CAREER: Quantum-Inspired Relaxation and Embedding Frameworks for Scalable Scientific Computing in High Dimensions

Many scientific and engineering problems are difficult because the number of possible configurations grows exponentially with system size. This challenge arises in quantum materials, molecular systems, statistical mechanics, imaging, and global optimization. Direct computation is often impossible, while purely local or black-box methods can miss the structure that makes the problem scientifically tractable. This project develops scalable mathematical algorithms for such high-dimensional problems by drawing on ideas from quantum physics, optimization, probability, and numerical linear algebra. The goal is to compute useful approximations, bounds, and reduced descriptions without discarding the interactions that define the original system. The work advances national priorities in quantum science and artificial intelligence by creating computational foundations for quantum simulation, scientific inference, and reliable high-dimensional computation. Its longer-term benefits include better tools for modeling materials, molecules, many-particle systems, and imaging data, with relevance to health, energy, and national technological capacity. The project also supports education and workforce development through undergraduate research, graduate mentorship, new course materials, open-source software, public lectures, and community-building activities in high-dimensional scientific computing. The project develops quantum-inspired relaxation and embedding frameworks for scalable scientific computing in high dimensions. One research direction focuses on entropy-regularized semidefinite programming relaxations for many-particle optimization and inference problems. The investigator develops fast solvers based on convex duality, mirror ascent, optimal transport subproblems, randomized numerical linear algebra, and matrix-free implementations, with the aim of obtaining rigorous convergence guarantees and near-optimal computational scaling. Target applications include molecular geometry optimization, group synchronization, cryo-electron microscopy imaging, and learning effective potentials from data. A second research direction develops embedding methods that reduce large quantum and classical systems to coupled lower-dimensional computations while preserving variational or self-consistency structure. These methods include variational impurity-bath embedding for fermionic systems, tensor slice belief propagation for two-dimensional classical and quantum models, and generalized Schmidt embedding for quantum systems and classical Boltzmann distributions. Together, these tools connect semidefinite relaxations, graphical models, tensor networks, randomized algorithms, and variational principles, with the goal of producing scalable and systematically improvable methods for quantum many-body physics, statistical mechanics, optimization, and scientific machine learning. 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. NSF Award ID: 2543506 | Program: 01003031DB NSF RESEARCH & RELATED ACTIVIT,01002930DB NSF RESEARCH & RELATED ACTIVIT,01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Michael Lindsey | Institution: University of California-Berkeley, BERKELEY, CA | Award Amount: $296,354 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2543506 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2543506.html