Quantum methods in mathematics and computer science

该项目的目的是探索数学和计算机科学中与量子力学相关的理论思想。量子力学是一种物理理论，值得注意的是，它需要一种新的概率论模型，可以称为量子概率论，这反过来又导致了量子计算理论和构建量子计算机的可能性。该项目将研究量子算法（可以在量子计算机上运行的算法）解决拓扑问题，例如确定两个结是等价的还是不同的。该项目还将在一种新的几何结构“量子度量空间”中研究量子纠错（量子计算机需要的一种新型纠错），在这种几何结构中，距离用量子概率的语言定义。最后，该项目将探索量子代数，这是一种具有非交换变量的现代代数，同样受到量子力学的启发，与量子计算有关。该项目将产生至少两个，甚至可能三个更广泛的影响。首先，所有工作将在arXiv上传播，arXiv是一个广泛使用的预印本服务器，包括量子计算的大多数工作，首席研究员将帮助支持arXiv。其次，该项目将包括开发关于量子概率、量子计算和量子力学的课堂讲稿，这些讲稿可能会被制作成未来的研究生教科书。第三，如果量子计算机建成，它们将产生极其广泛的影响，而这个项目中的想法将发挥作用。

The aim of this project is to explore theoretical ideas in mathematics and computer science that are related to quantum mechanics. Quantum mechanics is a physical theory that, notably, requires a new model of probability theory that can be called quantum probability, and this in turn leads to the theory of quantum computing and the possibility of building quantum computers. The project will investigate quantum algorithms (algorithms that could be run on quantum computers) for problems in topology, for example determining whether two knots are equivalent or different. The project will also investigate quantum error correction (a new kind of error correction that would be needed by quantum computers) in the setting of a new kind of geometry, "quantum metric spaces", in which distances are defined in the language of quantum probability. Finally the project will explore quantum algebra, which is a kind of modern algebra with non-commuting variables also inspired by quantum mechanics and which has relations to quantum computing.This project will have at least two and potentially three broader impacts. First, all work will be disseminated on the arXiv, a widely used preprint server that includes most work in quantum computing, and the principal investigator will help support the arXiv. Second, the project will include development of lecture notes, which could be made into a future graduate textbook, on quantum probability, quantum computing, and quantum mechanics. Third, if quantum computers are ever build, they will have an extremely broad impact, and the ideas in this project will play a role.