FET: Small: A triangle of quantum mathematics, computational complexity, and geometry

FET：小：量子数学、计算复杂性和几何的三角关系

• 批准号: 2317280
• 负责人: Greg Kuperberg
• 金额: $ 59.92万
• 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: 2023
• 资助国家: 美国
• 起止时间: 2023-12-01 至 2026-11-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2317280&HistoricalAwards=false
• 关键词: FET; Small; triangle; quantum; mathematics

The research aims of this project are organized as a triangle whose corners are three different research areas: the mathematics that underlies quantum reality; the existence or non-existence of fast algorithms for computational problems; and the modern geometry of knots, curvature, and higher-dimensional spaces and their symmetries. Considering each leg of the research triangle, the project will explore the existence or non-existence of quantum algorithms, meaning algorithms that can only be run on a quantum computer. The project will explore the existence or non-existence of fast algorithms for geometric and topological questions; for instance, when are two different-looking knots actually the same knot? And the project will explore the geometric properties of networks that arise as interaction diagrams in quantum physics. As part of the broader impacts outside of pure mathematics and theoretical computer science, the project will shed light on the capabilities and limitations of quantum computers and traditional classical computers, particularly for geometric calculations, but also concerning how to protect data from future quantum computers. Other broader impacts will include training graduate students and developing public expository materials.The project will explore quantum algorithms for algebraic problems such as the hidden subgroup problem, which generalizes Shor's algorithm for factoring integers and more generally for period-finding. The project will explore the computational complexity of classification problems for manifolds, including the homeomorphism problem for 3-manifolds and for higher-dimensional simply-connected manifolds. The project will also explore the geometry of planar tensor networks bases for tensor invariants of Lie groups, generalizing prior work by the investigator that established that non-positively-curved tensor networks yield bases of tensor invariants for rank 2 Lie groups.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目的研究目的是一个三角形组织的，其角落是三个不同的研究领域：构成量子现实的数学；对于计算问题的快速算法的存在或不存在；以及结的现代几何形状，曲率和较高的空间及其对称性。 考虑到研究三角形的每个腿，该项目将探索量子算法的存在或不存在，这意味着只能在量子计算机上运行的算法。 该项目将探讨快速算法的几何和拓扑问题的存在或不存在；例如，什么时候两个不同的结实际上是相同的结？该项目将探索作为量子物理学中相互作用图的网络的几何特性。 作为纯数学和理论计算机科学之外的更广泛影响的一部分，该项目将阐明量子计算机和传统古典计算机的能力和局限性，尤其是用于几何计算，但也涉及如何保护数据免受未来量子计算机的侵害。 其他更广泛的影响将包括培训研究生和开发公共说明性材料。该项目将探索代数问题的量子算法，例如隐藏的子组问题，该问题将Shor的算法推广用于整数的算法，更全面地进行周期性调查。 该项目将探讨流形分类问题的计算复杂性，包括3个manifolds的同态性问题以及更高维度简单相关的流形。 The project will also explore the geometry of planar tensor networks bases for tensor invariants of Lie groups, generalizing prior work by the investigator that established that non-positively-curved tensor networks yield bases of tensor invariants for rank 2 Lie groups.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.