Complexity and entanglement in non-local computation and holography

TL;DR

This paper explores how gravity, via the AdS/CFT correspondence, constrains the complexity of non-local computations by relating it to entanglement and the geometry of bulk regions, specifically the ridge surface.

Contribution

It establishes a relationship between the complexity of local operations and the entanglement cost for their non-local implementation in holographic theories.

Findings

Complexity and entanglement cost are polynomially related.

Gravity constrains the complexity of operations to be polynomial in the ridge area.

Non-local computation in holography is governed by entanglement and geometric surfaces.

Abstract

Does gravity constrain computation? We study this question using the AdS/CFT correspondence, where computation in the presence of gravity can be related to non-gravitational physics in the boundary theory. In AdS/CFT, computations which happen locally in the bulk are implemented in a particular non-local form in the boundary, which in general requires distributed entanglement. In more detail, we recall that for a large class of bulk subregions the area of a surface called the ridge is equal to the mutual information available in the boundary to perform the computation non-locally. We then argue the complexity of the local operation controls the amount of entanglement needed to implement it non-locally, and in particular complexity and entanglement cost are related by a polynomial. If this relationship holds, gravity constrains the complexity of operations within these regions to be polynomial in the area of the ridge.