Collaborative Research: Mathematical Aspects of Quantum Information Science

该项目研究量子信息科学中出现的数学问题。 其目的是发展理论和计算技术，以解决有关可扩展量子计算机发展的问题。 研究人员将（1）使用完全正线性映射理论研究连接指定量子态的量子操作存在的准则，并提供相关量子操作的构造方案;（2）使用广义数值范围研究量子纠错码的理论和实现;（3）利用基于Lie理论的优化技术研究量子态的最佳逼近，改进量子控制理论。 该项目将涉及理论家和实验家之间的合作，以便可以进行实验来测试理论和假设，并为理论发展提供见解。 量子计算是一个快速发展的多学科研究领域。 如果能够建造大规模量子计算机，它们将能够解决当前经典计算机能力之外的重要问题。 虽然量子计算机的发展有望产生深远的影响，但仍有许多尚未解决的理论问题和实验挑战必须克服。 例如，量子计算机必须采用一种方法来纠正量子系统与其环境之间不可避免的不可控相互作用所导致的数据错误，这种现象称为退相干。 本项目旨在进一步发展量子信道中纠错的理论和算法，并研究连接指定量子态的量子操作的构造。 该项目还将研究量子控制理论中的数学问题，例如通过更容易获得的量子态来近似所需的量子态。

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.