Quantum methods in mathematics and computer science

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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.