On The Fourier Transform Approach To Quantum Error Control

Abstract

Quantum mechanics is the physics of the very small. Quantum computers are devices that utilize the power of quantum mechanics for their computational primitives. Associated to each quantum system is an abstract space known as the Hilbert space. A subspace of the Hilbert space is known as a quantum code. Quantum codes allow to protect the computational state of a quantum computer against decoherence errors. The well-known classes of quantum codes are stabilizer or additive codes, non-additive codes and Clifford codes. This thesis aims at demonstrating a general approach to the construction of the various classes of quantum codes. The framework utilized is the Fourier transform over finite groups. The thesis is divided into four chapters. The first chapter is an introduction to basic quantum mechanics, quantum computation and quantum noise. It lays the foundation for an understanding of quantum error correction theory in the next chapter. The second chapter introduces the basic theory behind quantum error correction. Also, the various classes and constructions of active quantum error-control codes are introduced. The third chapter introduces the Fourier transform over finite groups, and shows how it may be used to construct all the known classes of quantum codes, as well as a class of quantum codes as yet unpublished in the literature. The transform domain approach was originally introduced in (Arvind et al., 2002). In that paper, not all the classes of quantum codes were introduced. We elaborate on this work to introduce the other classes of quantum codes, along with a new class of codes, codes from idempotents in the transform domain. The fourth chapter details the computer programs that were used to generate and test for the various code classes. Code was written in the GAP (Groups, Algorithms, Programming) computer algebra package. The fifth and final chapter concludes, with possible directions for future work. References cited in the thesis are attached at the end of the thesis.