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A universal, scalable quantum computer will require the use of quantum error correction in order to achieve fault tolerance. The assessment and comparison of error-correcting strategies is performed by classical simulation. However, due to the prohibitive exponential scaling of general quantum circuits, simulations are restrained to specific subsets of quantum operations. This creates a gap between accuracy and efficiency which is particularly problematic when modeling noise, because most realistic noise models are not efficiently simulable on a classical computer. We have introduced extensions to the Pauli channel, the traditional error channel employed to model noise in simulations of quantum circuits. These expanded error channels are still computationally tractable to simulate, but result in more accurate approximations to realistic error channels at the single qubit level. Using the Steane [[7,1,3]] code, we have also investigated the behavior of these expanded channels at the logical error-corrected level. We have found that it depends strongly on whether the error is incoherent or coherent. In general, the Pauli channel will be an excellent approximation to incoherent channels, but an unsatisfactory one for coherent channels, especially because it severely underestimates the magnitude of the error. Finally, we also studied the honesty and accuracy of the expanded channels at the logical level. Our results suggest that these measures can be employed to generate lower and upper bounds to a quantum code's threshold under the influence of a specific error channel.

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.