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From Quantum Entanglement to Tensor Decomposition by Global Optimization

从量子纠缠到全局优化的张量分解

基本信息

批准号：
1912816
负责人：
Moody Chu
金额：
$ 47.08万
依托单位：
North Carolina State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1912816&HistoricalAwards=false
关键词：
Quantum Entanglement Tensor Decomposition Global

项目摘要

Entanglement and separability are twins. Entanglement is the most basic mode when characterizing the coupling or interaction of multiple parts within a system; separability is to represent the complicated system in an equivalent but more manifesting relationship for understanding and control. This project aims to develop methods to numerically measure the "absolute" gap between an entangled state and its nearest separable state with the new tool of global optimization techniques. The initial goal is to establish a basal paradigm for gauging entanglement and separability with global optimization technologies in the context of quantum informatics. With modest modification, the paradigm can be applied across different fields. Results from this research will make it possible to address separability issues in many other contexts, such as economic development, agricultural production, industrial manufacture, environmental evolution, social networks, and applied mechanics, where constituents, factors, parts, or subsystems are regularly intertwined.Quantum entanglement is regarded as an indispensable resource for many applications due to the potential of quantum computing for fast, concurrent computation. The nonlinear correlations among subsystems make it difficult to analyze by traditional decomposition techniques. On the other hand, the notion of tensors has also gained new attention thanks to its great descriptive flexibility. Both structures share similar features concerning entanglement and separability. There have been many activities and achievements on both fronts. Yet, the avenue of numerically measuring the "absolute" gap between an entangled state and its nearest separable state has never been fully undertaken. This project aims to tackle both quantum entanglement and low-rank tensor approximation under one framework by global optimization techniques. When global optimization is finished, within the prescribed error tolerance we have in hand the metric between a given state and the set of separable states, by which we can gauge the quality of entanglement, draw conclusions on whether the given system is robustly entangled, and extend the knowledge to other applications. This project aims to establish theoretic and algorithmic foundations to: 1) exploit the geometric properties of entanglement; 2) develop a common platform for new algorithms effective in robustness, speed, and accuracy; and 3) explore the generalization to applications with additional constraints. This research together with the resulting software package is expected to find wide applicability extending from quantum mechanics to data analysis, network analysis, and other fields. The work will solidify study of many features under one unified framework.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.

纠缠性和可分离性是双胞胎。纠缠态是表征系统内多个部分耦合或相互作用的最基本模式；可分性是将复杂的系统用一种等价但更明显的关系来表示，以便于理解和控制。本项目旨在利用全局优化技术的新工具，开发一种方法来数值测量纠缠态与其最近可分离态之间的“绝对”差距。最初的目标是在量子信息学的背景下，用全局优化技术建立一个测量纠缠和可分离性的基本范式。稍加修改，该范式可以应用于不同的领域。这项研究的结果将使解决许多其他背景下的可分离性问题成为可能，例如经济发展，农业生产，工业制造，环境演变，社会网络和应用力学，其中成分，因素，零件或子系统经常交织在一起。由于量子计算具有快速、并发计算的潜力，量子纠缠被认为是许多应用中不可或缺的资源。子系统之间的非线性相关性使得传统的分解技术难以进行分析。另一方面，张量的概念由于其在描述上的极大灵活性也获得了新的关注。这两种结构在纠缠和可分离性方面具有相似的特征。在这两条战线上开展了许多活动，取得了许多成就。然而，对纠缠态与其最近的可分离态之间的“绝对”间隙进行数值测量的途径从未完全进行过。本项目旨在利用全局优化技术在一个框架下解决量子纠缠和低秩张量逼近问题。当全局优化完成后，在规定的误差容限内，我们掌握了给定状态和可分离状态集之间的度量，通过它我们可以衡量纠缠的质量，得出给定系统是否鲁棒纠缠的结论，并将知识扩展到其他应用。本项目旨在建立理论和算法基础：1)利用纠缠的几何性质；2)为鲁棒性、速度和准确性有效的新算法开发一个通用平台；3)探索在附加约束条件下的推广应用。本研究以及由此产生的软件包有望从量子力学扩展到数据分析、网络分析等领域。这项工作将把许多特征的研究巩固在一个统一的框架下。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。