The geometry of quantum stabiliser codes

Abstract

The aim of this project is to bring together quantum error-correcting codes theory and the study of finite geometries. A quantum code is used to protect quantum information from errors that may occur due to quantum decoherence. We give a geometric interpretation of the codes as sets of lines in certain finite projective spaces. We exploit the geometric aspect of codes to rewrite proofs in a more intuitive way and explore their properties through visualization. Some examples of stabiliser codes and their associated quantum sets of lines are presented. We also discuss how to build nonadditive codes as the union of stabiliser codes. Finite geometry has proved to be a powerful tool to work on quantum error-correcting codes. Some of its applications include finding new codes or proving the non-existence of codes with certain parameters.