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

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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.