  In this work we develop a quantum algorithm to simulate the dynamics of the density matrix governed by the von Neumann equation for time-dependent Hamiltoinans. The method relies on the vectorization of the density matrix through the properties of the structure constants of a given Lie algebra. Even though we have used the algebra formed by the Pauli strings, the algorithm can be easily adapted to other algebras. One of the main advantages of this approach is that it yields real density matrix coefficients that are easy to determine through phase kickback. The algorithm is demonstrated using the IBM noisy quantum circuit simulator. 
Quantum simulation of the von Neumann equation of time-dependent   Hamiltonians