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arxiv: 2510.22658 · v3 · submitted 2025-10-26 · ✦ hep-th · gr-qc· quant-ph

Toward Krylov-based holography in double-scaled SYK

Yichao Fu , Hyun-Sik Jeong , Keun-Young Kim , Juan F. Pedraza This is my paper

Pith reviewed 2026-05-18 04:13 UTC · model grok-4.3

classification ✦ hep-th gr-qcquant-ph
keywords Krylov complexitydouble-scaled SYKholographic dictionarywormhole velocityreplica wormholesbaby universesfirewallsthird quantization
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The pith

The growth rate of Krylov complexity in double-scaled SYK equals the wormhole velocity in its holographic dual.

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

The paper builds a holographic dictionary that translates observables from the Krylov subspace of the double-scaled SYK model into features of the dual two-dimensional gravity theory with baby universes. It establishes that the rate at which Krylov state complexity grows tracks the speed of a wormhole connecting two boundaries. Expectation values of this rate in coherent states can detect the presence of firewall-like structures by reconstructing bulk information. Higher Krylov complexities relate to replica wormhole contributions, and in the third-quantized description, Krylov entropy measures the entropy flow when tracing over baby universes. A sympathetic reader would care because this turns an abstract quantum information quantity into a concrete probe of gravitational dynamics and topology in a solvable model.

Core claim

Building on the duality between Krylov complexity and geodesic length in Jackiw-Teitelboim and sine-dilaton gravity, the authors develop a precise holographic dictionary for quantities in the Krylov subspace of the double-scaled Sachdev-Ye-Kitaev model. They show that the growth rate of Krylov state complexity corresponds to the wormhole velocity and that its expectation value in coherent states diagnoses firewall-like structures via bulk reconstruction. An alternative bulk picture uses the proper momentum of an infalling particle, forming a threefold duality. Higher-order complexities capture connected bulk contributions from replica wormholes, and Krylov entropy equals the von Neumann ent

What carries the argument

The threefold duality linking the growth rate of Krylov complexity, wormhole velocity, and the proper momentum of an infalling particle, which maps boundary Krylov-space quantities to bulk gravitational dynamics.

If this is right

  • The expectation value of the Krylov complexity growth rate in coherent states serves as a boundary diagnostic for firewall-like structures.
  • Higher-order Krylov complexities capture connected bulk contributions encoded by replica wormholes.
  • The logarithmic variant of Krylov complexity probes the replica saddle structure.
  • Krylov entropy quantifies information flow into the baby universe sector in the third-quantized setting.

Where Pith is reading between the lines

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

  • If this dictionary holds, Krylov complexity could be used to simulate aspects of bulk wormhole physics in quantum circuits based on SYK Hamiltonians.
  • Extending this to other models might connect Krylov complexity to other holographic complexity proposals like circuit complexity.
  • The approach suggests that ensemble-averaged gravity with baby universes admits a boundary description via Krylov entropy that tracks information loss to extra sectors.

Load-bearing premise

The duality between Krylov complexity and geodesic length in Jackiw-Teitelboim and sine-dilaton gravity extends reliably to the double-scaled SYK model and ensemble-averaged 2D gravity.

What would settle it

Numerically computing the Krylov complexity growth rate in the double-scaled SYK model and comparing it directly to the analytically predicted wormhole velocity from the dual gravity theory would test the proposed correspondence.

read the original abstract

Building on the duality between Krylov complexity and geodesic length in Jackiw-Teitelboim and sine-dilaton gravity, we develop a precise holographic dictionary for quantities in the Krylov subspace of the double-scaled Sachdev-Ye-Kitaev model (DSSYK). First, we demonstrate that the growth rate of Krylov state complexity corresponds to the wormhole velocity, and show that its expectation value in coherent states serves as a boundary diagnostic of firewall-like structures via bulk reconstruction. We also delineate an alternative bulk description in terms of the proper momentum of an infalling particle at early times, establishing a threefold duality between the Krylov complexity growth rate, wormhole velocity, and proper momentum, with clear regimes of validity. Beyond the first moments, we argue that higher-order Krylov complexities capture connected bulk contributions encoded by replica wormholes, while the logarithmic variant probes the replica saddle structure. Finally, within a third-quantized setting incorporating baby universes, we show that the Krylov entropy equals the von Neumann entropy of the parent-geometry density matrix obtained after tracing out baby universes, thereby quantifying information flow into the baby universe sector. Together, these results elevate Krylov-space observables to sharp probes of bulk dynamics and topology in ensemble-averaged 2D gravity.

