On the Dual of the Lovász Theta Prime Number for the Disjunctive Product of Graphs

The Lovász theta function, and the variants of it given by Schrijver and Szegedy are upper bounds on the independence number of a graph. These functions play an important role in several optimization problems, such as the Cohn-Elkies bound for optimal sphere packing densities.<br/><br/>This thesis covers the properties of these functions. The...

QAOA Mixing Hamiltonians for MinVertexCover

The minimum vertex cover problem (MinVertexCover) is an important optimization problem in graph theory, with applications in numerous fields outside of mathematics. As MinVertexCover is an NP-hard problem, there currently exists no efficient algorithm to find an optimal solution on arbitrary graphs. We consider quantum optimization algorithms,...

Convergence of the mixing method: An iterative algorithm for solving diagonally constrained semidefinite programs

This thesis explores the convergence of the mixing method, an iter- ative algorithm for solving diagonally constrained semidefinite programs. In this paper we first give an exposition of the convergence proof for the mixing method based on the proof by Wang, Chang, and Kolter , where we restructure some parts of the proof and provide extra de-...

Clebsch-Gordan Coefficients: A Quantum Mechanical and Mathematical Perspective

In this work, Clebsch-Gordan coefficients are studied from both a quantum mechanical and a mathematical perspective. In quantum mechanics, Clebsch-Gordan coefficients arise when two quantum systems with a certain angular momentum are combined and the total angular momentum is to be found. We start by discussing the relevant postulates of quantum...

Learning based hardware-centric quantum circuit generation

Large fault­tolerant universal gate quantum computers will provide a major speed­up to a variety of common computational problems. While such computers are years away, we currently have noisy intermediate­scale quantum (NISQ) computers at our disposal. In this project we present two quantum machine learning approaches that can be used to find...

Geometry and Reconstruction of Bipartite Quantum Correlations

The first part of this thesis provides a mathematical description for bipartite quantum correlations, aiming to analyze the geometry of several sets of correlations. We explain why quantum entanglement can be used to simulate shared randomness: C_loc(Γ) ⊆ C_q^d(Γ) for a sufficiently large d. The known bound for this...