Complexity and the Hilbert space dimension of 3D gravity

TL;DR

The paper addresses the size of the gravity Hilbert space for near-extremal black holes by applying spread complexity to evolving thermofield double states in near-extremal $AdS_3$ gravity, extracting Lanczos data that matches an $SL(2,\mathbb{R})$-type structure and Laguerre polynomials. Non-perturbative gravitational effects via wormholes introduce a universal sine-kernel spectral correlator, curing divergences and producing a finite late-time saturation of $C_S(t)$ to $\sim e^{S_0}$. This establishes a direct link between spread complexity saturation and the finite black-hole Hilbert space dimension, providing a method to infer the entropy $S_0$ from complexity and supporting the view of gravity as encoding ensemble-averaged physics. The results connect near-extremal 3d gravity to JT/Schwarzian dynamics and illustrate how non-perturbative gravity effects determine observable complexity saturation.

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.