  We perform a detailed comparison of two Matrix Product States (MPS) based
time evolution algorithms for Anderson Impurity Models. To describe the bath,
we use both the star-geometry as well as the commonly employed Wilson chain
geometry. For each bath geometry, we use either the Time Dependent Variational
Principle (TDVP) or the Time Evolving Block Decimation (TEBD) to perform the
time evolution. To apply TEBD for the star-geometry, we use a specially adapted
algorithm that can deal with the long-range coupling terms. Analyzing the major
sources of errors, one expects them to be proportional to the system size for
all algorithms. Surprisingly, we find errors independent of system size except
for TEBD in chain geometry. Additionally, we show that the right combination of
bath representation and time evolution algorithm is important. While TDVP in
chain geometry is a very precise approach, TEBD in star geometry is much
faster, such that for a given accuracy it is superior to TDVP in chain
geometry. This makes the adapted version of TEBD in star geometry the most
efficient method to solve impurity problems.
