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arxiv: 2510.19436 · v2 · submitted 2025-10-22 · 🪐 quant-ph · cond-mat.stat-mech· hep-th

Krylov Complexity Under Hamiltonian Deformations and Toda Flows

Kazutaka Takahashi , Pratik Nandy , Adolfo del Campo This is my paper

Pith reviewed 2026-05-18 05:17 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mechhep-th
keywords Krylov complexityHamiltonian deformationsToda flowsspread complexityKrylov entropyquantum dynamicscoherent Gibbs statesrandom matrices
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The pith

Certain Hamiltonian deformations leave the Krylov subspace unchanged while reorganizing its basis via generalized Toda equations.

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

The paper establishes that for a specific class of Hamiltonian deformations the minimal subspace containing the quantum dynamics stays exactly the same. The deformations only rearrange the basis inside that subspace, and the coefficients of the reorganized basis obey generalized Toda equations that depend on the deformation parameters. This relation lets imaginary-time-like evolutions be rewritten as real-time unitary dynamics. A reader would care because the construction gives a systematic route from known solvable models to new ones and supplies explicit formulas for survival probability, spread complexity, and Krylov entropy in thermodynamic and random-matrix settings.

Core claim

For a certain class of deformations, the resulting Krylov subspace is unchanged, and we observe time evolutions with a reorganized basis. The tridiagonal form of the generator in the Krylov space is maintained, and we obtain generalized Toda equations as a function of the deformation parameters. The imaginary-time-like evolutions can be described by real-time unitary ones.

What carries the argument

The Krylov subspace method applied to Hamiltonian deformations, which preserves the subspace and produces generalized Toda equations for the basis coefficients.

Load-bearing premise

There exists a nontrivial class of Hamiltonian deformations that leave the Krylov subspace invariant while merely reorganizing its basis.

What would settle it

Take a concrete quadratic deformation of a known Hamiltonian such as the harmonic oscillator, compute the Lanczos coefficients before and after the deformation, and check whether the Krylov subspace is identical and the new coefficients satisfy the generalized Toda equations.

Figures

Figures reproduced from arXiv: 2510.19436 by Adolfo del Campo, Kazutaka Takahashi, Pratik Nandy.

Figure 1
Figure 1. Figure 1: The spread complexity for systems with SL(2,R) symmetry. We set [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The spread complexity for systems described by [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The Lanczos coefficients an(β) and bn(β) of the two-dimensional Ising model. We take a 6 × 5 lattice with open boundary conditions, which gives the Krylov dimension d = 48. At β = 0, the diagonal components an(0) are zero and the off-diagonal components βn(0) are plotted in the inset of the right panel. The critical point at the thermodynamic limit is given by βJ = 1 2 ln(√ 2 + 1) ≈ 0.4407. -1.0 -0.8 -0.6 … view at source ↗
Figure 4
Figure 4. Figure 4: The Lanczos coefficients an and bn of the fully￾connected Ising model with N = 2000. The Krylov dimension is given by d = N/2 + 1 = 1001 and we plot the coefficients at intervals of 10. The point βJ = 1 represents the critical point at the thermodynamic limit. we consider “classical” Ising models generally written as H = − X (i,j) Jijσiσj , (76) where σi = ±1 represents spin variable at site i. In the stan… view at source ↗
Figure 6
Figure 6. Figure 6: The complexity K and the Krylov entropy S of the 6 × 5 Ising model. They are periodic in t with the period J t = π. The singularity in the β-axis appears at β = 2βc where βc represents the critical point of the corresponding ther￾modynamic system. C. Spread complexity and Krylov entropy To study how the thermodynamic and dynamical sin￾gularities are incorporated into other quantities defined in the Krylov … view at source ↗
Figure 8
Figure 8. Figure 8: The ensemble average of the spread complexity [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The result for d = 2 and τ = (0, τ2). See the caption in [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 12
Figure 12. Figure 12: The Lanczos coefficients for τ = (0, τ2) with τ2∆2 ∼ d. See the caption in [PITH_FULL_IMAGE:figures/full_fig_p011_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The Lanczos coefficients for τ = (0, τ2) with τ2∆2 ∼ d 2 . See the caption in [PITH_FULL_IMAGE:figures/full_fig_p011_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The time-averaged complexity K(τ ) (solid lines) and ⟨n⟩ = P n ne−f(ϵn,τ)+Sn /Z(τ ) (dashed lines) for random matrices of the unitary ensemble. We take d = 100 and d = 1000, and each result is averaged over 100 samples. The left panel denotes the case τ = (β, 0) and the right denotes τ = (0, τ2). accidental degeneracy and have D = d. For |ψ0(τ )⟩, we apply the Krylov algorithm to construct L(τ ). Then, th… view at source ↗
Figure 15
Figure 15. Figure 15: The upper panels represent ⟨K(t, τ )⟩ for the chiral random matrix ensembles. We take d = 1000, τ = (0, τ2), and the average is taken over 100 samples. The lower panels represent the Lanczos coefficients. We note an = 0 in the present setting. to write ⟨K(τ2)⟩ d − 1 ∼ 2 d R Emax 0 dE e−τ2E 2 ρ(E) R E 0 dE′ ρ(E′ ) R Emax 0 dE e−τ2E2 ρ(E) ∼ 2 ∆ R ∞ 0 dE e−τ2E 2 E R ∞ 0 dE e−τ2E2 ∝ 1 (τ2∆2) 1/2 . (98) C. Ran… view at source ↗
read the original abstract

