#### Date of Award

Spring 1-1-2016

#### Document Type

Thesis

#### Degree Name

Master of Arts (MA)

#### Department

Mathematics

#### First Advisor

Katherine Stange

#### Second Advisor

Bengt Fornberg

#### Third Advisor

Judith Packer

#### Fourth Advisor

Divya E. Vernerey

#### Abstract

The intent of this thesis is to elucidate the quantum computing algorithm developed by Peter Shor called Shor's algorithm. We will provide a detailed description and simulation of the algorithm using MATLAB. Precursory information regarding quantum phenomena such as superposition, entanglement, and Dirac notation, will be described in great detail so that the reader may have a better understanding of the operations in Shor's algorithm.

Quantum computers store and transport information quite differently than their classical counterparts. We will provide a quick overview of these differences to high- light the benefit of utilizing quantum phenomena in a computer in order to create massive parallel computations. Thus, reducing the complexity time for classical algorithms used to solve problems such as the prime factorization problem and the period finding problem.

The Quantum Fourier Transform is a principle component in Shor's algorithm. We will explicitly define the Quantum Fourier Transform and show that it is a unitary transformation. We will also show how the Quantum Fourier Transform, as well as another unitary transformation called the Hadamard transform, functions in Shor's algorithm.

One of the initial parameters in Shor's algorithm is to select a random variable. We will examine the erratic effects of this random variable as well as how it effects the probability of us successfully reducing an integer into a product of two primes. We will provide a thorough analysis of the randomness in Shor's algorithm. We will also show how measuring the state of our quantum system as well as selecting a suitable random variable impacts finding the period of the Quantum Fourier Transform which in turn will either give a high or low probability of obtaining a factor of some integer.

#### Recommended Citation

Parsons, Elizabeth Ellen, "Simulation of a Quantum Prime Factoring Algorithm" (2016). *Mathematics Graduate Theses & Dissertations*. 39.

https://scholar.colorado.edu/math_gradetds/39