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We show that quantum dynamics of any systems with $SU(1,1)$ symmetry giverise to emergent Anti-de Sitter spacetimes in 2+1 dimensions (AdS$_{2+1}$).Using the continuous circuit depth, a quantum evolution is mapped to atrajectory in AdS$_{2+1}$. Whereas the time measured in laboratories becomeseither the proper time or the proper distance, quench dynamics follow geodesicsof AdS$_{2+1}$. Such a geometric approach provides a unified interpretation ofa wide range of prototypical phenomena that appear disconnected. For instance,the light cone of AdS$_{2+1}$ underlies expansions of unitary fermions releasedfrom harmonic traps, the onsite of parametric amplifications, and theexceptional points that represent the $PT$ symmetry breaking in non-Hermitiansystems. Our work provides a transparent means to optimize quantum controls byexploiting shortest paths in the emergent spacetimes. It also allowsexperimentalists to engineer emergent spacetimes and induce tunnelings betweendifferent AdS$_{2+1}$.

Emergent spacetimes from Hermitian and non-Hermitian quantum dynamics