Quantum error-correcting codes and their geometries

TL;DR

This paper introduces the mathematical and geometric foundations of quantum error-correcting codes, detailing their construction, properties, and the geometric tools used to analyze and determine their parameters.

Contribution

It provides a comprehensive exposition of quantum code geometries, including stabilizer and non-stabilizer codes, and introduces geometric methods for analyzing code parameters and equivalences.

Findings

Geometric methods facilitate efficient parameter deduction.

Construction of various quantum codes with different local dimensions.

Analysis of quantum MDS codes and MacWilliams identities.

Abstract

This is an expository article aiming to introduce the reader to the underlying mathematics and geometry of quantum error correction. Information stored on quantum particles is subject to noise and interference from the environment. Quantum error-correcting codes allow the negation of these effects in order to successfully restore the original quantum information. We briefly describe the necessary quantum mechanical background to be able to understand how quantum error-correction works. We go on to construct quantum codes: firstly qubit stabilizer codes, then qubit non-stabilizer codes, and finally codes with a higher local dimension. We will delve into the geometry of these codes. This allows one to deduce the parameters of the code efficiently, deduce the inequivalence between codes that have the same parameters, and presents a useful tool in deducing the feasibility of certain parameters. We also include sections on quantum maximum distance separable codes and the quantum MacWilliams identities.