Complexity Growth for AdS Black Holes

TL;DR

This paper investigates the Complexity-Action duality for stationary AdS black holes, deriving exact formulas for the growth rate of action in the Wheeler-DeWitt patch, linking it to the quantum complexity growth of holographic states.

Contribution

It provides a universal formula for the action growth rate in various AdS black holes, extending the duality conjecture to multiple black hole solutions.

Findings

Derived exact late-time action growth formulas for various AdS black holes.

Established a universal relation between action growth and thermodynamical quantities.

Confirmed the conjecture that stationary AdS black holes are the fastest computers in nature.

Abstract

Recently a Complexity-Action (CA) duality conjecture has been proposed, which relates the quantum complexity of a holographic boundary state to the action of a Wheeler-DeWitt (WDW) patch in the anti-de Sitter (AdS) bulk. In this paper we further investigate the duality conjecture for stationary AdS black holes and derive some exact results for the growth rate of action within the Wheeler-DeWitt (WDW) patch at late time approximation, which is supposed to be dual to the growth rate of quantum complexity of holographic state. Based on the results from the general $D$-dimensional Reissner-Nordstr\"{o}m (RN)-AdS black hole, rotating/charged Ba\~{n}ados-Teitelboim-Zanelli (BTZ) black hole, Kerr-AdS black hole and charged Gauss-Bonnet-AdS black hole, we present a universal formula for the action growth expressed in terms of some thermodynamical quantities associated with the outer and inner horizons of the AdS black holes. And we leave the conjecture unchanged that the stationary AdS black hole in Einstein gravity is the fastest computer in nature.