Circuit and Krylov complexity of primordial perturbations of modified gravity in inflation

Hai-Bing Fu; Tao Li

REVIEW 4 major objections 6 minor 1 cited by

Circuit and Krylov complexity of primordial perturbations of modified gravity in inflation

T0 review · 4 major / 6 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read The paper claims that in f(φ,R) modified-gravity inflation, the squeezed strength of primordial curvature perturbations is suppressed after horizon exit, making Krylov complexity much smaller while circuit complexity becomes larger than in

desk verdict New ODEs for the f(phi,R) squeezed parameters, but the quadratic action they come from is asserted without derivation and fails the B=0 consistency check against the canonical result. read the letter →

arxiv 2607.09408 v2 pith:TMUJWX7C submitted 2026-07-10 hep-th gr-qcquant-ph

Tao Li , Hai-Bing Fu This is my paper · ORCID

classification hep-thgr-qcquant-ph MSC 83F0581P4583D05

keywords Krylovcomplexitycircuittwo-modesqueezedstateinflationmodifiedgravityf(φR)primordialperturbationsLanczoscoefficients

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

This paper compares two inflationary models—canonical scalar-field inflation and a modified-gravity f(φ,R) inflation—through the lens of quantum complexity of primordial perturbations. The perturbations are modeled as a two-mode squeezed state, whose evolution is governed by the squeezed strength r_k and angle φ_k. The central claim is that the f(φ,R) coupling suppresses r_k after horizon exit, so Krylov complexity (K = sinh² r_k) grows far less than in canonical inflation, while the circuit complexity grows substantially more. If true, this means different gravitational theories leave distinct quantum-complexity fingerprints on the seeds of cosmic structure, offering a new way to probe modified gravity in the early universe.

What carries the argument

The central object is the two-mode squeezed state |ψ⟩_sq = (1/cosh r_k) Σ (−1)^n e^{2in φ_k} tanh^n r_k |n;n⟩, parameterized by the squeezed strength r_k and squeezed angle φ_k. Its evolution is determined by first-order differential equations (49)–(53) derived from the quadratic action. The Krylov complexity reduces exactly to K = sinh² r_k, so all Krylov diagnostics (complexity, entropy, Lanczos coefficients) are controlled by r_k alone. The modified-gravity contribution B enters the evolution equations and is the mechanism that suppresses r_k after horizon exit, thereby reducing Krylov complexity while increasing circuit complexity.

What would settle it

Re-derive the quadratic action for scalar perturbations in f(φ,R) inflation from the action (44) by explicitly expanding the metric and scalar field to second order in a fixed gauge (e.g., comoving curvature perturbation). If the resulting equation of motion differs from Eq. (47) in the coefficient (B−A), then the predicted suppression of r_k after horizon exit—and hence the complexity hierarchy—does not hold for generic f(φ,R) theories.

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Extended reading notes

Core claim

Working in the Jordan frame, the paper derives a quadratic action S^(2) = ∫dτ d³x [−π² − (∂v)² + (B−A)v²] for curvature perturbations in f(φ,R) inflation, where B = n² a² R/(κ² β²) encodes the modified-gravity correction and A encodes the potential. From this action it obtains evolution equations for the squeezed strength r_k and squeezed angle φ_k. The central result is that the f(φ,R) correction changes these parameters dramatically: before horizon exit it enhances the entanglement and causes oscillations, but after horizon exit it suppresses r_k relative to canonical inflation. Since the Krylov complexity of the two-mode squeezed state equals K = sinh² r_k, the paper finds that canonical

Load-bearing premise

Everything hinges on the asserted quadratic action (47) for f(φ,R) perturbations, which is written down without derivation—no metric perturbation, gauge choice, or sound-speed term is given; if the correct action contains extra terms, the squeezed parameters and all complexity outputs change.

