A recent study has explored the quantum complexity of primordial curvature perturbations, fundamental for understanding the origin of cosmic structures, within the inflationary paradigm. The research compares the canonical scalar-field inflation model with a 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. This work sheds new light on how modified gravity theories can influence the quantum properties of the early universe.

The researchers started from the quadratic action for curvature perturbations and derived the evolution equations for the squeezed strength $r_k$ and squeezed angle $\phi_k$. Using these parameters, they evaluated both circuit complexity and Krylov-space diagnostics. Specifically, they computed Krylov complexity, Krylov entropy, Lanczos coefficients $b_n$, and an effective dissipative contribution $c_n$ within an open-system extension. Numerical analysis revealed that the $f(\phi,R)$ coupling in modified gravity enhances the squeezed strength relative to canonical scalar-field inflation.

This enhancement in squeezed strength has direct implications for quantum complexity. Since the Krylov complexity of the two-mode squeezed state is directly controlled by the mean pair number ($K=\sinh^2 r_k$), the observed enhancement leads to a smaller growth in Krylov complexity and related Krylov-space quantities. Conversely, circuit complexity displayed a more pronounced evolution within the $f(\phi,R)$ framework, particularly after the horizon exit regime. These findings suggest that modifications to gravity can significantly alter how quantum information is processed and evolves in the primordial universe.