Complexity Equals Action

TL;DR

This paper proposes that quantum complexity in holography is dual to the gravitational action of a specific spacetime region, tested across various black hole scenarios, suggesting black holes are the universe's fastest computers.

Contribution

It introduces the complexity=action conjecture, linking quantum complexity to gravitational action, improving upon the earlier volume-based duality, and tests it in diverse black hole contexts.

Findings

Complexity is dual to the action of the Wheeler-DeWitt patch.

The conjecture holds for various black hole types and perturbations.

Black holes may be the fastest computational entities in nature.

Abstract

We conjecture that the quantum complexity of a holographic state is dual to the action of a certain spacetime region that we call a Wheeler-DeWitt patch. We illustrate and test the conjecture in the context of neutral, charged, and rotating black holes in AdS, as well as black holes perturbed with static shells and with shock waves. This conjecture evolved from a previous conjecture that complexity is dual to spatial volume, but appears to be a major improvement over the original. In light of our results, we discuss the hypothesis that black holes are the fastest computers in nature.