Editorial extensions

If this is right

In f(φ,R) inflation, Krylov complexity K is much smaller than in canonical inflation because r_k is suppressed after horizon exit, implying slower operator growth of curvature perturbations.
Circuit complexity is significantly larger in f(φ,R) inflation, and the Lyapunov exponent is nearly unchanged while post-exit oscillations persist, suggesting more intricate quantum evolution.
Lanczos coefficients b_n grow monotonically and are greatly amplified by f(φ,R), indicating enhanced chaos in the modified-gravity model.
The dissipative coefficient c_n, introduced in an open-system extension, also grows after horizon exit; when dissipation is included, the paper infers that Krylov complexity can decrease rather than grow monotonically.
The paper connects larger Krylov complexity to larger curvature perturbations, implying that canonical inflation produces stronger late-time structure seeds than f(φ,R) inflation.

Reading between the lines

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

A proper second-order perturbation of the f(φ,R) action, with explicit gauge fixing, might reveal additional terms (e.g., sound-speed or non-minimal coupling) that could alter or even invert the hierarchy between canonical and f(φ,R) complexity; this is worth testing.
Because K = sinh² r_k equals the mean particle number per mode, the suppression of r_k suggests f(φ,R) inflation produces fewer particles after horizon exit; this quantitative prediction could be checked against the amplitude of the curvature power spectrum.
The open-system treatment with dissipation c_n implies that environmental effects during inflation could reduce observable complexity growth; extending the model to compute the resulting power spectrum would give a testable cosmological signature.
The opposite behaviors of circuit complexity (higher in f(φ,R)) and Krylov complexity (lower in f(φ,R)) indicate that these two measures are not interchangeable probes of early-universe dynamics, and future work should clarify which one, if either, connects to observables.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First thing to know: this is a routine application of the authors' established squeezed-state complexity machinery to f(phi,R) inflation. The new equations — the squeezed-parameter ODEs (49)–(53) and the Lanczos/dissipation coefficients (69) — are new for this action, and the paper is honest about borrowing the machinery from earlier work. The problem is that the quadratic action it all rests on, Eq. (47), is asserted in a single sentence, and the B=0 limit doesn't reduce to the canonical action (28): the gradient term has the wrong sign and A is a Fourier-space coefficient sitting in a position-space action. That's not a minor gap; the ODEs and all complexity outputs inherit whatever is wrong with that seed. The numerical section also omits initial conditions, the exact n/beta, and which Lanczos index is plotted, so the figures are not reproducible as presented. And the claims are not internally consistent: the abstract says f(phi,R) enhances r_k, the Sec. III summary says it suppresses r_k after horizon exit, and Sec. IV says Krylov complexity is amplified while Fig. 6 shows the opposite. The open-system 'negative growth' conclusion is not supported by the computed exponential growth either.

What the paper does well: it frames the question clearly, engages the existing literature (including the authors' own prior work, which is a legitimate foundation), and the formal definition of Krylov complexity for the two-mode squeezed state is standard and correctly recalled. If the quadratic action were derived properly, the subsequent steps would be straightforward. That's why this is fixable in principle.

Who is this for? Someone working in the quantum-information-in-cosmology subfield who wants to see another Lagrangian put through the complexity machine. Not a general hep-th audience. I would send it to a referee because the question is legitimate and the flaws are specific enough to be addressed, but the referee should demand the full derivation of (47) from the action (44), and until that happens the paper should not be cited as evidence about f(phi,R) complexity.

Referee Report

4 major / 6 minor

Summary. The paper compares the quantum-complexity history of primordial curvature perturbations in canonical single-field inflation with that in a Jordan-frame f(φ,R) modified-gravity inflation model. It derives evolution equations for the two-mode squeezed parameters (r_k, φ_k) from a claimed quadratic action, solves them numerically, and then computes circuit complexity, Krylov complexity K=sinh² r_k, Krylov entropy, Lanczos coefficients b_n, and a putative dissipative coefficient c_n. The central advertised result is that the f(φ,R) coupling changes the squeezing dynamics, enhancing r_k before horizon exit while suppressing it afterward, thereby reversing the hierarchy between circuit and Krylov complexity relative to canonical inflation.

