Exploring symmetric cones with quantum

Quantum algorithms for optimization often achieve speedups in the problem dimension. Yet, their error dependence and sensitivity to scale makes it challenging to identify broad classes of optimization problems for which thei r is a clear advantage over classical algorithms. This dissertation is comprisedof multiple projects spanning three parts that seek to reducethis gap. Part I concerns quantum linear algebra. We provide a construction for implementing matrix arithmetic operations, such as Kronecker and Hadamard products, on a quantum computer. Then, we demonstrate how Iterat ive Refinement can be leveraged to exponentialy improve the dependence on precision in the overall running time associated with classicaly solving linear systems of equations using quantum computers.