Hamiltonian Simulation: From Quantum Spin Systems to Nonlinear Dispersive Equations
Abstract: Precision and efficiency in Hamiltonian system simulations are essential for the advancement of quantum technologies and numerical analysis. This talk presents two recent developments in algorithms for quantum spin systems and broader Hamiltonian systems. First, we introduce a fourth-order Magnus-based algorithm for simulating many-body systems under highly oscillatory, time-dependent pulses. These integrators achieve high accuracy despite taking large time-steps, enabling faster computations on classical computers and reduced circuit depths on quantum devices, making them suitable for near-term quantum applications. Additionally, we discuss a novel iterative linearization framework for nonlinear dispersive equations, preserving key structural properties, including the L2 norm, momentum, and Hamiltonian energy. Both techniques leverage the Magnus expansion as a foundational component.