  We present a simple, natural #P-complete problem. Let G be a directed graph,
and let k be a positive integer. We define q(G;k) as follows. At each vertex v,
we place a k-dimensional complex vector x_v. We take the product, over all
edges (u,v), of the inner product <x_u,x_v>. Finally, q(G;k) is the expectation
of this product, where the x_v are chosen uniformly and independently from all
vectors of norm 1 (or, alternately, from the Gaussian distribution). We show
that q(G;k) is proportional to G's cycle partition polynomial, and therefore
that it is #P-complete for any k>1.
