  The first quantum algorithm to offer an exponential speedup (in the query
complexity setting) over classical algorithms was Simon's algorithm for
identifying a hidden exclusive-or mask. Here we observe how part of Simon's
algorithm can be interpreted as a Clebsch-Gordan transform. Inspired by this we
show how Clebsch-Gordan transforms can be used to efficiently find a hidden
involution on the group G^n where G is the dihedral group of order eight (the
group of symmetries of a square.) This problem previously admitted an efficient
quantum algorithm but a connection to Clebsch-Gordan transforms had not been
made. Our results provide further evidence for the usefulness of Clebsch-Gordan
transform in quantum algorithm design.
