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

本项目将探索一个三角形的三个角之间的关系研究领域：量子数学与信息，理论计算机科学和几何拓扑。该项目将探索与数论和密码学相关的计算问题的新量子算法；它将探索计算复杂性的结果，即有效的量子算法不存在的证据。它还将探讨与“拓扑”领域及其与量子信息的联系有关的难题。这项研究的更广泛影响将包括更好地理解量子计算机有朝一日建成后能做什么；更好地理解拓扑问题的计算机算法；加强数学、计算机科学和量子物理学之间的跨学科联系，包括让学生和广大公众更容易获得思想。在更高的技术层面上，该项目将研究量子算法和代数问题（如隐藏子群问题）的硬度结果。它将研究技术，如法向曲面理论，以表明拓扑问题是NP或其他类似的复杂性类；或者它们是NP困难的。该项目将研究量子拓扑不变量的结构问题，如建筑几何和编织群表示的图像。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.