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Khovanov homology and quantum error-correcting codes

Rostislav Akhmechet, Milena Harned, Pranav Venkata Konda, Felix Shanglin Liu, Nikhil Mudumbi, Eric Yuang Shao, Zheheng Xiao

TL;DR

This work systematically builds quantum error-correcting CSS codes from Khovanov homology and its extensions, investigating how diagrammatic operations (Reidemeister moves, connected sums, tensor products) affect code distance. It proves that reduced and unreduced Khovanov distances coincide under a specific basis, while showing that RII/III moves do not generally double or preserve distance; connected sums correspond to tensor products and can saturate distance bounds in several families. By extending the framework to annular Khovanov homology and sl3 link homology, the authors derive new infinite families of codes, with explicit asymptotics for Hopf-link, torus-link, and unknot constructions, including unknot codes in sl3 that scale as 3^ℓ in distance and have lengths growing as ~25^ℓ. The results provide both theoretical insights into distance behavior under key topological operations and concrete code families with provable asymptotics, highlighting the potential of link-homology-inspired constructions for 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.

Khovanov homology and quantum error-correcting codes

TL;DR

This work systematically builds quantum error-correcting CSS codes from Khovanov homology and its extensions, investigating how diagrammatic operations (Reidemeister moves, connected sums, tensor products) affect code distance. It proves that reduced and unreduced Khovanov distances coincide under a specific basis, while showing that RII/III moves do not generally double or preserve distance; connected sums correspond to tensor products and can saturate distance bounds in several families. By extending the framework to annular Khovanov homology and sl3 link homology, the authors derive new infinite families of codes, with explicit asymptotics for Hopf-link, torus-link, and unknot constructions, including unknot codes in sl3 that scale as 3^ℓ in distance and have lengths growing as ~25^ℓ. The results provide both theoretical insights into distance behavior under key topological operations and concrete code families with provable asymptotics, highlighting the potential of link-homology-inspired constructions for 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 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.

Paper Structure

This paper contains 26 sections, 38 theorems, 143 equations, 26 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Homological algebra and CSS codes
  4. Khovanov homology
  5. Properties of distance from Khovanov homology
  6. Distance in reduced homology
  7. 1. Multiplication of unmarked circles:
  8. 2. Multiplication with marked circle:
  9. 3. Comultiplication of unmarked circle:
  10. 4. Comultiplication of marked circle:
  11. Behavior under connected sums
  12. Behavior under Reidemeister II/III moves
  13. Annular Khovanov homology
  14. Tensor products of Khovanov chain complexes
  15. Setting up notation
  16. ...and 11 more sections

Key Result

Theorem 1.1

The distance of a Khovanov chain complex is the same in its unreduced and reduced forms (under the basis introduced in Audoux).

Figures (26)

  • Figure 1: The three Reidemeister moves, where we distinguish two variants of RI.
  • Figure 2: Conventions in Khovanov homology.
  • Figure 3: A connected sum being taken between two pointed diagrams.
  • Figure 4: We show that $r$ commutes with the 4 different types of edge maps of the differential.
  • Figure 5: Examples of how distance behaves under RII and RII
  • ...and 21 more figures

Theorems & Definitions (80)

  • Theorem 1.1: detailed version in Theorem \ref{['thm:reduced equals unreduced']}
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5: Theorem \ref{['main theorem sl3']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • ...and 70 more