  Quantum many-body systems divide into a variety of phases with very different
physical properties. The question of what kind of phases exist and how to
identify them seems hard especially for strongly interacting systems. Here we
make an attempt to answer this question for gapped interacting quantum spin
systems whose ground states are short-range correlated. Based on the local
unitary equivalence relation between short-range correlated states in the same
phase, we classify possible quantum phases for 1D matrix product states, which
represent well the class of 1D gapped ground states. We find that in the
absence of any symmetry all states are equivalent to trivial product states,
which means that there is no topological order in 1D. However, if certain
symmetry is required, many phases exist with different symmetry protected
topological orders. The symmetric local unitary equivalence relation also
allows us to obtain some simple results for quantum phases in higher dimensions
when some symmetries are present.
