Angular $k$-uniformity and the Hyperinvariance of Holographic Codes

TL;DR

This paper introduces angular k-uniformity, a geometric criterion that helps design and analyze hyperinvariant holographic codes in higher dimensions, advancing understanding of bulk-boundary duality in quantum error correction.

Contribution

It proposes angular k-uniformity as a new geometric principle to construct and analyze hyperinvariant holographic codes beyond 2D regular tilings.

Findings

Enables systematic construction of hyperinvariant codes in arbitrary dimensions.

Extends hyperinvariance concepts to heterogeneous networks and qLEGO architectures.

Provides a geometry-aware framework for analyzing boundary correlations and state dependence.

Abstract

Holographic quantum error-correcting codes, often realized through tensor network architectures, have emerged as compelling toy models for exploring bulk-boundary duality in AdS-CFT. By encoding bulk information into highly entangled boundary degrees of freedom, they capture key features of holography such as subregion duality, operator reconstruction, and complementary recovery. Among them, hyperinvariant tensor networks-characterized by the inclusion of edge tensors and the enforcement of multi-tensor isometries-offer a promising platform for realizing features such as state dependence and nontrivial boundary correlations. However, existing constructions are largely confined to two-dimensional regular tilings, and the structural principles underlying hyperinvariance remain poorly understood, especially in higher dimensions. To address this, we introduce a geometric criterion called angular k-uniformity, which refines standard k-uniformity and its planar variants by requiring isometric behavior within angular sectors of a tensor's rotationally symmetric layout. This condition enables the systematic identification and construction of hyperinvariant holographic codes on regular hyperbolic honeycombs in arbitrary dimension, and extends naturally to heterogeneous networks and qLEGO architectures beyond regular tilings. Altogether, angular k-uniformity provides a versatile, geometry-aware framework for analyzing and designing holographic tensor networks and codes with hyperinvariant features such as nontrivial boundary correlations and state-dependent complementary recovery.