  We study the preparation of a quantum state using a circuit of depth $t$ from
a factorized state of $N$ sites. We argue that in the appropriate scaling limit
of large $t$ and $N$, the overlap between states evolved under generic
many-body chaotic dynamics belongs to a family of universal distribution that
generalizes the celebrated Porter-Thomas distribution. This is a consequence of
a mapping in the space of replicas to a model of dilute domain walls. Our
result provides a rare example in which analysis at an arbitrary number of
replicas is possible, giving rise to the complete overlap distribution. Our
general picture is derived and corroborated by the exact solution of the random
phase model and of an emergent random matrix model given by the Ginibre
ensemble. Finally, numerical simulations of two distinct random circuits show
excellent agreement, thereby demonstrating universality.
