FET: Small: Entanglement Estimation for Quantum Computing

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.