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Degenerate Codes and Capacities of Quantum Channels

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Title: Degenerate Codes and Capacities of Quantum Channels
Author: Jackson, Tyler Benjamin
Department: Department of Mathematics and Statistics
Program: Mathematics and Statistics
Advisor: Zeng, Bei
Abstract: No quantum system can be perfectly isolated from the environment and, as a result, no physical implementation of quantum information processing tasks can be completely free from noise. The best tool for combating such noise is the use of quantum error correcting codes (QECCs). The general requirements for QECCs have been documented for a while and yet construction of good codes and understanding their effect remains a difficult and active area of study. In this thesis I outline the work I have completed looking into both of these problems. My general technique throughout this work is to first reduce the problem space as much as possible through use of group theory. Then to use numerical methods and bring computational power to bear on the problem. In chapter 2, I investigate the construction of good codes for the amplitude damping channel, using the codeword stabilized quantum code(CWS) framework. Through an exhaustive search method many new codes with better parameters than previously known are found. In chapter 3, I continue constructing good codes for the amplitude damping channel, this time using code concatenation techniques to find results that would be unfeasible to find via an exhaustive search. Finally, chapter 4 broaches the difficult problem of determining if a quantum channel has capacity; the ultimate use of QECCs in a sense. Expanding on the key works done on the problem, I develop the theory surrounding effective noise channels obtained from applying a QECC on multiple uses of a channel in order to determine if capacity exists. Using this framework and making use of computational power available today, these techniques allow us to find many very noisy non-Pauli channels that have positive capacity which previously had not been shown to have capacity.
Date: 2017-04
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