An introduction to quantum: computing, communication complexity protocols, nonlocality and graph parameters

• Autor(a) principal: Arenstein, Lucas Silva
• Data de Publicação: 2022
• Tipo de documento: Dissertação
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
• Texto Completo: https://www.teses.usp.br/teses/disponiveis/45/45134/tde-19062023-143615/
• topic: Algoritmo de Grover; Algoritmos quânticos; Complexidade de comunicação quântica; Computação quântica; Grovers algorithm; Não-localidade; Nonlocality; Número cromático quântico; Parâmetros quânticos de grafos; Quantum algorithms; Quantum chromatic number; Quantum communication complexity; Quantum computing; Quantum graph parameters

What are the advantages quantum computing can provide when compared to classical computing? In this thesis, we aim to answer this question from different perspectives. To achieve this goal we divided this work into three parts. In Part I we start by studying the basic principles of quantum computing. Next, we introduce two quantum algorithms: the Deutsch-Jozsa and Grovers algorithms. The first algorithm has a lower query complexity and the second has a lower time complexity when compared with the best classical algorithm for the same problems. The final topic of this initial part is a detailed explanation and example of how to use Grovers algorithm to solve a Boolean formula in conjunctive normal form. In the second part, we focus on communication complexity problems. These problems are usually stated as follows: two spatially separated parties Alice and Bob receive an input from a referee. Their goal is to compute the value of a function that depends on both of their inputs with the least amount of communication between them. In Part II we will introduce protocols in which the transmission of quantum bits (qubits), instead of bits, can reduce the amount of communication necessary to solve these problems. We also study how Alice and Bob can use quantum entanglement to solve two communication complexity problems without communicating something that can not be done classically. In Part III we study nonlocal games inspired by standard graph theory parameters. A nonlocal game is usually defined as a game in which players that can share and do computations in an entangled state have some sort of advantage over classical players. We begin this final part by introducing the quantum chromatic number of a graph, which is the minimal number of colors necessary in a nonlocal game in which Alice and Bob can convince a referee with certainty that they have a proper coloring of the graph. We end this thesis by introducing other two quantum graph parameters, one related to graph homomorphism and the other to the independence number of a graph.