  Graph states, which include for example Bell states, GHZ states and cluster
states, form a well-known class of quantum states with applications ranging
from quantum networks to error-correction. Deciding whether two graph states
are equivalent up to single-qubit Clifford operations is known to be decidable
in polynomial time and have been studied both in the context of producing
certain required states in a quantum network but also in relation to stabilizer
codes. The reason for the latter this is that single-qubit Clifford equivalent
graph states exactly corresponds to equivalent stabilizer codes. We here
consider the computational complexity of, given a graph state |G>, counting the
number of graph states, single-qubit Clifford equivalent to |G>. We show that
this problem is #P-Complete. To prove our main result we make use of the notion
of isotropic systems in graph theory. We review the definition of isotropic
systems and point out their strong relation to graph states. We believe that
these isotropic systems can be useful beyond the results presented in this
paper.
