Khovanov homology and quantum error-correcting codes

TL;DR

This paper explores how advanced Khovanov homology theories can be used to generate and analyze new quantum error-correcting codes with desirable properties for quantum computing.

Contribution

It introduces new families of quantum codes derived from various extensions of Khovanov homology and studies their properties, such as distance behavior under topological moves.

Findings

Generated new quantum codes from reduced, annular, and sl3 Khovanov homology.

Analyzed how code properties change under Reidemeister moves and connected sums.

Established theoretical properties of these codes in quantum error correction.

Abstract

Error-correcting codes for quantum computing are crucial to address the fundamental problem of communication in the presence of noise and imperfections. Audoux used Khovanov homology to define families of quantum error-correcting codes with desirable properties. We explore Khovanov homology and some of its many extensions, namely reduced, annular, and $\mathfrak{sl}_3$ homology, to generate new families of quantum codes and to establish several properties about codes that arise in this way, such as behavior of distance under Reidemeister moves or connected sums.