AF: Small: Quantum Theory, Computational Complexity, and Geometry/Topology

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The aim of the project is to explore relations between quantum theory, computational complexity, and geometry and topology.In the intersection of geometry/topology and computational complexity, the project will research the computational difficulty of topological properties of knots and links in ordinary 3-dimensional space, and of 3-dimensional manifolds. For instance, when are two 3-manifolds the same? Or, given a diagram of a knot, is it the unknot? The project will explore both easiness results, that certain answers can be computed quickly, possibly with artificial help; and hardness results, that certain answers are intrinsically difficult to compute.In the intersection of quantum theory and computational complexity, the project will consider both easiness results and hardness results for core problems in quantum computation. One of the most exciting mathematical discoveries of our era is the concept of a new type of computer, a quantum computer, that would be able to run new algorithms that cannot be run on a standard computer. A fundamental question which will be addressed by this research is what quantum computers would be able to do, if they were built.In the intersection of quantum theory and geometry and topology, the project will consider geometric problems with implications for quantum non-locality as described by Einstein-Podolsky-Rosen and by John Bell. The project will also consider new geometric spaces motivated by the theory of quantum error correction in quantum computation.The broader impacts of the project begin with the importance of its research areas. In particular, if quantum computers are eventually built, then their impact on society will be substantial. The proposed research has various implications for quantum computation. The project will disseminate its results through the arXiv e-print server, and help support the arXiv. The project will also develop expositions in quantum computation and computational complexity.