  Topological numbers can characterize the transition between different
topological phases, which are not described by Landau's paradigm of symmetry
breaking. Since the discovery of quantum Hall effect, more topological phases
have been theoretically predicted and experimentally verified. However, it is
still an experimental challenge to directly measure the topological number of
various predicted topological phases. In this paper, we demonstrate quantum
simulation of topological phase transition of a quantum wire (QW) using a
single nitrogen-vacancy (NV) center in diamond. Deploying quantum algorithm of
finding eigenvalues, we can reliably extract both the dispersion relations and
topological numbers.