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 paper extends the established Krylov complexity–geodesic length duality from Jackiw-Teitelboim and sine-dilaton gravity to the double-scaled SYK model. It claims that the growth rate of Krylov state complexity equals the wormhole velocity, that its coherent-state expectation value diagnoses firewall-like structures, and that a threefold duality holds with the proper momentum of an infalling particle. Higher Krylov complexities are argued to capture replica-wormhole contributions, while the logarithmic variant probes replica saddles; in a third-quantized baby-universe setting, Krylov entropy is shown to equal the traced von Neumann entropy of the parent geometry.

Significance. If the central mappings are rigorously justified, the work supplies new boundary observables that probe wormhole dynamics, replica saddles, and information flow into baby universes within ensemble-averaged 2D gravity. The delineation of regimes of validity and the explicit connection to third-quantized entropy are concrete strengths that could make Krylov-space quantities useful holographic diagnostics.

major comments (2)
  1. [§3] §3 (Krylov-geodesic dictionary in DSSYK): the claim that the growth rate of Krylov complexity equals wormhole velocity rests on transferring the inner-product definition from the JT/sine-dilaton derivation. In the double-scaling limit the Krylov basis inner product is constructed from ensemble-averaged two-point functions governed by the DSSYK density of states; no explicit check is given that this reproduces the geodesic-length functional used in the prior JT work. If the leading-order mapping fails, the threefold duality with wormhole velocity and proper momentum is not established.
  2. [§5] §5 (higher Krylov complexities and replica wormholes): the statement that higher-order Krylov complexities capture connected bulk contributions encoded by replica wormholes is presented as an argument rather than a derivation. The manuscript does not show how the ensemble-averaged moments of the Krylov basis reproduce the replica-wormhole saddles of the gravitational path integral; an explicit computation or saddle-point matching at the level of the first few moments is required to support the claim.
minor comments (2)
  1. [Abstract and §4] The regimes of validity for the threefold duality are stated in the abstract but lack quantitative error estimates or explicit bounds on the double-scaling parameter; adding these would strengthen the presentation.
  2. [§2] Notation for the coherent states used in the firewall diagnostic should be defined once in §2 and used consistently thereafter.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which help strengthen the rigor of our holographic dictionary. We address each major comment point by point below, indicating the revisions incorporated into the updated manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (Krylov-geodesic dictionary in DSSYK): the claim that the growth rate of Krylov complexity equals wormhole velocity rests on transferring the inner-product definition from the JT/sine-dilaton derivation. In the double-scaling limit the Krylov basis inner product is constructed from ensemble-averaged two-point functions governed by the DSSYK density of states; no explicit check is given that this reproduces the geodesic-length functional used in the prior JT work. If the leading-order mapping fails, the threefold duality with wormhole velocity and proper momentum is not established.

    Authors: We agree that an explicit verification in the double-scaling limit strengthens the presentation. The Krylov basis inner product is defined directly from the ensemble-averaged two-point functions of DSSYK, whose density of states is known to reproduce the sine-dilaton geometry in this limit. This ensures the inner product inherits the structure leading to the geodesic-length functional via the same saddle-point analysis as in prior JT work. To make the mapping fully explicit, we have added a derivation in the revised §3 computing the leading moments explicitly from the DSSYK spectral density and confirming they reproduce the wormhole velocity at leading order, thereby establishing the threefold duality with proper momentum in the delineated regimes of validity. revision: yes

  2. Referee: [§5] §5 (higher Krylov complexities and replica wormholes): the statement that higher-order Krylov complexities capture connected bulk contributions encoded by replica wormholes is presented as an argument rather than a derivation. The manuscript does not show how the ensemble-averaged moments of the Krylov basis reproduce the replica-wormhole saddles of the gravitational path integral; an explicit computation or saddle-point matching at the level of the first few moments is required to support the claim.

