Projective error models: Stabilizer codes, Clifford codes, and weak stabilizer codes

TL;DR

This paper explores the mathematical structure of stabilizer, Clifford, and weak stabilizer codes using projective representation theory, introducing weak stabilizer codes and analyzing their properties and relationships.

Contribution

It introduces weak stabilizer codes, analyzes their relationship with Clifford and stabilizer codes, and characterizes conditions for non-triviality using group cohomology.

Findings

Obstruction for stabilizer codes to be non-trivial is given by a group cohomology class.

Complete characterization of when Clifford codes are weak stabilizer codes under certain conditions.

Construction of infinite families of Clifford codes that are not stabilizer codes.

Abstract

By defining projective error models we study the mathematical structure of Clifford codes and stabilizer codes using tools from projective representation theory. Furthermore, we introduce a new class of codes which we have called weak stabilizer codes and we determine some relationships between these three classes of codes. We show that the obstruction for a stabilizer code to be non-trivial is given by a class in group cohomology, and we are able to determine similar obstructions for weak stabilizer codes to be non-trivial. In the case where the projective error model corresponds to a nice error basis we give a complete characterization of when a Clifford code is a weak stabilizer code in terms of the size of the group of logical operators and the size of the group of stabilizers of the code. Lastly, we produce two infinite families of Clifford codes that are not stabilizer codes, as well as a method of combining these examples into more examples of non-stabilizer Clifford codes.