Hi, good question, but the answer is kind of complicated and I'm not sure I know all of the details of how those two methods compare.

At a very high level, I would guess that TDVP is at this point a much better method than the MPO method overall. TDVP is very accurate as long as it does not run into "basis expansion" issues, while the MPO method has been found to be not very accurate.

If your question is about performance in terms of computation time or speed, then I think TDVP is also superior here, because each method takes about the same amount of time per time step, but with the MPO method you have to take small time steps, while in TDVP you can take very large time steps, as large as t=1.0 even! The actual discussion there is a bit complicated, because TDVP does have an higher cost for longer time steps, but still I think if you add up all the costs to reach some target time, TDVP is likely faster. Of course in some applications you are ok with taking small time steps because you want the information about the state at each small interval of time, so then time step size is less of a problem.

The only drawback to TDVP, really, is the basis expansion issue I mentioned earlier. This issue is discussed and a method to fix it proposed here:

https://journals.aps.org/prb/abstract/10.1103/PhysRevB.102.094315