Time evolution of controlled many-body quantum systems with matrix product operators

We present a method for describing the time evolution of many-body controlled quantum systems using matrix product operators (MPOs). Existing techniques for solving the time-dependent Schrödinger equation with an MPO Hamiltonian often rely on time discretization. In contrast, our approach uses the Magnus expansion and Chebyshev polynomials to model the time evolution and the MPO representation to efficiently encode the system's dynamics. This results in a scalable method that can be used efficiently for many-body controlled quantum systems. We apply this technique to quantum optimal control, specifically for a gate synthesis problem, demonstrating that it can be used for large-scale optimization problems that are otherwise impractical to formulate in a dense matrix representation.