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、计算复杂性和几何与拓扑之间的关系，在几何/拓扑学和计算复杂性的交叉中，研究普通三维空间中节点和链节的拓扑性质以及三维流形的计算难度。例如，两个三维流形什么时候是相同的？或者，给出一个结的图解，它是结吗？该项目将探索易性结果和难度结果，前者是某些答案可以快速计算，可能是在人工帮助下；后者是某些答案本质上很难计算。在量子理论和计算复杂性的交集中，该项目将同时考虑量子计算核心问题的易性结果和难度结果。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，如果量子计算机被建造起来，它们将能够做什么。在量子理论与几何和拓扑学的交叉中，该项目将考虑爱因斯坦-珀多尔斯基-罗森和约翰·贝尔所描述的与量子非定域性有关的几何问题。该项目还将考虑由量子计算中的量子纠错理论激发的新的几何空间。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。这项拟议的研究对量子计算有多方面的影响。该项目将通过arxiv电子印刷服务器传播其成果，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