Structure of 2D Topological Stabilizer Codes

TL;DR

This paper analyzes the fundamental structure of 2D topological stabilizer codes, revealing that they can be characterized by the homology of string operators and topological charges, with implications for code equivalence.

Contribution

It establishes a topological and homological framework for understanding 2D stabilizer codes under translational invariance, linking code equivalence to topological charge equivalence.

Findings

All 2D topological stabilizer codes can be described via string operator homology.

Code equivalence corresponds to topological charge equivalence under local transformations.

The approach emphasizes local properties over global code characteristics.

Abstract

We provide a detailed study of the general structure of two-dimensional topological stabilizer quantum error correcting codes, including subsystem codes. Under the sole assumption of translational invariance, we show that all such codes can be understood in terms of the homology of string operators that carry a certain topological charge. In the case of subspace codes, we prove that two codes are equivalent under a suitable set of local transformations if and only they have equivalent topological charges. Our approach emphasizes local properties of the codes over global ones.