Significance. The question of whether modified gravity leaves distinctive signatures in quantum-complexity observables of inflationary perturbations is a legitimate and timely one. The paper usefully reviews and applies standard squeezed-state and Krylov formalism, and the algebraic step K=sinh² r_k from the two-mode squeezed state is correct. However, the significance of the claimed result is not realized because every downstream conclusion rests on a quadratic action that is asserted rather than derived, and the manuscript contains direct contradictions about the sign of the f(φ,R) effect. The numerical outputs are therefore uncontrolled: a correct derivation of the f(φ,R) second-order action would in general change the ODEs, the squeezed parameters, and all complexity diagnostics. As it stands, the paper does not provide a reliable comparison between the two inflationary models.

major comments (4)

[§III B, Eq. (47)] The quadratic action S^(2)=∫dτ d³x[-π²-(∂_i v)²+(B-A)v²] is the load-bearing input for the entire paper, but it is not derived from the f(φ,R) action (44). The sentence 'we shall expand the correction for gravity to the second order' is not a derivation: no metric perturbation, gauge choice, field redefinition, or sound-speed term is provided. For a Jordan-frame f(φ,R) theory the standard second-order action is known to contain non-minimal coupling and sound-speed terms; a simple mass term B=n²a²R/(κ²β²) is not the generic result. Moreover, the B=0 limit fails an internal consistency check: setting n=0 in (47) gives -π²-(∂_i v)²-A v², whereas the canonical action (28) has +(∂_i v)² and -(8/3)αμ²a²v²; and A was defined in §III A as the Fourier-space coefficient 4αμ²a²/(3k), so (47) mixes position-space and Fourier-space objects. Any change in the correct action propagates directly into OD
[Abstract; §IV B 2; §V] The claimed physical effect is internally contradictory. The abstract states that the f(φ,R) coupling 'enhances the squeezed strength relative to the canonical scalar field inflation' and that 'this enhancement leads to a smaller growth in Krylov complexity.' But Eq. (70) gives K=sinh² r_k, which is strictly increasing in r_k; enhancing r_k cannot reduce K. The main text, in contrast, says r_k is suppressed after horizon exit (§III summary) and that canonical Krylov complexity is 'far bigger' than f(φ,R) (Fig. 6 discussion). The concluding bullet in §V says Krylov complexity is 'significantly amplified when the f(φ,R) coupling is included.' These statements cannot all be true. This contradicts the central quantitative message and prevents the reader from identifying what the paper actually claims.
[§IV B 1, Eqs. (63), (69)] The Lanczos and dissipation coefficients are presented as new physical diagnostics, but b_n and c_n are algebraic functions of the same A, B, a′/a inputs that already determine r_k; they add no independent information. The plots do not state which n is used, even though b_n ∝ n and c_n ∝ (2n+1); without this the numerical values in Figs. 4 and 5 are not reproducible. The coefficient c_n is called an 'effective dissipative contribution' and an 'open-system extension,' but no Lindblad/master equation is introduced; Eq. (57) is just the Liouvillian in tridiagonal form with a complex diagonal. The physical interpretation of c_n as energy exchange with the environment is therefore unsupported.
[§III, Eqs. (51)–(53); Figs. 1, 6] The numerical section omits essential data. No initial conditions r_k(y_i), φ_k(y_i) are given; the parameter n/β is only constrained as ≥1, not specified; and the de Sitter background a=-1/(ηH0) is assumed without derivation from the f(φ,R) background equations. The horizon exit is placed at y=-2 in Fig. 1 but at y=0 in Fig. 6. These omissions make the figures irreproducible and complicate any assessment of the claimed differences between the two models.

minor comments (6)

[Fig. 2 caption] The caption says 'The numerical solutions of r_k' but the plot shows φ_k; this should be corrected.
[§IV A] Two consecutive paragraphs are nearly verbatim duplicates ('Prior to the exit of the horizon...' and 'Prior to the horizon exit...'). The redundancy should be removed.
[§III and throughout] The text alternates between 'compression strength' and 'squeezed strength' for r_k. Use one consistent term.
[Eq. (33)] The notation for creation/annihilation operators is inconsistent: c_k vs c_{⃗k} and c†_{−⃗k}. Please standardize.
[Eqs. (45)–(46)] The approximation from f(φ)=1+ξ(e^{-βnφ}+e^{-β^{-1}nφ}) to f(φ)≈1+e^{-nφ/β} is not explained; the condition ξ=1 and the symmetry β↔1/β are stated but the approximation step needs justification.
[Figs. 4–6] The vertical axes are log-scaled but the axes labels still use 'y(Linear)' for the horizontal axis; the label is confusing and should be clarified.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction: complexity outputs are functions of the squeezed parameters solved from the model ODEs; the unsupported quadratic action (47) is a correctness risk, not a circular step.

