Code properties from holographic geometries

TL;DR

This paper explores the connection between holographic geometries and quantum error correction, introducing new concepts like 'price' and analyzing how bulk logical algebras are supported on boundary regions with different curvature properties.

Contribution

It derives general results about operator algebra quantum error correction and introduces the concept of 'price' to characterize logical support in holographic codes.

Findings

Holographic codes on negatively curved manifolds exhibit 'uberholography' with fractal boundary support.

Boundary physics in flat or positively curved holographic codes is highly nonlocal.

New constraints on the support ('price') and distance of logical subalgebras in holographic quantum codes.

Abstract

Almheiri, Dong, and Harlow [arXiv:1411.7041] proposed a highly illuminating connection between the AdS/CFT holographic correspondence and operator algebra quantum error correction (OAQEC). Here we explore this connection further. We derive some general results about OAQEC, as well as results that apply specifically to quantum codes which admit a holographic interpretation. We introduce a new quantity called `price', which characterizes the support of a protected logical system, and find constraints on the price and the distance for logical subalgebras of quantum codes. We show that holographic codes defined on bulk manifolds with asymptotically negative curvature exhibit `uberholography', meaning that a bulk logical algebra can be supported on a boundary region with a fractal structure. We argue that, for holographic codes defined on bulk manifolds with asymptotically flat or positive curvature, the boundary physics must be highly nonlocal, an observation with potential implications for black holes and for quantum gravity in AdS space at distance scales small compared to the AdS curvature radius.