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.
Highlights
Introduction.—The AdS=CFT correspondence [1] is the most powerful known approach to quantum gravity. It posits that every physical quantity in d þ 1–dimensional gravity with asymptotically anti–de Sitter (AdSdþ1) boundary conditions can be mapped to a corresponding quantity in a conformal field theory living on its asymptotic boundary (CFTd)
The present Letter is concerned with a conjectured translation of one important gravitational phenomenon: that a black hole grows deeper for an exponentially long time
In the most recent version of the conjecture, the depth of the black hole is quantified by the gravitational action A inside a Wheeler–de Witt patch—the part of spacetime that is spacelike separated from a time slice of the asymptotic boundary
Summary
Introduction.—The AdS=CFT correspondence (holographic duality) [1] is the most powerful known approach to quantum gravity. As a first step in the argument, I shall estimate the complexity of this class of states by examining the Euclidean path integrals which prepare them. The objective is to characterize the complexity of the path integral C1⁄2φ as a functional of φðz; xÞ.
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