full rationale

The paper's claimed derivation chain is: model action (44) -> asserted quadratic action (47) -> Hamiltonian (48) -> squeezed-parameter ODEs (49)-(53) -> r_k and phi_k -> circuit complexity (20) and Krylov-space quantities (56), (63), (69), (73). K = sinh^2 r_k is indeed a definitional relation for the two-mode squeezed state, but the paper does not infer r_k from K or fit K to data; r_k and phi_k are solved from model-dependent differential equations, and K is then evaluated from those solutions. Thus the model comparison does not reduce to the complexity formula; it reduces to the dynamics of r_k, which is the physical input being tested. The Lanczos coefficients and dissipation terms are algebraic outputs of the same Hamiltonian; the paper's assertion that b_n drives K even though the displayed K depends only on r_k is an internal-consistency problem, not a circularity. The self-citations [7,9,13,14,94] provide frameworks or motivation, but the central numerical comparison is computed in this paper from its own Hamiltonian; no uniqueness theorem or ansatz is imported from the authors' prior work as a load-bearing premise. The most serious weakness is Eq. (47): it is asserted with only "we shall expand the correction for gravity to the second order," without a derivation from (44), it places a k-dependent A = 4 alpha mu^2 a^2/(3k) into a position-space action, and setting B=0 does not return the canonical action (28). Those are unverified-assumption and consistency defects that would change the outputs, but they do not make the derivation circular: the outputs were not used to define or fit (47).

Assumptions & free parameters 6 free parameters · 5 assumptions · 1 invented entities

The paper's central outputs are determined by a handful of hand-chosen parameters (H0=1, k=0.01, αμ²=1) and an unstated initial-condition set, plus an ad hoc quadratic action whose derivation is missing. The 'dissipative' coefficient c_n is an invented diagnostic without independent support.

free parameters (6)

H0 = 1
Inflationary Hubble scale set to unity in all numerical integrations (Figs. 1–6); no sensitivity analysis or physical scale selection.
k = 0.01
Comoving wavenumber of the perturbation mode, fixed to 0.01 in all figures; determines horizon exit at y=-2.
αμ² = 1 (with κ²=1)
Potential-energy scale in the canonical action; chosen to equal 1, fixing the A term.
n/β (or n²/β² in B) = not specified (stated only as n/β ≥ 1)
Controls the modified-gravity contribution B=n²a²R/(κ²β²) in Eq. (47); no explicit value is used for the figures, so numerical results are not reproducible.
Initial conditions r_k(y_i), φ_k(y_i) = not stated
The ODE system (43)/(53) requires initial squeezed amplitude and angle; not given, though figures show φ_k around 89–94.
ξ = 1
Coupling constant in f(φ) of Eq. (45) set to 1 to simplify expressions.

assumptions (5)

domain assumption Standard Mukhanov–Sasaki formalism for curvature perturbations
The quadratic action in terms of v, π, and z is taken from standard inflationary perturbation theory (Ref. [91]); no derivation appears in the paper.
domain assumption De Sitter background approximation a = -1/(ηH0) with constant H0
Used to convert conformal time to y=log10 a and to replace z'/z with a'/a; assumes H≈H0 constant even in the f(φ,R) model.
ad hoc to paper Quadratic action S^(2)=∫dτd³x[-π²-(∂_i v)²+(B-A)v²] for f(φ,R)
Asserted after 'expanding the correction for gravity to the second order'; no actual perturbation of action (44) is shown. Load-bearing for all subsequent results.
ad hoc to paper Approximation f(φ)≈1+e^{-nφ/β}
Obtained from Eq. (45) by dropping terms; used to produce B. Higher-order exponential terms are ignored without justification.
domain assumption Two-mode squeezed state with wavefunction (37) describes the quantum state of perturbations
Standard in inflationary quantum optics; accepted from prior literature.

invented entities (1)

Effective dissipative coefficient c_n
purpose: Intended to quantify energy/information exchange with the environment in an 'open-system extension'.
Defined by splitting the Hamiltonian into H_close and H_open (Eqs. 59–60, 65–66), but no Lindblad/master equation is derived, and no open-system evolution is solved. c_n is not tied to any observable.

