On the Noether charge and the gravity duals of quantum complexity

TL;DR

This paper proposes a new gravity dual for quantum complexity called CA-2, based on the gravitational action on the Wheeler-DeWitt patch, and compares it with existing dualities through black hole examples.

Contribution

It introduces the CA-2 duality, relating complexity to the gravitational action, and analyzes its consistency with the Lloyd bound across various black holes.

Findings

CA-2 duality often respects the Lloyd bound but can violate it.

CA duality generally violates the Lloyd bound.

CV-2 duality always respects the Lloyd bound.

Abstract

The physical relevance of the thermodynamic volumes of AdS black holes to the gravity duals of quantum complexity was recently argued by Couch et al. In this paper, by generalizing the Wald-Iyer formalism, we derive a geometric expression for the thermodynamic volume and relate its product with the thermodynamic pressure to the non-derivative part of the gravitational action evaluated on the Wheeler-DeWitt patch. We propose that this action provides an alternative gravity dual of the quantum complexity of the boundary theory. We refer this to "complexity=action 2.0" (CA-2) duality. It is significantly different from the original "complexity=action" (CA) duality as well as the "complexity=volume 2.0" (CV-2) duality proposed by Couch et al. The latter postulates that the complexity is dual to the spacetime volume of the Wheeler-DeWitt patch. To distinguish our new conjecture from the various dualities in literature, we study a number of black holes in Einstein-Maxwell-Dilation theories. We find that for all these black holes, the CA duality generally does not respect the Lloyd bound whereas the CV-2 duality always does. For the CA-2 duality, although in many cases it is consistent with the Lloyd bound, we also find a counter example for which it violates the bound as well.