Novel Encodings of Homology, Cohomology, and Characteristic Classes

TL;DR

This paper introduces new topological quantum error-correcting codes that encode classical topological invariants like characteristic classes, expanding the capabilities of quantum codes to represent complex fiber bundle obstructions.

Contribution

It constructs and analyzes extensions of toric codes to encode obstruction classes such as Chern and Euler classes within quantum error correction.

Findings

Constructed explicit codes encoding Euler class of S^2

Analyzed topological error structures in extended toric codes

Demonstrated encoding of characteristic classes in QECC

Abstract

Topological quantum error-correcting codes (QECC) encode a variety of topological invariants in their code space. A classic structure that has not been encoded directly is that of obstruction classes of a fiber bundle, such as the Chern or Euler class. Here, we construct and analyze extensions of toric codes. We then analyze the topological structure of their errors and finally construct a novel code using these errors to encode the obstruction class to a fiber bundle. In so doing, we construct an encoding of characteristic classes such as the Chern and Pontryagin class in topological QECC. An example of the Euler class of $S^2$ is constructed explicitly.