  We investigate non-Abelian statistics of Majorana Kramers pairs (MKPs) in a
network system of one-dimensional time-reversal invariant topological
superconductors by using numerical simulations of braiding dynamics, and
examine the tolerance against various perturbations which may cause decoherence
of MKPs. We, first, consider effects of a magnetic field which breaks
time-reversal symmetry. In contrast to a naive expectation, the non-Abelian
braiding of MKPs is robust against applied magnetic fields provided that the
initial and final states of a braiding process are invariant under the
combination of a time reversal and a mirror reflection, even when intermediate
states break the combined symmetry. Secondly, we investigate the stability of
non-Abelian braidings in the case with gate-induced inhomogeneous potentials at
junctions between superconducting nanowires, which generally generate a
non-Majorana nearly zero-energy Andreeev bound state at the junctions. It is
found that the non-Majorana nearly zero-energy states interfere with MKPs,
resulting in the failure of non-Abelian braidings, when the length scale of the
inhomogeneous potentials is comparable to the coherence length of the
superconductors.
