  Unitary dynamics of a quantum system initialized in a selected basis state
yields, generically, a state that is a superposition of all the basis states.
This process, associated with the quantum information scrambling and intimately
tied to the resource theory of coherence, may be viewed as a gradual
delocalization of the system's state in the Hilbert space. This work analyzes
the Hilbert space delocalization under dynamics of random quantum circuits,
which serve as a minimal model of chaotic dynamics of quantum many-body
systems. We employ analytical methods based on the replica trick and Weingarten
calculus to investigate the time evolution of the participation entropies which
quantify the Hilbert space delocalization. We demonstrate that the
participation entropies approach, up to a fixed accuracy, their long-time
saturation value in times that scale logarithmically with the system size.
Exact numerical simulations and tensor network techniques corroborate our
findings.
