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arXiv · 2605.19298

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

Cassandra M. Hopkin·Victor V. Albert·Dominic J. Williamson
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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.

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Cassandra M. Hopkin, Victor V. Albert, Dominic J. Williamson. 2026-05-19. Translation-invariant quantum low-density parity-check codes from compactified fracton models. https://arxiv.org/abs/2605.19298

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