  The quantum hidden subgroup approach is an actively studied approach to solve
combinatorial problems in quantum complexity theory. With the success of the
Shor's algorithm, it was hoped that similar approach may be useful to solve the
other combinatorial problems. One such problem is the graph isomorphism problem
which has survived decades of efforts using the hidden subgroup approach. This
paper provides a systematic approach to create arbitrarily large classes of
classically efficiently solvable graph automorphism problems or easy graph
automorphism problems for which hidden subgroup approach is guaranteed to
always fail irrespective of the size of the graphs no matter how many copies of
coset states are used. As the graph isomorphism problem is believed to be at
least as hard as the graph automorphism problem, the result of this paper
entails that the hidden subgroup approach is also guaranteed to always fail for
the arbitrarily large classes of graph isomorphism problems. Combining these
two results, it is argued that the hidden subgroup approach is essentially a
dead end and alternative quantum algorithmic approach needs to be investigated
for the graph isomorphism and automorphism problems.
