Translation-invariant quantum low-density parity-check codes from compactified fracton models

TL;DR

This paper introduces a unifying framework for translation-invariant quantum error-correcting codes, including fracton and Abelian Two-Block Group Algebra codes, via compactified hypergraph product models, revealing their properties and bounds.

Contribution

It provides a unifying picture connecting various translation-invariant codes, including fracton and A2BGA codes, through parent hypergraph product fracton models and their compactifications.

Findings

All BB codes with the same check weight derive from a single parent model.

Extended code-parameter bounds for A2BGA codes based on parent models.

Conjecture on limitations of transversal gates and energy barriers.

Abstract

Quantum error-correcting codes with translation symmetry and local checks have been studied extensively, leading to a wide variety of fracton codes in three or more dimensions which lack a complete unifying picture. Recently, the study of translation-invariant codes with long-range checks has revealed impressive performance for small fixed-size instances in two dimensions. Here, we provide a unifying picture for a large family of translation-invariant codes, both local and long-range, that captures many fracton codes and all Abelian Two-Block Group Algebra (A2BGA) codes, including the Bivariate Bicycle (BB) codes. The balanced product structure of A2BGA codes leads to a local parent code that is a hypergraph product fracton model in a higher dimension. Different compactifications of a parent code produce a wide variety of descendant codes which provides a unifying picture for their properties. In particular, all BB codes with the same check weight are derived from a single parent hypergraph product fracton model. This construction allows us to extend Wang and Pryadko's code-parameter bounds for Generalized Bicycle codes to A2BGA codes. We conjecture that the transversal gates and energy barriers of the translation-invariant descendant codes are limited by those of their parent fracton models.