Finite Elements in the Quantum Era

About This Grant

Quantum scientific computing, a rapidly growing interdisciplinary field, integrates classical numerical methods with quantum technologies to tackle challenges across physics, chemistry, biology, engineering, industrial applications, and beyond, and has the potential to vastly outperform classical computers in solving such problems. In this project, the overall goal is to develop, analyze, and implement quantum and quantum-inspired numerical algorithms that leverage both well-studied classical methods and state-of-the-art quantum techniques to overcome bottlenecks in numerical simulations of complex problems, such as "the curse of dimensionality," which restricts the development of efficient methods and solvability for problems in high-dimensional space. The quantum methods developed in this work have the potential to translate into advancements in many practical applications, e.g., machine learning, data science, optimization, and electric grid simulations. All techniques will be implemented in an open-source software package and will be made available to the scientific community. The main focus of this project is to develop, analyze, and implement quantum finite-element methods (FEMs) for solving partial differential equations (PDEs). While the discretization steps of FEMs can be implemented optimally on classical computers, solving the resulting large-scale and often ill-conditioned linear systems remains the most computationally intensive step. To overcome this challenge, the project explores the use of quantum algorithms and investigates the conditions under which quantum methods offer polynomial versus exponential speedups, clarifying optimal cases for a quantum advantage. This work has two main research objectives: (1) To develop efficient and provable quantum-inspired classical FEMs for solving high-dimensional PDEs; and (2) To develop practical quantum FEMs for solving PDEs on both near-term and far-term quantum computers with theoretical guarantees. In the first objective, quantum principles are introduced into classical algorithms, creating quantum-inspired classical algorithms that achieve comparable speedups for FEM applications on classical computers. In the second objective, hybrid quantum-classical approaches are first developed for near-term quantum computers, which exist today. The long-term goal is then to extend these methods to fully quantum FEM implementations on far-term large-scale quantum computers, which have yet to be developed. This involves incorporating quantum techniques into adaptivity and mesh refinement, as well as developing structure-preserving quantum FEMs. The goal is to produce new fundamental theory and advanced numerical methods for solving PDEs in the quantum era. 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.