  We study circuit complexity for conformal field theory states in arbitrary
dimensions. Our circuits start from a primary state and move along a unitary
representation of the Lorentzian conformal group. We consider different choices
of distance functions and explain how they can be understood in terms of the
geometry of coadjoint orbits of the conformal group. Our analysis highlights a
connection between the coadjoint orbits of the conformal group and timelike
geodesics in anti-de Sitter spacetimes. We extend our method to study circuits
in other symmetry groups using a group theoretic generalization of the notion
of coherent states.