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Pith. "Pith review of Circuit and Krylov complexity of primordial perturbations of modified gravity in inflation." pith.science (2026). https://pith.science/paper/TMUJWX7C

@misc{pith2026260709408,
  author       = {Pith},
  title        = {Pith review of: Circuit and Krylov complexity of primordial perturbations of modified gravity in inflation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TMUJWX7C}},
  note         = {Machine review of arXiv:2607.09408}
}

abstract

In this work, we investigate quantum complexity diagnostics of primordial curvature perturbations within the inflationary paradigm. We compare canonical scalar-field inflation with the modified gravity model $f(\phi,R)$, focusing on the evolution of the two-mode squeezed state generated by the coupling between the $\vec{k}$ and $-\vec{k}$ momentum sectors. Starting from the quadratic action for curvature perturbations, we derive the evolution equations for the squeezed strength $r_k$ and squeezed angle $\phi_k$, utilizing them to evaluate both circuit complexity and Krylov-space diagnostics. Specifically, we compute the Krylov complexity, Krylov entropy, Lanczos coefficients $b_n$, and an effective dissipative contribution $c_n$ within an open-system extension. Our numerical results demonstrate that the $f(\phi,R)$ coupling enhances the squeezed strength relative to the canonical scalar field inflation. Since the Krylov complexity of the two-mode squeezed state is directly controlled by the mean pair number ($K=\sinh^2 r_k$), this enhancement leads to a smaller growth in Krylov complexity and related Krylov-space quantities. Furthermore, circuit complexity displays a more pronounced evolution in the $f(\phi,R)$ framework, particularly after the horizon exit regime. Ultimately, our work sheds new light on the quantum complexity of modified gravity $f(\phi,R)$.

Figures

Figures reproduced from arXiv: 2607.09408 by the authors.

**Figure 1.** illustrates the evolution of the squeezed parameter rk in canonical scalar field inflation and f(ϕ, R) inflation, and took into account their evolution following the introduction [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗

**Figure 1.** FIG. 1: The numerical solutions of [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗

**Figure 2.** FIG. 2: The numerical solutions of [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

Figures from the paper (4 more)

**Figure 3.** Figure 3: FIG. 3: The numerical solutions of circuit complexity in the canonical scalar field inflation and [PITH_FULL_IMAGE:figures/full_fig_p018_3.png]

**Figure 4.** Figure 4: FIG. 4: The numerical solutions of [PITH_FULL_IMAGE:figures/full_fig_p023_4.png]

**Figure 5.** Figure 5: FIG. 5: The numerical solutions of [PITH_FULL_IMAGE:figures/full_fig_p023_5.png]

**Figure 6.** Figure 6: FIG. 6: The numerical solutions of Krylov complexity in terms of log [PITH_FULL_IMAGE:figures/full_fig_p025_6.png]

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Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

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Lanczos coefficients and dissipative term To proceed, one only needs the corresponding operator wave function—or the squeezed strengthr k—along with the Hamiltonian ˆHk. In canonical scalar field inflation, these equa- tions are expressed as ˆH= Z d3k ˆHk = Z d3k (A−k) ˆc† −⃗kˆc−⃗k + ˆc⃗kˆc† ⃗k + (A+i a′ a )ˆc† ⃗kˆc† −⃗k + (A−i a′ a )ˆc⃗kˆc−⃗k .(58) Combi...

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Krylov complexity and Krylov entropy The Krylov complexity captures operator growth, and its definition 56 in the two-mode squeezed state has been established K= sinh 2 rk,(70) it is directly related only to squeezed strengthr k. In the canonical scalar field inflation and f(ϕ, R) inflation, the evolution equations for the squeezed strengthr k can be obta...

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