This project studies mathematical problems arising in quantum information science. The aim is to develop theoretical and computational techniques to solve problems concerning the development of scalable quantum computers. The investigators will (1) use the theory of completely positive linear maps to study the criteria for the existence of quantum operations connecting prescribed quantum states and to provide construction schemes for associated quantum operations; (2) use generalized numerical ranges to study the theory and implementation of quantum error correction codes; and (3) use optimization techniques based on Lie theory to study the best approximation of quantum states to improve quantum control theory. The project will involve collaborations between theorists and experimentalists so that experiments can be conducted to test the theory and conjectures as well as to provide insights for theoretical development.
Quantum computing is a rapidly-growing area of multidisciplinary research. If large-scale quantum computers can be built, they will be able to solve important problems that lie beyond the capabilities of current classical computers. While the development of quantum computers promises far-reaching implications, there are still many open theoretical questions and experimental challenges that must be overcome. For example, a quantum computer must employ a method to correct data errors that result from the inevitable uncontrollable interaction between a quantum system and its environment, a phenomenon known as decoherence. This project aims to further develop the theory and algorithms for error correction in a quantum channel and to investigate the construction of quantum operations connecting prescribed quantum states. The project will also investigate mathematical problems in quantum control theory such as approximating a desired quantum state by more accessible ones.
The purpose of the project is to study important problems arising in quantum information science, which concerns the use of quantum properties to store and process information. Advance of the proposed research will catalyze the development of quantum computer leading to a transformation of the computational environment. The change will significantly affect high speed computing as well as daily life activities such as the security of financial transactions. In terms of intellectual merit of the project, the principal investigators have made progress in the study of several topics in quantum computing including quantum error correction, quantum operations and quantum channels, entanglement, maps on quantum states. Under the support of this grant, fifty two papers on these and related topics were published or accepted for publications in international journals. New techniques have been established for the research, and some computer programs developed in the study were posted on line for public access. In terms of broader impact of the project, not only did the principal investigators solved and advanced individual research problems, they disseminated their results broadly to the general research community in several different ways. They published their finding in journals, research archives, and personal homepage so that they are accessible to the public. During the grant period, they did more than 100 presentations on their results and ideas in conferences and colloquium talks in different institutions. Furthermore, they organized 4 professional meetings, 3 workshops, 3 symposiums, and 4 summer schools on the topic to introduce the problems, results and techniques to researchers. The research project involved more than twenty collaborators from different disciplines and countries including USA, Canada, China, Hong Kong, Japan, Taiwan, and Slovenia, the results and techniques were disseminated to different circles. The project also involved more than ten undergraduate, and three graduate students, several beginning researchers, and researchers from different areas. These activities helped promote interdisciplinary research and human resource development.
