FET: Small: Entanglement Estimation for Quantum Computing

量子纠缠是实现量子计算和其他量子技术的物理资源。虽然估计量子态纠缠度的问题已经有了相对较长的历史，但这个问题及其后果还远远没有被完全理解。特别是，尽管众所周知的事实是纠缠使计算速度优于经典算法，但对可用纠缠的数量和类型如何与量子算法的性能相关的定量理解在很大程度上仍然难以捉摸。本项目旨在通过将算法性能与量子态纠缠程度联系起来，为量子算法的分析和设计开发一个新的框架，从而解决这一挑战。在文献中，量子电路是理解量子算法性能的最普遍的模型。然而，电路模型仅限于研究量子算法的计算复杂性，它不能直接利用或量化纠缠作为算法分析的基本资源。本项目提出的框架将补充现有的框架，为量子算法的发展带来新的视角。在项目过程中，将开发一门关于量子算法的新课程，增加量子计算研究生证书课程的长期目标，传统的计算机科学课程将更新为量子信息科学的先决材料或介绍。为了提高人们对量子科学及其多样化职业机会的认识，研究团队将继续为高中STEM教育工作者举办外展研讨会，以开发有效的量子科学课程，并提高向高中生教授这门学科的信心。本课题研究了基于一般随机状态概念的纠缠估计理论。一般态是根据一定的分布随机产生的量子态。通用随机态的使用在现代量子科学中变得越来越重要。随机态的集成是我们理解量子电路复杂性的基础，也是纠缠估计理论发展的基础。随机态在量子器件基准测试和量子优势测试中也有应用。推力1的重点是基于熵的估计和基于度量的估计。对于基于熵的估计，将得到广义熵降为包括Renyi熵、von Neumann熵和量子纯度在内的标准熵的精确统计性能。对于基于度量的估计，将研究量子计算中保真度和体积的关键纠缠度量的非渐近行为。在Thrust 2中，将使用主要的通用状态模型来揭示纠缠估计器和算法性能之间的深层联系。重点是量子电路切割算法在状态层析和量子模拟算法在量子优化算法的背景下。该项目的一个组成部分将是使用IBM Quantum模拟器评估和验证一些项目发现。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。

Quantum entanglement is the physical resource that enables quantum computing and other quantum technologies. Although the problem of estimating the degree of entanglement of quantum states has a relatively long history, this issue and its consequences are far from being fully understood. In particular, a quantitative understanding of how the amount and type of available entanglement relates to the performance of quantum algorithms remains largely elusive despite the well-known fact that entanglement empowers the computational speedups over classical algorithms. This project aims to address this challenge by developing a new framework for the analysis and design of quantum algorithms by connecting algorithm performance to the degree of entanglement of quantum states. In the literature, quantum circuits have been the most pervasive model to understand the performance of quantum algorithms. The circuitry models are, however, limited to the study of computational complexity of quantum algorithms, which do not directly exploit nor quantify entanglement as a fundamental resource for algorithm analysis. The framework presented in this project will complement the existing one leading to new perspectives in quantum algorithm development. During the course of the project a new course on quantum algorithms will be developed, adding to a long-term goal of a Graduate Certificate Program on quantum computing, traditional computer science courses will be updated with either prerequisite materials or introductions on quantum information science. To increase awareness of quantum science and its diverse career opportunities, the research team will continue to host outreach workshops for high school STEM educators to develop effective quantum science curricular and to improve confidence in teaching the subject to high school students.This project studies the theory of entanglement estimation, which is based on the concept of generic random states. Generic states are quantum states generated at random according to certain distributions. The use of generic random states has become increasingly important in modern quantum science. Ensembles of random states underlie our understanding of complexity of quantum circuits as well as the development of entanglement estimation theory. Random states also find applications in benchmarking quantum devices and testing quantum advantage. The focus of Thrust 1 is on entropy-based estimation and metric-based estimation. For entropy-based estimation, the exact statistical performance of the generalized entropy that reduces to the standard entropies including Renyi entropy, von Neumann entropy, and quantum purity will be obtained. For metric-based estimation, the non-asymptotic behavior of key entanglement metrics of fidelity and volumes in quantum computing will be investigated. In Thrust 2, major generic state models will be utilized to uncover the deep connection between entanglement estimators and algorithm performance. The focus is on quantum circuit cutting algorithms in the context of state tomography and quantum simulation algorithms in the context of quantum optimization algorithms. An integral part of the project will be the evaluation and verification of some of the project findings using IBM Quantum Simulators.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.