  Direct simulation of the von Neumann dynamics for a general (pure or mixed)
quantum state can often be expensive. One prominent example is the real-time
time-dependent density functional theory (rt-TDDFT), a widely used framework
for the first principle description of many-electron dynamics in chemical and
materials systems. Practical rt-TDDFT calculations often avoid the direct
simulation of the von Neumann equation, and solve instead a set of
Schr\"odinger equations, of which the dynamics is equivalent to that of the von
Neumann equation. However, the time step size employed by the Schr\"odinger
dynamics is often much smaller. In order to improve the time step size and the
overall efficiency of the simulation, we generalize a recent work of the
parallel transport (PT) dynamics for simulating pure states [An, Lin,
Multiscale Model. Simul. 18, 612, 2020] to general quantum states. The PT
dynamics provides the optimal gauge choice, and can employ a time step size
comparable to that of the von Neumann dynamics. Going beyond the linear and
near adiabatic regime in previous studies, we find that the error of the PT
dynamics can be bounded by certain commutators between Hamiltonians, density
matrices, and their derived quantities. Such a commutator structure is not
present in the Schr\"odinger dynamics. We demonstrate that the parallel
transport-implicit midpoint (PT-IM) method is a suitable method for simulating
the PT dynamics, especially when the spectral radius of the Hamiltonian is
large. The commutator structure of the error bound, and numerical results for
model rt-TDDFT calculations in both linear and nonlinear regimes, confirm the
advantage of the PT dynamics.
