  We develop a numerical method based on matrix product states for simulating
quantum many-body systems at finite temperatures without importance sampling
and evaluate its performance in spin 1/2 systems. Our method is an extension of
the random phase product state (RPPS) approach introduced recently [T. Iitaka,
arXiv:2006.14459]. We show that the original RPPS approach often gives
unphysical physical values for thermodynamic quantities even in the Heisenberg
chain. We find that by adding the operation of Trotter gates to the RPPS, the
sampling efficiency of the approach significantly increases and its results are
consistent with those of the purification approach. We also apply our method to
a frustrated spin 1/2 system to exemplify that it can simulate a system in
which the purification approach fails.
