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

这个项目将探索一个三角研究领域的三个角之间的关系：量子数学和信息，理论计算机科学，以及几何拓扑学。该项目将探索新的量子算法，以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的结果，即有效的量子算法不存在的证据。它还将探索与“拓扑学”及其与量子信息的联系相关的难题。这项研究的更广泛的影响将包括更好地理解量子计算机有朝一日建成后的功能；更好地理解计算机算法解决拓扑问题；数学、计算机科学和量子物理之间更多的跨学科联系，包括为学生和广大公众提供更多的想法。在更技术的层面上，该项目将研究量子算法和代数问题的困难结果，如隐藏子群问题。它将研究诸如正规表面理论之类的技术，以证明拓扑问题属于NP或其他类似的复杂类；或者它们是NP难的。该项目将调查有关量子拓扑不变性的结构问题，如建筑几何和辫子群表示的图像。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.