We compute the quantum circuit complexity of the evolution of scalar curvature perturbations on expanding backgrounds, using the language of squeezed vacuum states. In particular, we construct a simple cosmological model consisting of an early-time period of de Sitter expansion followed by a radiation-dominated era and track the evolution of complexity throughout this history. During early-time de Sitter expansion the complexity grows linearly with the number of e-folds for modes outside the horizon. The evolution of complexity also suggests that the Universe behaves like a chaotic system during this era, for which we propose a scrambling time and Lyapunov exponent. During the radiation-dominated era, however, the complexity decreases until it "freezes in" after horizon re-entry, leading to a "de-complexification" of the Universe.

19 pages, 8 figures. Published version