#### Date of Award

12-1-2021

#### Degree Name

Master of Science

#### Department

Physics

#### First Advisor

Byrd, Mark

#### Abstract

Errors in a quantum system is a problem for quantum computers. Entanglement of qubits can lead to the propagation of errors throughout the system. In order to tackle this issue, Quantum Error Correction Codes were developed. Quantum Error Correction Codes are elements of a subspace of the entire Hilbert space describing the system. This subset is called the codespace. All computation is done using the elements of the codespace. The elements of the basis of the codepace are called codewords. Errors cause the system to evolve to the outside the codespace, which we can detect and then recover the original state. Knill et al. \cite{Knill} developed a set of Error Correction Conditions (ECC) to ascertain whether a given Error Correction Code would be able to recognize and correct errors from a given error map.One of the ways to describe quantum states is using the bra and ket vectors and density matrices. A popular way to describe the evolution of quantum states is using the Operator Sum Representation (OSR) of the evolutionary map. The Knill and Laflamme ECC are described using the OSR and state vectors (bra and ket notation) of the codewords. While this expression is very useful, in previous research by Byrd et al. \cite{Alvin} into Error Correction of NCP error maps, they ran into problems where the Knill and Laflamme ECC did not work for Error Correction in this scenario. Thus, we looked for different forms of Error Correction Conditions where this problem would not arise. Although we haven't successfully obtained ECC for NCP maps, the result we obtained is required for an ECC for NCP maps.Another way to describe quantum states is using the polarization vector representation. The evolution of the quantum system can be described by the evolution of the polarization vector. This can be described by a vector transformation matrix called the affine map.The aim of this paper is to describe the Knill and Laflamme ECC in terms of the polarization vector representation of the quantum states and the affine map of the error map. We have also developed a novel method to describe subspaces of quantum systems in the polarization vector representation.

#### Access

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