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。虽然量子态纠缠度的估计问题有着相对较长的历史，但这个问题及其后果还远未被完全理解。特别是，尽管众所周知的事实是纠缠使经典算法的计算加速，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。该项目旨在通过将算法性能与量子态的纠缠程度联系起来，开发一种用于分析和设计量子算法的新框架来应对这一挑战。在文献中，量子电路一直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，不直接利用或量化纠缠作为算法分析的基本资源。该项目中提出的框架将补充现有的框架，从而为量子算法开发带来新的视角。在该项目的过程中，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的认识和多样化的就业机会，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，并提高对高中学生教授该学科的信心。本项目研究基于通用随机态概念的纠缠估计理论。类属态是按照一定的分布随机产生的量子态。一般随机态的使用在现代量子科学中变得越来越重要。随机态系综是理解量子电路复杂性和发展纠缠估计理论的基础。随机态也可以应用于量子器件的基准测试和量子优势的测试。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，将获得简化为包括Renyi熵、von Neumann熵和量子纯度的标准熵的广义熵的精确统计性能。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。在推力2中，主要的通用状态模型将被用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的背景下，状态层析成像和量子仿真算法的背景下，量子优化算法。该项目的一个组成部分将是使用IBM量子模拟器对一些项目结果进行评估和验证。该奖项反映了NSF的法定使命，并被认为值得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来支持。

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.