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Explore the mathematical foundations of simulating many-body quantum systems with Coulomb interactions in this 51-minute conference talk. Delve into the fundamental question of whether simulation costs can scale polynomially with system size when dealing with realistic interactions, particularly focusing on Coulomb potentials that are central to electronic and molecular dynamics. Learn about groundbreaking research proving that first-order Trotterization for unbounded Hamiltonians achieves polynomial dependence on particle number in the continuum limit, with a convergence rate of order 1/4 - a significant departure from prior analyses of bounded operators that diverge in this limit. Discover why this 1/4-order rate represents an optimal bound, as demonstrated by its saturation in the hydrogen atom ground state, and understand why higher-order Trotter formulas fail to improve worst-case scaling. Examine the additional regularity conditions on initial states that can restore original Trotter convergence rates, while gaining insight into the analytical challenges posed by many-body structures and the singular nature of Coulomb potentials in quantum simulation.
