# Einstein's Equations from Varying Complexity

## TL;DR

This paper proposes a novel derivation of Einstein's equations from the concept of quantum circuit complexity, linking holography, tensor networks, and gravitational dynamics in three-dimensional anti-de Sitter space.

## Contribution

It introduces a method to derive Einstein's equations by varying quantum complexity, connecting holography and tensor network techniques.

## Key findings

- Derived Einstein's equations from complexity variation in AdS space.
- Established a link between tensor network renormalization and gravitational dynamics.
- Provided a new perspective on the holographic principle and gravity.

## Abstract

A recent proposal equates the circuit complexity of a quantum gravity state with the gravitational action of a certain patch of spacetime. Since Einstein's equations follow from varying the action, it should be possible to derive them by varying complexity. I present such a derivation for vacuum solutions of pure Einstein gravity in three-dimensional asymptotically anti-de Sitter space. The argument relies on known facts about holography and on properties of Tensor Network Renormalization, an algorithm for coarse-graining (and optimizing) tensor networks.