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arXiv:2607.09408

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

Tao Li, Hai-Bing Fu

hep-thgr-qcquant-phSubmitted Jul 10, 2026• Announced Jul 13, 2026• Revised Jul 17, 2026 (v2)
v1 · Jul 10, 2026v2 · Jul 17, 2026
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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 $f(φ,R)$, 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 $φ_k$, 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 $f(φ,R)$ 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 $f(φ,R)$ framework, particularly after the horizon exit regime. Ultimately, our work sheds new light on the quantum complexity of modified gravity $f(φ,R)$.

Comments
37 pages, 6 figures

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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(φ,R)f(φ,R)f(φ,R), focusing on the evolution of the two-mode squeezed state generated by the coupling between the k⃗\vec{k}k and −k⃗-\vec{k}−k momentum sectors. Starting from the quadratic action for curvature perturbations, we derive the evolution equations for the squeezed strength rkr_krk​ and squeezed angle φkφ_kφk​, utilizing them to evaluate both circuit complexity and Krylov-space diagnostics. Specifically, we compute the Krylov complexity, Krylov entropy, Lanczos coefficients bnb_nbn​, and an effective dissipative contribution cnc_ncn​ within an open-system extension. Our numerical results demonstrate that the f(φ,R)f(φ,R)f(φ,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⁡2rkK=\sinh^2 r_kK=sinh2rk​), 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(φ,R)f(φ,R)f(φ,R) framework, particularly after the horizon exit regime. Ultimately, our work sheds new light on the quantum complexity of modified gravity f(φ,R)f(φ,R)f(φ,R).
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In this work, we systematically investigate the quantum-information diagnostics of cosmological perturbations with a nontrivial sound speed, utilizing a normalized open two-mode squeezed-state framework. Rather than introducing new observables, our analysis focuses on how a modified sound speed dynamically reshapes the Schrödinger evolution of the squeezing parameters (rkr_krk​ and φkφ_kφk​). We demonstrate how these dynamical changes are inherited by the reduced density matrix of the observable sector. By employing a sound-speed-resonance parametrization, we derive and evaluate the purity, von Neumann entropy, Rényi entropies, and logarithmic negativity. To overcome the intrinsic multiscale stiffness of the post-inflationary equations, we introduce a bounded variable x=tanh⁡rkx = \tanh r_kx=tanhrk​ as a partial regularization, which enables reliable numerical simulations exclusively within the inflationary regime. Our numerical results reveal that a nontrivial sound speed significantly suppresses the purity of the reduced state, indicating enhanced effective mixedness. Simultaneously, it strongly amplifies and modulates both the entropic and entanglement diagnostics. More precisely, a nontrivial sound speed postpones the onset of classicality by modulating the decoherence process. Ultimately, we show that a nontrivial sound speed leaves distinct and identifiable quantum-information signatures within the entanglement structure of the early universe.
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Primordial black holes (PBHs) form when large primordial curvature perturbations re-enter the Hubble radius and exceed the classical collapse threshold. These perturbations originate as quantum fluctuations of the inflationary vacuum, motivating a quantum-information description of the PBH-producing scalar sector. We develop a conservative extension of the standard PBH and scalar-induced gravitational-wave (SIGW) framework in which Gaussian quantum discord is used as a diagnostic of residual quantum correlations, not as a new PBH-formation criterion. We describe each (k,−k)(\bm k,-\bm k)(k,−k) pair as a two-mode Gaussian state and show that, in the pure squeezed limit, discord grows rapidly with the squeezing parameter, so low discord thresholds are automatically satisfied for strongly squeezed modes. The nontrivial regime is the mixed state produced by decoherence. Using a Lindblad open-system description, we motivate a Gaussian loss channel for the scalar covariance matrix and distinguish the discord from a covariance-survival factor Qdec(k)Q_{\rm dec}(k)Qdec​(k). If the decoherence channel suppresses the scalar two-point covariance, PBH abundance can be affected through the classical collapse variance, while the SIGW spectrum is modified more directly by the factors Qdec(ku)Qdec(kv)Q_{\rm dec}(ku)Q_{\rm dec}(kv)Qdec​(ku)Qdec​(kv) inside the radiation-era convolution. For a narrow scalar peak and slowly varying QdecQ_{\rm dec}Qdec​, this gives the benchmark scaling ΩGWeff≃Qdec2ΩGWclassΩ_{\rm GW}^{\rm eff}\simeq Q_{\rm dec}^2Ω_{\rm GW}^{\rm class}ΩGWeff​≃Qdec2​ΩGWclass​. Thus quantum discord and decoherence provide a controlled way to characterize the quantum-to-classical transition of PBH-producing perturbations, with the clearest imprint appearing in scalar-induced gravitational waves.
Krylov complexity is a powerful diagnostic of quantum dynamics, with clear connections to other measures of quantum chaos and operator growth. One such measure is the Loschmidt amplitude, defined as the overlap of initially identical states evolved under two slightly different Hamiltonians. Its decay in certain systems is controlled by the classical Lyapunov exponent. Using the algebraic properties of the Krylov complexity operator, we express Krylov complexity as the derivative of a Loschmidt amplitude whose perturbation is parameterized by an angular variable φφφ. This formulation allows us to define a spectral propagator that encodes the entire complexity distribution, which we characterize for specific types of systems. We study the two-dimensional quantum geometry spanned by time and φφφ where the original and deformed trajectories reside, demonstrating that Krylov complexity is upper-bounded by its volume. We also express the time derivative of Krylov complexity in terms of a distinct Loschmidt amplitude. Depending on the growth of the Lanczos coefficients, the perturbation term in this amplitude can be truncated. We propose that the strength of this perturbation provides a classification scheme for Krylov complexity dynamics and relate it to the φφφ-derivative of the spectral propagator. Using this analytical framework, we derive general relations between the time-dependence of the survival amplitude and Krylov space measures.
We investigate how interactions affect the quantum state of scalar perturbations during inflation and the quantum correlations they may exhibit. Focusing on the case of scalar perturbations in single-field inflation, we model interactions using a Lindblad equation with a non-unitary contribution quadratic in the scalar perturbations, and of parametrisable amplitude and time dependence. We compute the quantum state of these interacting perturbations, which is fully described by its purity and squeezing parameters. First, we show that, in most of the parameter space, not only the purity but also the squeezing parameter is significantly reduced by interactions. Second, we show that this de-squeezing induced by the interactions, on top of the purity loss, causes a further suppression of quantum correlations. We thus emphasise that the quantum or classical character of the correlations exhibited by the perturbations cannot be correctly determined by computing the effect of interactions on the purity alone. Since the phenomenological framework adopted in this paper encompasses a wide class of possible interactions, our results provide general insights into the nature of decoherence processes in primordial fluctuations.