Real-Time Iteration Scheme for Dynamical Mean-Field Theory: A Framework for Near-Term Quantum Simulation

We present a time-domain iteration scheme for solving the Dynamical Mean-Field Theory (DMFT) self-consistent equations using retarded Green's functions in real time. Unlike conventional DMFT approaches that operate in imaginary time or frequency space, our scheme operates directly with real-time quantities. This makes it particularly suitable for near-term quantum computing hardware with limited Hilbert spaces, where real-time propagation can be efficiently implemented via Trotterization or variational quantum algorithms. We map the effective impurity problem to a finite one-dimensional chain with a small number of bath sites, solved via exact diagonalization as a proof-of-concept. The hybridization function is iteratively updated through time-domain fitting until self-consistency. We demonstrate stable convergence across a wide range of interaction strengths for the half-filled Hubbard model on a Bethe lattice, successfully capturing the metal-to-insulator transition. Despite using limited time resolution and a minimal bath discretization, the spectral functions clearly exhibit the emergence of Hubbard bands and the suppression of spectral weight at the Fermi level as interaction strength increases. This overcomes major limitations of two-site DMFT approximations by delivering detailed spectral features while preserving efficiency and compatibility with quantum computing platforms through real-time dynamics.