Classical Holographic Codes

Abstract

In this work, we introduce classical holographic codes. These can be understood as concatenated probabilistic codes and can be represented as networks uniformly covering hyperbolic space. In particular, classical holographic codes can be interpreted as maps from bulk degrees of freedom to boundary degrees of freedom. Interestingly, they are shown to exhibit features similar to those expected from the AdS/CFT correspondence. Among these are a version of the Ryu-Takayanagi formula and intriguing properties regarding bulk reconstruction and boundary representations of bulk operations. We discuss the relation of our findings with expectations from AdS/CFT and, in particular, with recent results from quantum error correction.