Complexity geometry in Hermitian and non-Hermitian quantum dynamics

We show that quantum dynamics of any systems with SU(1,1) symmetry are governed by the same underlying geometry of an anti-de Sitter spacetime in 2+1 dimensions (AdS(2+1)). Using complexity geometry, quantum evolutions are mapped to trajectories in AdS(2+1). Whereas the time measured in laboratories becomes either the proper time or the proper distance, quench dynamics follow the geodesics of AdS(2+1). Such a geometric approach provides us with a unified interpretation of a wide range of prototypical phenomena that appear disconnected. For instance, the light cone of AdS(2+1) underlies expansions of unitary fermions released from harmonic traps, the onsite of parametric amplifications, and the exceptional points that represent the PT symmetry breaking in non-Hermitian systems. Our work provides a transparent means to optimize quantum controls by exploiting the shortest paths in the complexity geometry. It also allows experimentalists to engineer the corresponding geometries by manipulating the quantum dynamics.