Understanding holographic error correction via unique algebras and atomic examples

TL;DR

This paper provides a constructive method to analyze holographic quantum error-correcting codes, enabling explicit computation of RT formula terms and introducing simpler, circuit-based examples with proven algebraic uniqueness.

Contribution

It introduces a fully constructive framework for holographic codes, allowing explicit calculations and simpler circuit-based examples, along with a proof of algebraic uniqueness for complementary recovery.

Findings

Explicit recipe for RT formula terms

Construction of simpler holographic code examples

Proof of algebraic uniqueness for correctable systems

Abstract

We introduce a fully constructive characterisation of holographic quantum error-correcting codes. That is, given a code and an erasure error we give a recipe to explicitly compute the terms in the RT formula. Using this formalism, we employ quantum circuits to construct a number of examples of holographic codes. Our codes have nontrivial holographic properties and are simpler than existing approaches built on tensor networks. Finally, leveraging a connection between correctable and private systems we prove the uniqueness of the algebra satisfying complementary recovery. The material is presented with the goal of accessibility to researchers in quantum information with no prior background in holography.