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arxiv: 2605.28681 · v2 · pith:AEHVBF7Rnew · submitted 2026-05-27 · ✦ hep-th · quant-ph

Krylov complexity has it all

Wolfgang M\"uck This is my paper

Pith reviewed 2026-06-29 11:10 UTC · model grok-4.3

classification ✦ hep-th quant-ph
keywords Krylov complexityLanczos coefficientsoperator dynamicsreturn amplitudespectral densityspread complexityquantum systems
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0 comments X

The pith

Krylov complexity encodes the full dynamics of any quantum operator.

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 complexity contains every piece of information needed to describe how a quantum operator evolves. It demonstrates the claim by building a recursive algorithm that extracts all Lanczos coefficients directly from the short-time Taylor expansion of the complexity. This places Krylov complexity on the same footing as the Lanczos coefficients, the return amplitude, and the spectral density as complete descriptors of operator growth. A reader would care because it unifies several standard tools for tracking quantum dynamics into one quantity. The work also clarifies that spread complexity does not permit an analogous reconstruction without extra dynamical data.

Core claim

Krylov complexity contains the entire information about the dynamics of a quantum operator. An explicit recursive algorithm recovers the Lanczos coefficients from the Taylor expansion of Krylov complexity around t=0. This equivalence adds Krylov complexity to the list of quantities—the Lanczos coefficients, return amplitude, and spectral density—that fully characterize operator evolution.

What carries the argument

The recursive algorithm that extracts Lanczos coefficients from the Taylor expansion of Krylov complexity around t=0.

If this is right

  • Krylov complexity can be used in place of the Lanczos coefficients to characterize operator growth.
  • The return amplitude and spectral density are recoverable from Krylov complexity alone.
  • Operator evolution in quantum systems is fully described by Krylov complexity without further input.
  • No similar complete reconstruction is possible for spread complexity without additional dynamical information.

Where Pith is reading between the lines

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

  • Numerical studies could treat early-time Krylov complexity as a single sufficient observable for operator dynamics.
  • The equivalence may simplify calculations in holographic models where operator growth maps to bulk quantities.
  • Simple lattice models could serve as direct numerical tests of the recursive recovery procedure.
  • The distinction drawn with spread complexity suggests that completeness is not automatic for every notion of complexity.

Load-bearing premise

The Taylor expansion of Krylov complexity around t=0 encodes sufficient independent information to allow recursive recovery of all Lanczos coefficients without loss or additional dynamical input.

What would settle it

Compute the Krylov complexity Taylor series in a concrete model such as a free fermion chain, run the recursive algorithm, and check whether the recovered Lanczos coefficients match those obtained by direct orthogonalization.

read the original abstract

This paper establishes that Krylov complexity contains the entire information about the dynamics of a quantum operator, extending the list of equivalent quantities that can serve this purpose, such as the Lanczos coefficients, the return amplitude, and the spectral density. To demonstrate this equivalence, an explicit recursive algorithm is constructed to calculate Lanczos coefficients from the Taylor expansion of the Krylov complexity around $t=0$. Furthermore, the paper discusses the distinction between Krylov and spread complexity, clarifying why a similar recursive algorithm cannot exist for the latter without additional dynamical input. These results provide a ``proof of principle'' for using Krylov complexity as a complete characterization of operator evolution in quantum systems.

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

1 major / 1 minor

Summary. The manuscript claims that Krylov complexity fully encodes the dynamics of a quantum operator, placing it on equal footing with the Lanczos coefficients, return amplitude, and spectral density. It supports this by constructing an explicit recursive algorithm that extracts all Lanczos coefficients b_n from the Taylor coefficients of the Krylov complexity C_K(t) expanded around t=0, and it explains why an analogous recursion cannot exist for spread complexity without supplementary dynamical information.

Significance. If the recursion is bijective and free of singularities, the result would establish Krylov complexity as a complete, self-contained descriptor of operator evolution. This would add a new entry to the list of equivalent characterizations and could streamline analytic and numerical studies of operator growth and quantum chaos.

major comments (1)
  1. [recursive algorithm section (near the abstract's description of the Taylor expansion)] The central claim rests on the asserted bijectivity of the map from the infinite sequence {b_n} to the Taylor coefficients of C_K(t). The manuscript must demonstrate that the recursion uniquely determines every b_n at all orders without division by zero or implicit use of the return amplitude (or other external data). A concrete check—either an inductive proof or explicit computation through at least order 10—should be supplied in the section presenting the algorithm.
minor comments (1)
  1. Clarify the precise definition of C_K(t) used in the Taylor expansion (e.g., whether it is the normalized or un-normalized version) to avoid ambiguity when comparing with the return amplitude.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive suggestion regarding the recursive algorithm. We address the major comment below and will incorporate the requested demonstration in the revised version.

read point-by-point responses

  1. Referee: The central claim rests on the asserted bijectivity of the map from the infinite sequence {b_n} to the Taylor coefficients of C_K(t). The manuscript must demonstrate that the recursion uniquely determines every b_n at all orders without division by zero or implicit use of the return amplitude (or other external data). A concrete check—either an inductive proof or explicit computation through at least order 10—should be supplied in the section presenting the algorithm.

    Authors: We agree that an explicit verification of bijectivity, uniqueness at all orders, and the absence of division-by-zero singularities is essential to substantiate the claim. The recursive algorithm presented in the manuscript is derived solely from the Taylor expansion of C_K(t) at t=0 and does not invoke the return amplitude or any other external dynamical data. In the revised manuscript we will add, in the section describing the algorithm, either a short inductive proof establishing that each b_n is uniquely determined without singularities or an explicit computation of the first ten orders, confirming the recursion proceeds unambiguously from the Taylor coefficients alone. revision: yes

Circularity Check

0 steps flagged

No significant circularity; explicit recursion provides independent inverse map

full rationale

The paper's central claim is supported by constructing an explicit recursive algorithm that recovers all Lanczos coefficients from the Taylor coefficients of Krylov complexity at t=0. This constitutes a direct demonstration of bijective equivalence rather than a self-definitional loop, a fitted parameter renamed as prediction, or load-bearing self-citation. The algorithm is presented as the proof of principle that the Taylor series encodes the full dynamics without additional input, and no quoted step reduces the result to its own inputs by construction. The derivation remains self-contained against the definitions of the quantities involved.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review identifies no explicit free parameters, axioms, or invented entities; the claim rests on standard assumptions of quantum operator evolution and analyticity of the complexity function at t=0.

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Forward citations

Cited by 2 Pith papers

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

  1. Higher-loop wormhole length in sine-dilaton gravity from DSSYK Krylov complexity

    hep-th 2026-06 unverdicted novelty 6.0

    Five-loop perturbative expansion of DSSYK Krylov complexity is computed via singular perturbation and mapped to wormhole length in sine-dilaton gravity, with variance, cumulants, and large-time resummation also derived.

  2. Higher-loop wormhole length in sine-dilaton gravity from DSSYK Krylov complexity

    hep-th 2026-06 unverdicted novelty 6.0

    Five-loop perturbative computation of DSSYK Krylov complexity equaling wormhole length in sine-dilaton gravity, with cumulants and all-order large-time resummation.

Reference graph

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