Homological codes and abelian anyons

TL;DR

This paper generalizes Kitaev's toric code to CW complexes, linking ground states and excitations to topological properties, and demonstrates that certain code products exhibit abelian anyonic statistics.

Contribution

It introduces a homological framework for generalized abelian codes on CW complexes, connecting topological invariants to quantum error correction and anyon statistics.

Findings

Ground states depend only on the CW complex's homotopy type.

Homological product of CSS code with infinite toric code exhibits abelian anyons.

Charges and excitations are characterized by cellular homology and cohomology.

Abstract

We study a generalization of Kitaev's abelian toric code model defined on CW complexes. In this model qudits are attached to $n$ dimensional cells and the interaction is given by generalized star and plaquette operators. These are defined in terms of coboundary and boundary maps in the locally finite cellular cochain complex and the cellular chain complex. We find that the set of frustration free ground states and the types of charges carried by certain localized excitations depend only on the proper homotopy type of the CW complex. As an application we show that the homological product of a CSS code with the infinite toric code has excitations with abelian anyonic statistics.