Homological Quantum Error Correction with Torsion

TL;DR

This paper explores homological quantum error correction using algebraic topology, focusing on torsion in qudit codes, and introduces a new structure theorem for the logical space, advancing understanding of torsion effects.

Contribution

It introduces the Structure Theorem for the Qudit Logical Space, applying the Universal Coefficient Theorem to analyze torsion in quantum codes, and develops a specialized cell complex framework.

Findings

Proves the Structure Theorem for the Qudit Logical Space.

Provides insights into the role of torsion in logical qudit spaces.

Improves upon previous results regarding torsion in quantum error correction.

Abstract

Homological quantum error correction uses tools of algebraic topology and homological algebra to derive Calderbank-Shor-Steane quantum error correcting codes from cellulations of topological spaces. This work is an exploration of the relevant topics, a journey from classical error correction, through homology theory, to CSS codes acting on qudit systems. Qudit codes have torsion in their logical spaces. This is interesting to study because it gives us extra logical qudits, of possibly different dimension. Apart from examples and comments on the topic, we prove an original result, the Structure Theorem for the Qudit Logical Space, an application of the Universal Coefficient Theorem from homological algebra, which gives us information about the logical space when torsion is involved, and that improves on a previous result in the literature. Furthermore, this work introduces our own abstracted and restricted version of the general notion of a cell complex, suited exactly to our needs.