Efficient simulation of many-body quantum dynamics is central to advances in physics, chemistry, and quantum computing. A fundamental question is whether the simulation cost can scale polynomially with system size in the presence of realistic interactions. In this talk, we focus on many-body quantum systems with Coulomb interactions, which play a central role in electronic and molecular dynamics. We prove that first-order Trotterization for such unbounded Hamiltonians admits a polynomial dependence on the number of particles in the continuum limit, with a convergence rate of order $1/4$—in contrast to prior Trotter analyses for bounded operators, which diverge in this limit. The result holds for all initial wavefunctions in the domain of the Hamiltonian. This $1/4$-order rate is optimal, as previous work shows that it can be saturated by the ground state of the hydrogen atom. Moreover, higher-order Trotter formulas do not improve the worst-case scaling. We also discuss additional regularity conditions on the initial state under which the original Trotter convergence rate can be recovered. The main analytical challenges arise from the many-body structure and the singular nature of the Coulomb potential.
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