Quantum

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

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

Featured May 30, 2026

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Simply

By showing that many different quantum error-correcting codes are just simpler, higher-dimensional 'parent' codes squished down using twisted boundary conditions, the authors provide a unified view of their properties.

In depth

The paper introduces a unifying framework for a broad class of quantum low-density parity-check (qLDPC) codes, including many fracton codes and Abelian Two-Block Group Algebra (A2BGA) codes. This is achieved by demonstrating that these diverse codes can be understood as compactifications of simpler, higher-dimensional hypergraph product (HGP) fracton models. This perspective allows the extension of code-parameter bounds and the formulation of conjectures regarding logical gate restrictions and energy barriers.

Key Takeaways

1
A unifying picture is provided for a large family of translation-invariant qLDPC codes, including fracton codes and A2BGA codes, by relating them to higher-dimensional hypergraph product fracton models.
2
The concept of compactification (imposing twisted boundary conditions) is used to derive lower-dimensional descendant codes from a higher-dimensional parent HGP fracton model.
3
The framework allows for the extension of distance bounds for A2BGA codes and motivates conjectures about the inheritance of logical gate restrictions and energy barrier scaling from parent to descendant codes.

Conceptual Flow

HIGH LEVEL

Methodology: Unifying Diverse Codes

The paper shows that many different quantum codes can be made from one big 'parent' code by squishing it in different ways.

Big Parent Code

Squish and Twist

Code Type A

Code Type B

Code Type C

Results: A Family Tree of Codes

They found that these squished codes fall into three main families, helping us understand how they are related and what they can do.

Parent Code

Creates

Family 1 Codes

Family 2 Codes

Family 3 Codes