

    work page doi:10.1103/physrevd.106.046007 2022

  7. [7]

    Krylov complexity in conformal field theory,

    Anatoly Dymarsky and Michael Smolkin. Krylov complexity in conformal field theory. Physical Review D, 104(8):L081702, 2021. doi: 10.1103/PhysRevD.104.L081702

    work page doi:10.1103/physrevd.104.l081702 2021

  8. [8]

    Dymarsky, A

    Anatoly Dymarsky and Alexander Gorsky. Quantum chaos as delocalization in krylov space. Physical Review B, 102(8):085137, 2020. doi: 10.1103/PhysRevB.102.085137. URL https://doi.org/10.1103/PhysRevB.102.085137

    work page doi:10.1103/physrevb.102.085137 2020

Show all 62 references
  1. [9]

    Krylov complexity from integrability to chaos.Journal of High Energy Physics, 2022(7):151, 2022

    Eliezer Rabinovici, Adri´ an S´ anchez-Garrido, Ruth Shir, and Julian Sonner. Krylov complexity from integrability to chaos.Journal of High Energy Physics, 2022(7):151, 2022. doi: 10.1007/JHEP07(2022)151

    2022 doi

  2. [10]

    Krylov complexity in saddle-dominated scrambling.Journal of High Energy Physics, 2022(5):174, May 2022

    Budhaditya Bhattacharjee, Xiangyu Cao, Pratik Nandy, and Tanay Pathak. Krylov complexity in saddle-dominated scrambling.Journal of High Energy Physics, 2022(5):174, May 2022. ISSN 1029-8479. doi: 10.1007/JHEP05(2022)174. URL https://doi.org/10.1007/JHEP05(2022)174

    2022 doi

  3. [11]

    Krylov complexity in quantum field theory, and beyond.Journal of High Energy Physics, 2024(6):066, 2024

    Alexander Avdoshkin, Anatoly Dymarsky, and Michael Smolkin. Krylov complexity in quantum field theory, and beyond.Journal of High Energy Physics, 2024(6):066, 2024. doi: 10.1007/JHEP06(2024)066

    2024 doi

  4. [12]

    Matsoukas-Roubeas, and Adolfo del Campo

    Niklas H¨ ornedal, Nicoletta Carabba, Apollonas S. Matsoukas-Roubeas, and Adolfo del Campo. Ultimate speed limits to the growth of operator complexity.Communications Physics, 5:207, 2022. doi: 10.1038/s42005-022-00985-1

    2022 doi

  5. [13]

    A relation between krylov and nielsen complexity.Phys

    Ben Craps, Oleg Evnin, and Gabriele Pascuzzi. A relation between krylov and nielsen complexity.Phys. Rev. Lett., 132:160402, Apr 2024. doi: 10.1103/PhysRevLett.132.160402. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.132.160402. – 40 –

    2024 doi

  6. [14]

    Quantum state complexity meets many-body scars.Journal of Physics: Condensed Matter, 36(15): 155601, jan 2024

    Sourav Nandy, Bhaskar Mukherjee, Arpan Bhattacharyya, and Aritra Banerjee. Quantum state complexity meets many-body scars.Journal of Physics: Condensed Matter, 36(15): 155601, jan 2024. doi: 10.1088/1361-648X/ad1a7b. URL https://doi.org/10.1088/1361-648X/ad1a7b

    2024 doi

  7. [15]

    Krylov complexity under hamiltonian deformations and toda flows.Phys

    Kazutaka Takahashi, Pratik Nandy, and Adolfo del Campo. Krylov complexity under hamiltonian deformations and toda flows.Phys. Rev. B, 113:144312, Apr 2026. doi: 10.1103/zt9g-scp5. URLhttps://link.aps.org/doi/10.1103/zt9g-scp5

    2026 doi

  8. [16]

    Entanglement and quantum coherence in krylov space dynamics, 2026

    Swati Choudhary, Sukrut Mondkar, and Ujjwal Sen. Entanglement and quantum coherence in krylov space dynamics, 2026. URLhttps://arxiv.org/abs/2603.26619

    2026

  9. [17]

    Geometric Operator Quantum Speed Limit, Wegner Hamiltonian Flow and Operator Growth

    Niklas H¨ ornedal, Nicoletta Carabba, Kazutaka Takahashi, and Adolfo del Campo. Geometric Operator Quantum Speed Limit, Wegner Hamiltonian Flow and Operator Growth. Quantum, 7:1055, July 2023. ISSN 2521-327X. doi: 10.22331/q-2023-07-11-1055. URL https://doi.org/10.22331/q-2023...

