Complexity and the Hilbert space dimension of 3D gravity

TL;DR

This paper uses Krylov complexity to estimate the Hilbert space size of 2+1D black holes, showing complexity saturation at late times corresponds to the Bekenstein-Hawking entropy, offering a new computational approach.

Contribution

It introduces a novel method using Krylov complexity to determine the Hilbert space dimension of black holes in AdS3, connecting complexity saturation to entropy.

Findings

Complexity saturates at late times, matching the exponential of the Bekenstein-Hawking entropy.

Lanczos coefficients indicate chaotic motion on the SL(2,R) group.

The approach provides a new way to compute Hilbert space dimensions in complex quantum systems.

Abstract

A central problem in formulating a theory of quantum gravity is to determine the size and structure of the Hilbert space of black holes. Here we use a quantum dynamical Krylov complexity approach to calculate the Hilbert space dimension of a black hole in 2+1-dimensional Anti-de Sitter space. We achieve this by obtaining the spread of an initial thermofield double state over the Krylov basis. The associated Lanczos coefficients match those for chaotic motion on the $SL(2,\mathbb{R})$ group. By including non-perturbative effects in the path integral, which computes coarse-grained ensemble averages, we find that the complexity saturates at late times. The saturation value is given by the exponential of the Bekenstein-Hawking entropy. Our results introduce a new way to compute the Hilbert space dimension of complex interacting systems from the saturating value of spread complexity.