Krylov subspace method for quantum dynamics

For a given system, the structure of the minimal subspace where the state unfolds determines the static and dynamical properties of the state. The Krylov subspace method is a mathematical framework for constructing the space systematically and has been applied to a wide variety of problems. The method was applicable only for systems with time-indepedent generators. As applications to quantum dynamics with time-dependent Hamiltonians, we discuss the constrution of the adiabatic gauge potential and the generalization of the Krylov algorithm to time-dependent generators.