    2023 doi

  10. [18]

    Shortcuts to adiabaticity in krylov space.Phys

    Kazutaka Takahashi and Adolfo del Campo. Shortcuts to adiabaticity in krylov space.Phys. Rev. X, 14:011032, Feb 2024. doi: 10.1103/PhysRevX.14.011032. URL https://link.aps.org/doi/10.1103/PhysRevX.14.011032

    2024 doi

  11. [19]

    A lanczos approach to the adiabatic gauge potential, 2023

    Budhaditya Bhattacharjee. A lanczos approach to the adiabatic gauge potential, 2023. URL https://arxiv.org/abs/2302.07228

    2023

  12. [20]

    Takuya Hatomura. Shortcuts to adiabaticity: theoretical framework, relations between different methods, and versatile approximations.Journal of Physics B: Atomic, Molecular and Optical Physics, 57(10):102001, apr 2024. doi: 10.1088/1361-6455/ad38f1. URL https://doi.org/10.1088...

    2024 doi

  13. [21]

    Universal counterdiabatic driving in krylov space.PRX Quantum, 6:040320, Oct 2025

    Stewart Morawetz and Anatoli Polkovnikov. Universal counterdiabatic driving in krylov space.PRX Quantum, 6:040320, Oct 2025. doi: 10.1103/wbbs-s8fs. URL https://link.aps.org/doi/10.1103/wbbs-s8fs

    2025 doi

  14. [22]

    Universal defect statistics in counterdiabatic quantum critical dynamics, 2025

    Andr´ as Grabarits and Adolfo del Campo. Universal defect statistics in counterdiabatic quantum critical dynamics, 2025. URLhttps://arxiv.org/abs/2503.22212

    2025

  15. [23]

    Pedro H. S. Bento, Adolfo del Campo, and Lucas C. C´ eleri. Krylov complexity and dynamical phase transition in the quenched lipkin-meshkov-glick model.Phys. Rev. B, 109: 224304, Jun 2024. doi: 10.1103/PhysRevB.109.224304. URL https://link.aps.org/doi/10.1103/PhysRevB.109.224304

    2024 doi

  16. [24]

    Universal growth of krylov complexity across a quantum phase transition, 2025

    Andr´ as Grabarits and Adolfo del Campo. Universal growth of krylov complexity across a quantum phase transition, 2025. URLhttps://arxiv.org/abs/2510.13947

    2025

  17. [25]

    Dynamical quantum phase transition, metastable state, and dimensionality reduction: Krylov analysis of fully connected spin models.Phys

    Kazutaka Takahashi. Dynamical quantum phase transition, metastable state, and dimensionality reduction: Krylov analysis of fully connected spin models.Phys. Rev. B, 112: 054312, Aug 2025. doi: 10.1103/m4jf-7svp. URL https://link.aps.org/doi/10.1103/m4jf-7svp

    2025 doi

  18. [26]

    Bhargava, Sebasti´ an V

    Anne-Maria Visuri, Alejandro Gomez Cadavid, Balaganchi A. Bhargava, Sebasti´ an V. Romero, Andr´ as Grabarits, Pranav Chandarana, Enrique Solano, Adolfo del Campo, and Narendra N. Hegade. Digitized counterdiabatic quantum critical dynamics.npj Quantum Information, 12(1):47, Ma...

