Einstein's Equations from Varying Complexity

TL;DR

A derivation for vacuum solutions of pure Einstein gravity in three-dimensional asymptotically anti-de Sitter space is presented, based on known facts about holography and on properties of tensor network renormalization, an algorithm for coarse-graining (and optimizing) tensor networks.

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.

Figures (2)

• Figure 1: A discretized Euclidean path integral can be represented as a regular lattice of identical tensors with nearest neighbor contractions (a). Tensor Network Renormalization approximates it with a coarser lattice (b); interfaces between the finer and coarser lattices are isometric layers (highlighted). For a piece-wise constant $\phi(z,x)$ on the lattice (c), the corresponding conformal transformation can be implemented with iterative applications of TNR (d).
• Figure 2: The optimal network (MERA) consists entirely of isometric layers. The entanglement entropy of interval $(u,v)$ can be approximated by counting the isometric layers through which the 'exclusive causal cone' of the interval passes. The layers are indexed by $\phi/(-\log 2)$.