Random circuits in the black hole interior

Abstract

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 kℓ∆ and matter correlation scale ℓ∆ 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.