    Authors: We acknowledge that the original §5 framed the connection as an argument based on the structure of higher moments corresponding to multi-point functions that encode replica wormholes. To provide the requested explicit derivation, we have added a saddle-point analysis in the revised §5 for the first two higher-order moments. This computation demonstrates that the connected ensemble-averaged contributions match the replica-wormhole saddles of the gravitational path integral, with the logarithmic variant similarly aligned to the replica saddle structure. This strengthens the claim while remaining within the ensemble-averaged 2D gravity framework. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation extends external prior duality without reduction to inputs

full rationale

The paper explicitly states it builds on the established duality between Krylov complexity and geodesic length in Jackiw-Teitelboim and sine-dilaton gravity as a foundation for developing the dictionary in DSSYK. No equations or steps in the abstract or described claims reduce by construction to self-definitions, fitted parameters renamed as predictions, or self-citation chains within this work. The threefold duality, higher-order complexities, and third-quantized entropy relations are presented as extensions with stated regimes of validity, not tautological restatements of the input duality. The cited duality is external (from JT/sine-dilaton models) and treated as independent support rather than derived here, satisfying the criteria for non-circular extension.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Central claims rest on the assumed prior duality in JT/sine-dilaton gravity and on the standard setup of ensemble-averaged 2D gravity with third quantization and baby universes; no free parameters or new invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Duality between Krylov complexity and geodesic length holds in Jackiw-Teitelboim and sine-dilaton gravity
    Explicitly invoked as the foundation for extending the dictionary to DSSYK.

pith-pipeline@v0.9.0 · 5769 in / 1444 out tokens · 51543 ms · 2026-05-18T04:13:50.267030+00:00 · methodology

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Forward citations

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

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    Exact Krylov correlators in sl(2,R) models are proportional to proper radial momenta of infalling particles in BTZ black holes, extending the complexity-momentum correspondence to include fluctuations.

  2. q-Askey Deformations of Double-Scaled SYK

    hep-th 2026-05 unverdicted novelty 7.0

    q-Askey deformations of double-scaled SYK yield transfer matrices for orthogonal polynomials whose semiclassical chord dynamics map to ER bridges and new geometric transitions in sine dilaton gravity.

  3. Emergent States and Algebras from the Double-Scaling limit of Pure States in SYK

    hep-th 2026-04 unverdicted novelty 7.0

    In double-scaled SYK, state-adapted dressed chord operators change the emergent algebra from Type II1 to Type I∞ and restore purity of KM states, unlike generic operators.

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

    hep-th 2026-03 unverdicted novelty 7.0

    LogK complexity via replicas distinguishes genuine scrambling from saddle effects in quantum and classical systems and refines the measure for integrable cases.

  5. Krylov Subspace Dynamics as Near-Horizon AdS$_2$ Holography

    hep-th 2026-02 unverdicted novelty 7.0

    In the continuum limit the discrete Krylov chain becomes a Klein-Gordon field in AdS2, with Lanczos growth rate α identified as πT, recovering the maximal chaos bound and requiring the Breitenlohner-Freedman bound for...

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

    hep-th 2026-05 unverdicted novelty 6.0

    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.

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

    hep-th 2026-03 unverdicted novelty 6.0

    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.

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

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  9. Cosmological Entanglement Entropy from the von Neumann Algebra of Double-Scaled SYK & Its Connection with Krylov Complexity

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  10. Krylov state complexity for BMN matrix model

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    Holographic complexity of CFTs in global dS_d is computed via volume and action prescriptions in AdS foliation and brane setups, then compared to results from static and Poincare patches.

  12. Krylov complexity for Lin-Maldacena geometries and their holographic duals

    hep-th 2026-04 unverdicted novelty 5.0

    In the BMN matrix model and its holographic duals, Krylov basis states and Lanczos coefficients are uniquely fixed by the model's mass parameter.

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