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.
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