The quantum dynamics of a complex system can be efficiently described in Krylov space, the minimal subspace in which the dynamics unfolds. We apply the Krylov subspace method for Hamiltonian deformations, which provides a systematic way of constructing solvable models from known instances. In doing so, we relate the evolution of deformed and undeformed theories and investigate their complexity. For a certain class of deformations, the resulting Krylov subspace is unchanged, and we observe time evolutions with a reorganized basis. The tridiagonal form of the generator in the Krylov space is maintained, and we obtain generalized Toda equations as a function of the deformation parameters. The imaginary-time-like evolutions can be described by real-time unitary ones. As possible applications, we discuss coherent Gibbs states for thermodynamic systems, for which we analyze the survival probability, spread complexity, Krylov entropy, and associated time-averaged quantities. We further discuss the statistical properties of random matrices and supersymmetric systems for quadratic deformations.

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

0 major / 3 minor

Summary. The paper applies the Krylov subspace method to Hamiltonian deformations of quantum systems. For a certain class of deformations the Krylov subspace remains invariant while its basis is reorganized; the tridiagonal generator is preserved and generalized Toda equations are derived in terms of the deformation parameters. Imaginary-time-like evolution is shown to be equivalent to real-time unitary dynamics. Concrete applications are given to coherent Gibbs states (survival probability, spread complexity, Krylov entropy and their time averages) and to quadratic deformations in random-matrix and supersymmetric ensembles.

Significance. If the invariance and Toda-flow constructions hold, the work supplies a systematic route to solvable models whose Krylov complexity can be related directly to that of the undeformed theory. The real-time/unitary description of imaginary-time evolution and the explicit thermodynamic and random-matrix applications are concrete strengths that could be useful for both analytic and numerical studies of quantum dynamics.

minor comments (3)
  1. The precise definition of the class of deformations that leave the Krylov subspace invariant should be stated explicitly in the main text (rather than only in the abstract) so that readers can verify the scope of the Toda-flow construction.
  2. In the applications to Gibbs states, the time-averaged quantities are introduced without an explicit formula or reference to the averaging procedure; adding the relevant expression would improve reproducibility.
  3. Figure captions for the random-matrix and supersymmetric examples would benefit from a short statement of the ensemble parameters and the quadratic deformation used.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript and for recommending minor revision. The referee's summary correctly identifies the core results: the invariance of the Krylov subspace under a class of Hamiltonian deformations, the emergence of generalized Toda equations, the equivalence between imaginary-time-like and real-time unitary dynamics, and the concrete applications to coherent Gibbs states and quadratic deformations in random-matrix and supersymmetric ensembles. We are pleased that the potential utility for constructing solvable models and for both analytic and numerical studies is recognized.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper constructs Hamiltonian deformations that preserve the Krylov subspace and tridiagonal generator, then derives generalized Toda equations directly as functions of the deformation parameters. This is presented as an algebraic consequence of the invariance rather than a fit to complexity data. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the abstract or described chain. Applications to Gibbs states and random-matrix statistics follow as direct implications of the same invariance. The derivation chain remains independent of the target observables.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on the existence of a special class of deformations that preserve the Krylov subspace and on the applicability of the Krylov method itself; deformation parameters appear as inputs that generate the Toda flows.

free parameters (1)
  • deformation parameters
    Parameters that label the class of deformations and enter the generalized Toda equations; their specific values are not derived from first principles in the abstract.
axioms (1)
  • domain assumption The Krylov subspace method can be applied to Hamiltonian deformations while preserving the minimal subspace for a certain class of deformations.
    Invoked when stating that the subspace remains unchanged and the tridiagonal form is maintained.

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