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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
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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