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A Menagerie Of Symmetry Testing Quantum Algorithms, Margarite Lynn Laborde
A Menagerie Of Symmetry Testing Quantum Algorithms, Margarite Lynn Laborde
LSU Doctoral Dissertations
In Chapter 1, we establish the mathematical background used throughout this thesis. We review concepts from group and representation theory. We further establish fundamental concepts from quantum information. This will allow us to then define the different notions of symmetry necessary in the following chapters. In Chapter 2, we investigate Hamiltonian symmetries. We propose quantum algorithms capable of testing whether a Hamiltonian exhibits symmetry with respect to a group. Furthermore, we show that this algorithm is that this algorithm is DQC1-Complete. Finally, we execute one of our symmetry-testing algorithms on existing quantum computers for simple examples. In Chapter 3, we …
A Permutational Triadic Approach To Jazz Harmony And The Chord/Scale Relationship, John Bishop
A Permutational Triadic Approach To Jazz Harmony And The Chord/Scale Relationship, John Bishop
LSU Doctoral Dissertations
This study provides an original triadic theory that combines existing jazz theory, in particular the chord/scale relationship, and mathematical permutation group theory to analyze repertoire, act as a pedagogical tool, and provide a system to create new music. Permutations are defined as group actions on sets, and the sets used here are the constituent consonant triads derived from certain scales. Group structures provide a model by which to understand the relationships held between the triadic set elements as defined by the generating functions. The findings are both descriptive and prescriptive, as triadic permutations offer new insights into existing repertoire. Further, …