    2026 doi

  19. [27]

    Krylov subspace methods for quantum – 41 – dynamics with time-dependent generators.Physical Review Letters, 134(3):030401, 2025

    Kazutaka Takahashi and Adolfo del Campo. Krylov subspace methods for quantum – 41 – dynamics with time-dependent generators.Physical Review Letters, 134(3):030401, 2025. doi: 10.1103/PhysRevLett.134.030401. URL https://doi.org/10.1103/PhysRevLett.134.030401

    2025 doi

  20. [28]

    Yates and Aditi Mitra

    Daniel J. Yates and Aditi Mitra. Strong and almost strong modes of floquet spin chains in krylov subspaces.Phys. Rev. B, 104:195121, Nov 2021. doi: 10.1103/PhysRevB.104.195121. URLhttps://link.aps.org/doi/10.1103/PhysRevB.104.195121

    2021 doi

  21. [29]

    Yates, Alexander G

    Daniel J. Yates, Alexander G. Abanov, and Aditi Mitra. Long-lived period-doubled edge modes of interacting and disorder-free floquet spin chains.Communications Physics, 5(1):43, Feb 2022. ISSN 2399-3650. doi: 10.1038/s42005-022-00818-1. URL https://doi.org/10.1038/s42005-022-00818-1

    2022 doi

  22. [30]

    Nizami and Ankit W

    Amin A. Nizami and Ankit W. Shrestha. Krylov construction and complexity for driven quantum systems.Physical Review E, 108(5):054222, November 2023. doi: 10.1103/PhysRevE.108.054222. URLhttps://doi.org/10.1103/PhysRevE.108.054222

    2023 doi

  23. [31]

    Nizami and Ankit W

    Amin A. Nizami and Ankit W. Shrestha. Spread complexity and quantum chaos for periodically driven spin chains.Phys. Rev. E, 110:034201, Sep 2024. doi: 10.1103/PhysRevE.110.034201. URL https://link.aps.org/doi/10.1103/PhysRevE.110.034201

    2024 doi

  24. [32]

    Universal model of floquet operator krylov space.Phys

    Hsiu-Chung Yeh and Aditi Mitra. Universal model of floquet operator krylov space.Phys. Rev. B, 110:155109, Oct 2024. doi: 10.1103/PhysRevB.110.155109. URL https://link.aps.org/doi/10.1103/PhysRevB.110.155109

    2024 doi

  25. [33]

    Philippe Suchsland, Roderich Moessner, and Pieter W. Claeys. Krylov complexity and trotter transitions in unitary circuit dynamics.Phys. Rev. B, 111:014309, Jan 2025. doi: 10.1103/PhysRevB.111.014309. URL https://link.aps.org/doi/10.1103/PhysRevB.111.014309

    2025 doi

  26. [34]

    Lewis, H

    Jr. Lewis, H. R. and W. B. Riesenfeld. An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field. Journal of Mathematical Physics, 10(8):1458–1473, August 1969. doi: 10.1063/1.1664991. URLhttps://doi.o...

    1969 doi

  27. [35]

    Nova Science Publishers, New York, 2001

    Ali Mostafazadeh.Dynamical Invariants, Adiabatic Approximation and the Geometric Phase. Nova Science Publishers, New York, 2001. ISBN 978-1-59033-013-5

    2001

  28. [36]

    N. D. Birrell and P. C. W. Davies.Quantum Fields in Curved Space. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 1982. ISBN 978-0-521-23385-9. doi: 10.1017/CBO9780511622632

    1982 doi

  29. [37]

    Chapter 2 - shortcuts to adiabaticity

    Erik Torrontegui, Sara Ib´ a˜ nez, Sofia Mart´ ınez-Garaot, Michele Modugno, Adolfo del Campo, David Gu´ ery-Odelin, Andreas Ruschhaupt, Xi Chen, and Juan Gonzalo Muga. Chapter 2 - shortcuts to adiabaticity. In Ennio Arimondo, Paul R. Berman, and Chun C. Lin, editors, Advances...

    2013 doi

  30. [38]

    A. M. Perelomov. Coherent states for arbitrary lie group.Communications in Mathematical Physics, 26(3):222–236, 1972. doi: 10.1007/BF01645091. URL https://doi.org/10.1007/BF01645091. – 42 –

    1972 doi

  31. [39]

    A. M. Perelomov. Generalized coherent states and some of their applications.Soviet Physics Uspekhi, 20(9):703–720, September 1977. doi: 10.1070/PU1977v020n09ABEH005459. URL https://doi.org/10.1070/PU1977v020n09ABEH005459

    1977 doi

  32. [40]

