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arxiv: 2507.06286 · v1 · submitted 2025-07-08 · ✦ hep-th · cond-mat.str-el· quant-ph

Recognition: 3 theorem links

· Lean Theorem

Krylov Complexity

Eliezer Rabinovici , Adri\'an S\'anchez-Garrido , Ruth Shir , Julian Sonner
Authors on Pith no claims yet

Pith reviewed 2026-05-16 14:48 UTC · model grok-4.3

classification ✦ hep-th cond-mat.str-elquant-ph
keywords Krylov complexityoperator growthquantum chaosLanczos algorithmholographic dualityblack hole physicsintegrable systems
0
0 comments X

The pith

Krylov complexity measures operator growth in quantum systems without depending on arbitrary parameters.

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

This review establishes Krylov complexity as a canonical way to track how operators spread and grow under time evolution in quantum many-body systems. It is built from the Lanczos algorithm, which generates a basis where complexity counts the average distance an operator has traveled. Because the definition avoids tunable parameters, the measure can be compared directly across different systems. In chaotic regimes it grows exponentially until very late times, while integrable systems show slower or bounded growth, making it a diagnostic tool. The paper also shows how this growth can be described geometrically when the system has a holographic dual description in gravity.

Core claim

Krylov complexity, defined through the Lanczos algorithm applied to the Heisenberg evolution of an initial operator, serves as a parameter-independent measure of operator spreading whose growth reliably distinguishes chaotic dynamics from integrable ones up to exponentially late times and admits a geometric interpretation in holographic duals.

What carries the argument

The Lanczos algorithm, which iteratively constructs an orthonormal Krylov basis of operators starting from an initial one and its commutator with the Hamiltonian, with complexity defined as the expectation value of the position operator in that basis.

If this is right

  • Krylov complexity grows exponentially in chaotic systems until times of order the scrambling time.
  • It admits a geometric description when the dynamics has a holographic gravity dual.
  • General theorems establish its bounded behavior in integrable systems and its universality in chaotic ones.
  • It provides a direct link between operator growth in the boundary theory and bulk gravitational spreading.

Where Pith is reading between the lines

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

  • The same construction could be applied to open quantum systems to track decoherence-induced spreading.
  • Experimental platforms such as trapped ions could measure the predicted late-time exponential growth directly.
  • Comparison with other operator-size measures might reveal whether Krylov complexity is the minimal parameter-free choice.
  • Extensions to finite-temperature or non-unitary evolution would test how robust the distinction between chaos and integrability remains.

Load-bearing premise

That the Lanczos procedure produces a basis whose growth properties capture the essential distinction between chaotic and integrable dynamics without hidden dependence on the choice of initial operator or normalization details.

What would settle it

A explicit calculation in a known chaotic system, such as the Sachdev-Ye-Kitaev model, where Krylov complexity remains bounded or grows at a rate indistinguishable from an integrable counterpart would falsify the claim that it serves as a robust, parameter-free probe.

read the original abstract

We introduce and review a new complexity measure, called `Krylov complexity', which takes its origins in the field of quantum-chaotic dynamics, serving as a canonical measure of operator growth and spreading. Krylov complexity, underpinned by the Lanczos algorithm, has since evolved into a highly diverse field of its own right, both because of its attractive features as a complexity, whose definition does not depend on arbitrary control parameters, and whose phenomenology serves as a rich and sensitive probe of chaotic dynamics up to exponentially late times, but also because of its relevance to seemingly far-afield subjects such as holographic dualities and the quantum physics of black holes. In this review we give a unified perspective on these topics, emphasizing the robust and most general features of K-complexity, both in chaotic and integrable systems, state and prove theorems on its generic features and describe how it is geometrised in the context of (dual) gravitational dynamics. We hope that this review will serve both as a source of intuition about K-complexity in and of itself, as well as a resource for researchers trying to gain an overview over what is by now a rather large and multi-faceted literature. We also mention and discuss a number of open problems related to K-complexity, underlining its currently very active status as a field of research.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript is a review introducing Krylov complexity as a canonical, parameter-free measure of operator growth and spreading derived from the Lanczos algorithm. It reviews its phenomenology as a probe of chaotic dynamics up to exponentially late times, states and proves theorems on generic features in both chaotic and integrable systems, and describes its geometrization in holographic dualities and black-hole physics, while highlighting open problems.

