Thesis (Ph.D.) - Indiana University, Department of Mathematics and the Department of Computer Science, 2019Most quantum computing models are based on the continuum of real numbers, while classical digital computers faithfully realize only discrete computational models. Analog computers appear to be an option, but in reality are far weaker than would be needed for computational models requiring real numbers. One approach to resolving this conflict is to find consistent mathematical ways to limit measurement precision to computable contexts that do not require incomputable real numbers. Our goal is to build a more philosophically consistent models by investigating discrete quantum comput- ing using finite number systems, and, alternatively, by incorporating finite precision measurement using intervals into quantum theory. We begin by replacing the continuum of complex numbers by discrete finite fields in quantum theory. The simplest theory, defined over unrestricted finite fields, is so weak that it cannot ex- press Deutsch’s algorithm, but, paradoxically, is also so powerful that it can be used to solve the UNIQUE-SAT problem, which is as hard as a general NP-complete problem. (See the thesis for full Abstract)