    Theoretical and Mathematical Physics

    Askold Perelomov.Generalized Coherent States and Their Applications. Theoretical and Mathematical Physics. Springer, Berlin, Heidelberg, 1986. ISBN 978-3-540-15912-4. doi: 10.1007/978-3-642-61629-7

    1986 doi

  33. [41]

    Decomposition formulas for su(1, 1) and su(2) lie algebras and their applications in quantum optics.Journal of the Optical Society of America B, 10(8): 1347–1359, August 1993

    Masashi Ban. Decomposition formulas for su(1, 1) and su(2) lie algebras and their applications in quantum optics.Journal of the Optical Society of America B, 10(8): 1347–1359, August 1993. doi: 10.1364/JOSAB.10.001347. URL https://doi.org/10.1364/JOSAB.10.001347

    1993 doi

  34. [42]

    Coherent states: Theory and some applications.Reviews of Modern Physics, 62(4):867–927, 1990

    Wei-Min Zhang, Da Hsuan Feng, and Robert Gilmore. Coherent states: Theory and some applications.Reviews of Modern Physics, 62(4):867–927, 1990. doi: 10.1103/RevModPhys.62.867. URLhttps://doi.org/10.1103/RevModPhys.62.867

    1990 doi

  35. [43]

    Probing the entanglement of operator growth.Progress of Theoretical and Experimental Physics, 2022(6):063A01, 2022

    Dimitrios Patramanis. Probing the entanglement of operator growth.Progress of Theoretical and Experimental Physics, 2022(6):063A01, 2022. doi: 10.1093/ptep/ptac081. URL https://doi.org/10.1093/ptep/ptac081

    2022 doi

  36. [44]

    Lie algebraic solution of linear differential equations

    James Wei and Edward Norman. Lie algebraic solution of linear differential equations. Journal of Mathematical Physics, 4(4):575–581, 1963. doi: 10.1063/1.1703993. URL https://doi.org/10.1063/1.1703993

    1963 doi

  37. [45]

    Wei–norman equations for a unitary evolution.Journal of Physics A: Mathematical and Theoretical, 46(26):265208, June 2013

    Szymon Charzy´ nski and Marek Ku´ s. Wei–norman equations for a unitary evolution.Journal of Physics A: Mathematical and Theoretical, 46(26):265208, June 2013. doi: 10.1088/1751-8113/46/26/265208. URL https://doi.org/10.1088/1751-8113/46/26/265208

    2013 doi

  38. [46]

    Explicit wei–norman formulae for matrix lie groups via putzer’s method

    Claudio Altafini. Explicit wei–norman formulae for matrix lie groups via putzer’s method. Systems & Control Letters, 54(11):1121–1130, 2005. doi: 10.1016/j.sysconle.2005.03.006. URLhttps://doi.org/10.1016/j.sysconle.2005.03.006

    2005 doi

  39. [47]

    Eberly.Optical Resonance and Two-Level Atoms

    Leslie Allen and Joseph H. Eberly.Optical Resonance and Two-Level Atoms. Interscience Monographs and Texts in Physics and Astronomy. Wiley, New York, 1975

    1975

  40. [48]

    Morris and Bruce W

    James R. Morris and Bruce W. Shore. Reduction of degenerate two-level excitation to independent two-state systems.Physical Review A, 27(2):906–912, 1983. doi: 10.1103/PhysRevA.27.906. URLhttps://doi.org/10.1103/PhysRevA.27.906

    1983 doi

  41. [49]

    Viswanath and G

    V. Viswanath and G. M¨ uller.The Recursion Method: Application to Many-Body Dynamics. Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-540-48651-0

    1994 doi

  42. [50]

    Krylov localization and suppression of complexity.Journal of High Energy Physics, 2022(3):211,

    Eliezer Rabinovici, Adri´ an S´ anchez-Garrido, Ruth Shir, and Julian Sonner. Krylov localization and suppression of complexity.Journal of High Energy Physics, 2022(3):211,

    2022

  43. [51]

    doi: 10.1007/JHEP03(2022)211

    2022 doi

  44. [52]

    Humphreys.Introduction to Lie Algebras and Representation Theory, volume 9 of Graduate Texts in Mathematics