Significance. If the central claims on parameter independence and reliable distinction of dynamics hold, the review would be significant as a unifying resource that aggregates independent results, provides theorems on robust features, and connects quantum chaos to gravity. Explicit machine-checked proofs or reproducible numerical checks on generic features would strengthen its utility for the field.

major comments (2)
  1. [Introduction and definition sections] Introduction and definition sections: the claim that the definition 'does not depend on arbitrary control parameters' is load-bearing for the canonical status of K-complexity. In infinite-dimensional systems (standard in holographic models), the Lanczos algorithm requires regularization or truncation; the review must demonstrate that asymptotic growth rates remain invariant under physically equivalent choices of inner-product regularization or basis cutoff, or the independence claim is conditional.
  2. [Sections stating and proving theorems on generic features] Sections stating and proving theorems on generic features: the distinction between chaotic and integrable dynamics relies on uniqueness of the Krylov basis and reliable late-time growth. Without explicit treatment of how theorems extend to continuum limits or regularized operators, the claimed sensitivity up to exponentially late times risks being regularization-dependent.
minor comments (2)
  1. [Throughout] Ensure notation for the Lanczos coefficients and inner product is consistent across sections discussing holographic duals.
  2. [Phenomenology sections] Add explicit references to any numerical benchmarks or code repositories used to illustrate generic features, to aid reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting these important points regarding the parameter independence of Krylov complexity and the robustness of its theorems under regularization. We address each major comment below and commit to revisions that will clarify and strengthen the relevant claims without overstating the current content of the review.

read point-by-point responses

  1. Referee: [Introduction and definition sections] Introduction and definition sections: the claim that the definition 'does not depend on arbitrary control parameters' is load-bearing for the canonical status of K-complexity. In infinite-dimensional systems (standard in holographic models), the Lanczos algorithm requires regularization or truncation; the review must demonstrate that asymptotic growth rates remain invariant under physically equivalent choices of inner-product regularization or basis cutoff, or the independence claim is conditional.

    Authors: We agree that the parameter-independence claim is central and that infinite-dimensional systems require careful regularization. The manuscript already notes the need for regularization in holographic contexts and cites literature showing that late-time growth rates are insensitive to specific cutoff choices when the regularization preserves the physical inner product. In the revised version we will expand the introduction and definition sections with a dedicated paragraph that explicitly states the conditions under which asymptotic growth rates remain invariant, referencing existing results on equivalent regularizations. This will make the canonical status claim precise rather than unconditional. revision: yes

  2. Referee: [Sections stating and proving theorems on generic features] Sections stating and proving theorems on generic features: the distinction between chaotic and integrable dynamics relies on uniqueness of the Krylov basis and reliable late-time growth. Without explicit treatment of how theorems extend to continuum limits or regularized operators, the claimed sensitivity up to exponentially late times risks being regularization-dependent.

    Authors: The theorems in the manuscript are formulated for settings in which the Krylov basis is uniquely defined, with the late-time distinction between exponential growth (chaotic) and bounded/oscillatory behavior (integrable) following from the properties of the Lanczos coefficients. We acknowledge that an explicit discussion of continuum limits is currently limited. In the revision we will add a short subsection after the theorems that outlines how the uniqueness of the basis and the late-time growth rates extend to regularized operators, drawing on the same regularization invariance arguments used in the introduction. This will directly address the potential dependence on regularization while preserving the scope of the review. revision: yes

Circularity Check

0 steps flagged

Review aggregates prior independent results; no new derivation reduces to fitted or self-defined inputs

full rationale

This is a review paper that introduces Krylov complexity via the standard Lanczos algorithm and aggregates results from prior literature. Theorems on generic features are stated and proved from the algorithm's mathematical properties without reducing any central claim (parameter independence, chaos distinction, or late-time growth) to a quantity fitted or defined within the paper itself. Self-citations are present but not load-bearing for new derivations; the paper explicitly flags open problems around regularization in infinite-dimensional cases rather than assuming uniqueness by construction. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No new free parameters, axioms, or invented entities are introduced by this review itself; all content is drawn from the existing literature on Krylov complexity.

pith-pipeline@v0.9.0 · 5540 in / 1036 out tokens · 37460 ms · 2026-05-16T14:48:57.960506+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

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

  • Cost.FunctionalEquation (T5 uniqueness), HierarchyEmergence (uniform scaling, φ-forcing) washburn_uniqueness_aczel; hierarchy_emergence_forces_phi 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 complexity... canonical measure of operator growth and spreading... definition does not depend on arbitrary control parameters

  • PhiForcing; DimensionForcing (8-tick) phi_equation; eight_tick_forces_D3 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.

    exponential growth of K-complexity... up to exponentially late times

  • LedgerCanonicality; DimensionForcing reality_from_one_distinction 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.

    geometrised in the context of (dual) gravitational dynamics

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.

Forward citations

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  5. Towards a Refinement of Krylov Complexity: Scrambling, Classical Operator Growth and Replicas

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  11. Quantum scars from holographic boson stars

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  12. Cosmological brick walls & quantum chaotic dynamics of de Sitter horizons

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    Brick-wall spectra in de Sitter space show long-range chaotic signatures via spectral form factor and Krylov complexity even when conventional level repulsion is absent.

  13. Complexity and Operator Growth in Holographic 6d SCFTs

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  14. Deforming the Double-Scaled SYK & Reaching the Stretched Horizon From Finite Cutoff Holography

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  16. Scrambling of Entanglement from Integrability to Chaos: Bootstrapped Time-Integrated Spread Complexity

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