Abstract
In this paper, we present a quantitative holographic relation between a
microscopic measure of randomness and the geometric length of the wormhole in
the black hole interior. To this end, we perturb an AdS black hole with
Brownian semiclassical sources, implementing the continuous version of a random
quantum circuit for the black hole. We use the random circuit to prepare
ensembles of states of the black hole whose semiclassical duals contain
Einstein-Rosen (ER) caterpillars: long cylindrical wormholes with large numbers
of matter inhomogeneities, of linearly growing length with the circuit time. In
this setup, we show semiclassically that the ensemble of ER caterpillars of
average length $k\ell_{\Delta}$ and matter correlation scale $\ell_{\Delta}$
forms an approximate quantum state $k$-design of the black hole. At
exponentially long circuit times, the ensemble of ER caterpillars becomes
polynomial-copy indistinguishable from a collection of random states of the
black hole. We comment on the implications of these results for holographic
circuit complexity and for the holographic description of the black hole
interior.