Einstein Equations from Varying Complexity

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.

Figure 1

(a) A discretized Euclidean path integral can be represented as a regular lattice of identical tensors with nearest-neighbor contractions. (b) TNR approximates it with a coarser lattice; interfaces between the finer and coarser lattices are isometric layers (highlighted). (c),(d) For a piecewise constant $\varphi (z,x)$ 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 $\varphi /(-\log 2)$.