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

Jonas Eidesen

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.

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

Paper Structure

This paper contains 15 sections, 30 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some notation for common groups
  4. Projective representation theory: definitions and notation
  5. Projective representation theory: irreducible representations
  6. Projective representation theory: induction and Clifford theory
  7. Error models and projective error models
  8. Examples of projective error models
  9. Stabilizer codes, Clifford codes, and weak stabilizer codes: definitions
  10. Clifford codes: error detection
  11. Weak stabilizer Clifford codes
  12. Examples of non-stabilizer Clifford codes
  13. Examples of non-Clifford weak stabilizer codes
  14. Product codes
  15. Further questions and remarks

Key Result

Proposition 3.3

Let $V$ be a Hilbert space. If $(E,\lambda) \in \mathbf{EM}_V$ then the quotient $G := E/Z(E)$ admits a projectively faithful irreducible projective representation $\pi$ on $V$. Furthermore, the group homomorphism $q \circ \pi$ is uniquely determined by the representation $\lambda$. Conversely, if $

Theorems & Definitions (70)

  • Remark 2.1
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Definition 3.4
  • Proposition 3.5
  • proof
  • Example 3.6
  • Proposition 4.1
  • ...and 60 more