  It has been known for some time that graph isomorphism reduces to the hidden
subgroup problem (HSP). What is more, most exponential speedups in quantum
computation are obtained by solving instances of the HSP. A common feature of
the resulting algorithms is the use of quantum coset states, which encode the
hidden subgroup. An open question has been how hard it is to use these states
to solve graph isomorphism. It was recently shown by Moore, Russell, and
Schulman that only an exponentially small amount of information is available
from one, or a pair of coset states. A potential source of power to exploit are
entangled quantum measurements that act jointly on many states at once. We show
that entangled quantum measurements on at least \Omega(n log n) coset states
are necessary to get useful information for the case of graph isomorphism,
matching an information theoretic upper bound. This may be viewed as a negative
result because highly entangled measurements seem hard to implement in general.
Our main theorem is very general and also rules out using joint measurements on
few coset states for some other groups, such as GL(n, F_{p^m}) and G^n where G
is finite and satisfies a suitable property.
