  We study the quantum circuit complexity of cosmological perturbations in
different models of the early universe. A natural measure for the complexity of
cosmological perturbations is based on the symplectic group, allowing us to
identify complexity with geodesics in the hyperbolic plane. We investigate the
complexity of both the mode functions and the physical perturbations, arguing
that the latter often provides a more insightful description of the physics
involved. In all models the total complexity reached is rather large.
Inflationary perturbations may be represented by a comparatively simple quantum
circuit, while the perturbations during a matter-dominated contracting phase
present the most rapid growth in complexity. Ekpyrotic perturbations reside in
the middle and are distinguished by the smallest growth of complexity before
horizon exit. Our analysis serves to highlight how different cosmological
models achieve the same end result for the perturbations via different routes
and how all models show a pronounced sensitivity to initial conditions.
