FET: A research triangle of quantum mathematics, computational complexity, and geometric topology

FET：量子数学、计算复杂性和几何拓扑的研究三角

• 批准号: 2009029
• 负责人: Greg Kuperberg
• 金额: $ 49.45万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-10-01 至 2024-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2009029&HistoricalAwards=false
• 关键词: FET; research; triangle; quantum; mathematics

This project will explore relations among the three corners of a triangle of research areas: quantum mathematics and information, theoretical computer science, and geometric topology. The project will explore new quantum algorithms for computational problems related to number theory and cryptography; and it will explore computational complexity results, i.e., evidence that efficient quantum algorithms do not exist. It will also explore hard problems relating to the field of 'topology' and its connections to quantum information. The broader impacts of this research will include a better understanding of what quantum computers can do when they are one day built; a better understanding of computer algorithms for topological problems; and greater interdisciplinary connections among mathematics, computer science, and quantum physics, including greater access to ideas for students and the public at large.At a more technical level, the project will investigate quantum algorithms and hardness results for algebraic problems such as the hidden subgroup problem. It will investigate techniques such as normal surface theory to show that topological problems are in NP or other similar complexity classes; or that they are NP hard. And the project will investigate structural questions about quantum topological invariants such as building geometry and images of braid group representations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目将探索三角形研究领域三个角之间的关系：量子数学和信息、理论计算机科学和几何拓扑。该项目将探索新的量子算法，解决与数论和密码学相关的计算问题；它将探索计算复杂性结果，即有效量子算法不存在的证据。它还将探索与“拓扑”领域及其与量子信息的联系相关的难题。这项研究的更广泛影响将包括更好地了解量子计算机建成后可以做什么；更好地理解拓扑问题的计算机算法；以及数学、计算机科学和量子物理学之间更广泛的跨学科联系，包括为学生和广大公众提供更多获得想法的机会。在技术层面上，该项目将研究代数问题（例如隐藏子群）的量子算法和硬度结果问题。 它将研究法线表面理论等技术，以证明拓扑问题属于 NP 或其他类似的复杂性类别；或者说它们是 NP 难的。 该项目将研究有关量子拓扑不变量的结构问题，例如建筑几何形状和辫子群表示的图像。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。