Abstract
Quantum computers are expected to dramatically outperform classical computers for certain computational problems. There has been extensive previous work for linear dynamics and discrete models, including Hamiltonian simulations and systems of linear equations. However, for more complex realistic problems, one fundamental challenge is the substantial difference between the linear dynamics of a system of qubits and real-world systems with continuum, nonlinear, and stochastic behaviors. I focus mainly on the design and analysis of quantum algorithms for scientific computational problems, including topics such as linear and nonlinear differential equations, quantum dynamics, and stochastic processes, with applications in areas such as biology and epidemiology, fluid dynamics, quantum chemistry, finance, and machine learning.