Quantum walks may be described as the natural counterparts of classical random walks, governed by the principles of quantum mechanics. In recent times, because of their potential applications to the development of a new species of super-efficient algorithms, quantum walks have been the focus of extensive research interest. However, largely due to the mathematical challenges involved, some important aspects of the theory have remained relatively unexplored. These challenges have persisted, especially in connection with attempts to treat jointly the effects of the characteristically quantum-mechanical phenomena known as "entanglement" and "decoherence".
Through this project, PI aims to elucidate the effects of decoherence on quantum walks over certain networks. His study will address two questions: 1) How is the limiting distribution of the quantum walk related to the level of decoherence to which the quantum walk is exposed? and 2) How does the measure of entanglement between the subsystems (coin and position) depend upon the level of decoherence? In this approach, the measure of entanglement is reflected by the mutual information between the subsystems, which, in turn, is defined in terms of the von Neumann entropy.
The outcomes to flow from this proposal will be of interest to a wide audience of active investigators at the forefront of research in the fields of quantum computing and quantum information. Last but not least, through this project, PI aims to introduce his students, both graduate and undergraduate, to the emerging field of Quantum Information Science.
Chaobin Liu Quantum walks may be regarded as the natural counterparts of classical random walks, governed by the principles of quantum mechanics. In recent times, because of their potential applications to the development of a new species of super-efficient algorithms, and their applications to understanding the dynamics of various physical systems, quantum walks have been the focus of extensive research interest. However, due largely to the mathematical challenges involved, some important aspects of the theory have remained relatively unexplored. These challenges have persisted, especially in connection with attempts to treat jointly the effects of the characteristically quantum-mechanical phenomena known as "entanglement" and "decoherence". Through this project, we have attempted to elucidate the effects of decoherence on quantum walks over certain networks. Our study has addressed two basic questions: 1) How is the limiting distribution of the quantum walk affected by the level of decoherence to which the quantum walk is exposed? and 2) How does the measure of entanglement between the subsystems (coin and position) depend upon the level of decoherence? In this approach, the degree of entanglement is reflected by a measure of the mutual information between the subsystems, which, in turn, is defined in terms of the von Neumann entropy. The research outcomes of this proposal are of interest to a wide audience of active investigators in the fields of quantum computing and quantum information. The following is an outline, in chronological order, of the specific research outcomes of this project: For a quantum walk on the N-cycle, Liu et al have managed to describe precisely how the von Neumann entropy of the total quantum system responds to decoherence, and how the mutual information between subsystems is affected. This may be one of the first, if not the very first, published study which manages to treat both entanglement and decoherence in a unified theoretical framework. The findings have been published in Mathematical Structures in Computer Science. For a quantum walk on the N-cycle, Liu et al derived analytic confirmation of some conjectures concerning the effects of noise (decoherence) on the position probability distribution. These conjectures had been formulated and published by a world-renowned expert on quantum walks. The findings have been published in Phys. Rev. A . For a one-dimensional quantum walk with two entangled coins, Liu et al proved a significant result pertaining to the localization property of quantum walks. The result had been conjectured in 2005 by a research group in the United Kingdom. This contribution has been published in Quantum Information Processing. Going well beyond the initial research agenda of the project, we have published contributions to the development of "quantum Markov chain theory", which probes the limiting states of a quantum system governed by a sequence of stochastic quantum operations. The findings have been published in International Journal of Mathematics and Mathematical Sciences. Progressing even further beyond the initial research aims of the project, we have formulated and proved a weak limit theorem describing the terminal behavior of the transition probabilities of a discrete two-state quantum walk on the half-line subject to a general condition at the boundary. Unitary quantum walks of this kind are recognized to be very rare and difficult to analyze. The findings have been submitted for publication. Last but not least, through this project, our students, both graduate and undergraduate, have been introduced to the emerging field of quantum information sciences. The involvement of students in high-level research has been one of this project’s most important outcomes. Over the past three years, Liu has supervised research projects with five mathematics majors, both graduate and undergraduate. To the student participants, this project has provided a unique opportunity to practice basic research and to experience the power of mathematics as a means of modeling various fundamental processes of physics. From a more pedestrian perspective, the student participants have learned to work as team, in cooperation with their mentor and their peers, subject to the constraints of various formal guidelines and deadlines. To prove their proficiency, the students conducted numerous poster and oral presentations at various national and regional professional conferences.
