Fracton models from product codes

TL;DR

This paper establishes a connection between fracton order and product codes, showing how classical seed codes can generate various fracton models, including local and nonlocal types, through systematic constructions.

Contribution

It introduces conditions on classical seed codes that lead to fracton order in quantum product codes, including novel local codes on aperiodic tilings.

Findings

Nonlocal lineon models derived from hypergraph product codes.

Construction of local fracton models using planar aperiodic tilings.

Demonstration of systematic methods to realize fracton order from classical codes.

Abstract

We explore a deep connection between fracton order and product codes. In particular, we propose and analyze conditions on classical seed codes which lead to fracton order in the resulting quantum product codes. Depending on the properties of the input codes, product codes can realize either Type-I or Type-II fracton models, in both nonlocal and local constructions. For the nonlocal case, we show that a recently proposed model of lineons on an irregular graph can be obtained as a hypergraph product code. Interestingly, constrained mobility in this model arises only from glassiness associated with the graph. For the local case, we introduce a novel type of classical LDPC code defined on a planar aperiodic tiling. By considering the specific example of the pinwheel tiling, we demonstrate the systematic construction of local Type-I and Type-II fracton models as product codes. Our work establishes product codes as a natural setting for exploring fracton order.