  The graph isomorphism problem is theoretically interesting and also has many
practical applications. The best known classical algorithms for graph
isomorphism all run in time super-polynomial in the size of the graph in the
worst case. An interesting open problem is whether quantum computers can solve
the graph isomorphism problem in polynomial time. In this paper, an algorithm
is shown which can decide if two rooted trees are isomorphic in polynomial
time. Although this problem is easy to solve efficiently on a classical
computer, the techniques developed may be useful as a basis for quantum
algorithms for deciding isomorphism of more interesting types of graphs. The
related problem of quantum state symmetrization is also studied. A polynomial
time algorithm for the problem of symmetrizing a set of orthonormal states over
an arbitrary permutation group is shown.