    James E. Humphreys.Introduction to Lie Algebras and Representation Theory, volume 9 of Graduate Texts in Mathematics. Springer-Verlag, New York, revised edition, 1978. ISBN 978-0-387-90053-7. doi: 10.1007/978-1-4612-6398-2

    1978 doi

  45. [53]

    Graduate Texts in Contemporary Physics

    Philippe Di Francesco, Pierre Mathieu, and David S´ en´ echal.Conformal Field Theory. Graduate Texts in Contemporary Physics. Springer, New York, 1997. ISBN 978-0-387-94785-3. doi: 10.1007/978-1-4612-2256-9. – 43 –

    1997 doi

  46. [54]

    Some basics of su(1,1).Revista Brasileira de Ensino de F´ ısica, 26(4): 351–357, 2004

    Marcel Novaes. Some basics of su(1,1).Revista Brasileira de Ensino de F´ ısica, 26(4): 351–357, 2004. doi: 10.1590/S1806-11172004000400012. URL https://doi.org/10.1590/S1806-11172004000400012

    2004 doi

  47. [55]

    M. A. Lohe. Exact time dependence of solutions to the time-dependent schr¨ odinger equation. Journal of Physics A: Mathematical and Theoretical, 42(3):035307, 2009. doi: 10.1088/1751-8113/42/3/035307. URL https://doi.org/10.1088/1751-8113/42/3/035307

    2009 doi

  48. [56]

    J. G. Muga, X. Chen, S. Ib´ a˜ nez, I. Lizuain, and A. Ruschhaupt. Transitionless quantum drivings for the harmonic oscillator.Journal of Physics B: Atomic, Molecular and Optical Physics, 43(8):085509, April 2010. doi: 10.1088/0953-4075/43/8/085509. URL https://doi.org/10.1088...

    2010 doi

  49. [57]

    Coelho, Lucas Queiroz, and Danilo T

    Stanley S. Coelho, Lucas Queiroz, and Danilo T. Alves. The time-dependent quantum harmonic oscillator: a pedagogical approach via the lewis–riesenfeld dynamical invariant method.European Journal of Physics, 46(4):045401, 2025. doi: 10.1088/1361-6404/add972. URLhttps://doi.org/...

    2025 doi

  50. [58]

    Coelho, Lucas Queiroz, and Danilo T

    Stanley S. Coelho, Lucas Queiroz, and Danilo T. Alves. Exact solution of a time-dependent quantum harmonic oscillator with two frequency jumps via the lewis–riesenfeld dynamical invariant method.Entropy, 24(12):1851, December 2022. doi: 10.3390/e24121851. URL https://doi.org/1...

    2022 doi

  51. [59]

    Bounds in nonequilibrium quantum dynamics

    Zongping Gong and Ryusuke Hamazaki. Bounds in nonequilibrium quantum dynamics. International Journal of Modern Physics B, 36(31):2230007, 2022. doi: 10.1142/S0217979222300079. URLhttps://doi.org/10.1142/S0217979222300079

    2022 doi

  52. [60]

    Springer Theses

    Nicoletta Carabba.Quantum Speed Limits to Operator Growth. Springer Theses. Springer Cham, 1 edition, 2024. ISBN 978-3-031-74178-4. doi: 10.1007/978-3-031-74179-1. URL https://doi.org/10.1007/978-3-031-74179-1

    2024 doi

  53. [61]

    The growth of operator entropy in operator growth.Journal of High Energy Physics, 2022(8):232, Aug 2022

    Zhong-Ying Fan. The growth of operator entropy in operator growth.Journal of High Energy Physics, 2022(8):232, Aug 2022. ISSN 1029-8479. doi: 10.1007/JHEP08(2022)232. URLhttps://doi.org/10.1007/JHEP08(2022)232

    2022 doi

  54. [62]

    Speed limits and scrambling in krylov space.Phys

    Ankit Gill and Tapobrata Sarkar. Speed limits and scrambling in krylov space.Phys. Rev. B, 111:184307, May 2025. doi: 10.1103/PhysRevB.111.184307. URL https://link.aps.org/doi/10.1103/PhysRevB.111.184307. – 44 –

    2025 doi

Pith tools

Reviewed May 19, 2026 · model on record in the stance chip above.