Finding Paths via Quantum Systems and its Application for Quantum Algorithms

Abstract

The field of Quantum Information Theory provides the theoretical foundation for the pursuit of quantum computers. The ongoing questions of how quantum computers will be realized and what they will achieve, are both very uncertain. However, worldwide efforts are beginning to converge on some answers, and the future of quantum computers is looking brighter than ever. In contribution to the grand goal that is quantum computing, this thesis serves as a demonstration to the usefulness of quantum over classical computing. The central theme of my work, and my collaborators, is the exploration of using quantum systems as a tool for path finding and search algorithms. First, we explore a specific problem of sequential measurements in Quantum Information Theory, namely quantum retrodiction. In experiments involving quantum systems with numerous intermediate interactions, often times the only thing that matters to the experimenters is the final measurement. However, quantum retrodiction is the problem of extrapolating information about the past of a quantum system, by studying the present state. The case studied in this dissertation is the retrodiction of a series of two-outcome measurements, the double and triple qubit-interferometers. In both cases, we are presented with the problem of inferring the results of the two-outcome measurements, using only the final state of the system. Specifically, all of the possible outcomes from the measurements result in a different final state, and we are tasked with distinguishing which final state we have. One version of this problem can be seen as light traversing a series of half-silver mirrors, and we would like to know the path the light took. To distinguish which final state we are given, we employ the use of POVMs (Positive-Operator Valued Measurements) that reflect the symmetry of the system. The effectiveness of these POVMs in solving the retrodiction problem are then presented, along with a few comments as to other types of information that can be extracted. Second, we examine a different type of path finding via a quantum system. For a given graph G that can be expressed by connections and nodes, the same graph can be mapped onto a quantum system with a Hilbert space H. All of the connections in the graph can be represented as states in H, and the nodes act as sites for local unitary operations. The sum of all these local operations give rise to U, which is the unitary operator that acts on H, and describes the behavior of the system. Using this formalism, we explore a particular type of U called a scattering quantum random walk. The quantum random walk is the quantum mechanical analogy to the classical random walk, and serves as a powerful tool for searching on a graph. In this dissertation, we present the case of searching for a marked final node, on a graph that consists of a series of stars. The initial state of the system has an equal probability distribution among all states, reflecting no a priori knowledge about the location of the final node. But by applying the quantum random walk, the final state of the system reaches a superposition where nearly all of the probability is concentrated along the path leading to the marked final node. Lastly, we extend this idea of using a quantum random walk to reveal a path even further. We examine a specific type of graph G, called an “N-tree” graph, and apply our quantum random walk scheme. An N-tree graph can be described as a maze where the solver is presented with a junction consisting of N choices. Beyond each choice, is an identical junction of N choices, and so on. The depth of the maze is categorized by a parameter M, thus, to correctly solve the maze one must make M correct choices in succession. Labeling the marked final nodes as F, we present a quantum algorithm that uses a quantum random walk to locate F faster than any classical algorithm. Specifically, the quantum random walk results in a final state of the system with a high probability concentration on all the states leading to F. Two methods for searching for F are then presented. The first approach mimics the ideology of a Grover Search, relying on repetitive searches until F is found. The second approach presents a novel use of past measurements to dictates how the algorithm proceeds next. Specifically, we take advantage of the fact that states along the path leading to F hold most of the probability in the system. In the end, the second quantum algorithm approach provides an speedup over the classical algorithm of the order O().