Random Circuits in the Black Hole Interior

TL;DR

This paper establishes a quantitative relation between randomness measures and wormhole length in black hole interiors using holographic duality and random quantum circuits, revealing how complexity relates to geometric features.

Contribution

It introduces a model connecting random quantum circuits to wormhole geometry and shows how ensemble states approximate random states at long circuit times.

Findings

Long circuit times produce wormholes with linearly growing length.

Ensembles of states form approximate quantum state k-designs.

Black hole interior complexity relates to the length of ER wormholes.

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.