CAREER: Quantum Simulation with Unbounded Operators: Algorithms, Analysis, and Applications

Simulation of quantum dynamics, also known as Hamiltonian simulation, served as the original motivation for quantum computers and remains a core task in quantum computing today. It has wide-ranging potential in fields such as quantum physics, quantum chemistry, and biological molecular dynamics. This project aims to advance quantum simulation by addressing challenges posed by so-called unbounded operators, which frequently arise in scientific and engineering context due to the discretization of differential operators. By tackling these issues, the project seeks to enhance computational techniques, deepen theoretical understanding, and broaden the practical impact of quantum computing while strengthening its connections with mathematics. Graduate and undergraduate students involved will gain valuable interdisciplinary training at the intersection of mathematics and quantum information science. This project focuses on developing innovative quantum algorithms and advancing mathematical frameworks for quantum simulation with unbounded Hamiltonians. Key contributions include the development of quantum algorithms based on techniques such as Trotterization, randomization, Linear Combination of Unitaries, and Magnus expansion. These methods aim to overcome computational challenges and significantly accelerate the application of quantum simulation across various fields. Rigorous numerical analysis will be integrated to estimate the quantum complexity and provide performance guarantees for these algorithms. Special attention will be given to understanding and constructing superconvergence, particularly in relation to spatial discretization, leveraging mathematical tools from continuous and discrete microlocal analysis. By applying these advancements to real-world applications, this project will contribute to scalable quantum computing solutions and establish new benchmarks for algorithmic efficiency and accuracy. 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.