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

AF：小：量子理论、计算复杂性和几何/拓扑

• 批准号: 1716990
• 负责人: Greg Kuperberg
• 金额: $ 47.19万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-09-01 至 2021-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1716990&HistoricalAwards=false
• 关键词: AF; Small; Quantum; Theory; Computational

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.

该项目的目的是探索量子理论、计算复杂性以及几何和拓扑之间的关系。在几何/拓扑与计算复杂性的交叉领域，该项目将研究普通3-中结和连杆的拓扑性质的计算难度。维空间和 3 维流形。 例如，两个 3 流形何时相同？或者，给定一个结的图表，它是未结的吗？ 该项目将探索两个简单的结果，即可以在人工帮助下快速计算某些答案；在量子理论和计算复杂性的交叉点上，该项目将同时考虑量子计算核心问题的简单性结果和困难性结果。 我们这个时代最令人兴奋的数学发现之一是一种新型计算机的概念，即量子计算机，它将能够运行标准计算机无法运行的新算法。 这项研究将解决的一个基本问题是，如果量子计算机建成的话，它们能够做什么。在量子理论与几何和拓扑的交叉领域，该项目将考虑对量子非定域性产生影响的几何问题：由爱因斯坦-波多尔斯基-罗森和约翰·贝尔描述。 该项目还将考虑由量子计算中的量子纠错理论激发的新几何空间。该项目的更广泛影响始于其研究领域的重要性。特别是，如果量子计算机最终建成，那么它们对社会的影响将是巨大的。拟议的研究对量子计算具有多种影响。该项目将通过 arXiv 电子打印服务器传播其结果，并帮助支持 arXiv。该项目还将对量子计算和计算复杂性进行阐述。

项目成果

Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization

作为几何化推论的 3 流形的算法同胚

• DOI: 10.2140/pjm.2019.301.189
• 发表时间: 2019
• 期刊: Pacific Journal of Mathematics
• 影响因子: 0.6
• 作者: Kuperberg, Greg

Coloring invariants of knots and links are often intractable

结和链接的颜色不变量通常很棘手

• DOI: 10.2140/agt.2021.21.1479
• 发表时间: 2021
• 期刊: Algebraic & Geometric Topology
• 影响因子: 0.7
• 作者: Kuperberg, Greg;Samperton, Eric
• 通讯作者: Samperton, Eric

Computational complexity and 3–manifolds andzombies

计算复杂性和 3 流形和僵尸

• DOI: 10.2140/gt.2018.22.3623
• 发表时间: 2018
• 期刊: Geometry & Topology
• 影响因子: 2
• 作者: Kuperberg, Greg;Samperton, Eric
• 通讯作者: Samperton, Eric