  In the 1960s, statistical physicists discovered a fascinating algorithm for
counting perfect matchings in planar graphs. Valiant later showed that the same
problem is #P-hard for general graphs. Since then, the algorithm for planar
graphs was extended to bounded-genus graphs, to graphs excluding $K_{3,3}$ or
$K_{5}$, and more generally, to any graph class excluding a fixed minor $H$
that can be drawn in the plane with a single crossing. This stirred up hopes
that counting perfect matchings might be polynomial-time solvable for graph
classes excluding any fixed minor $H$. Alas, in this paper, we show #P-hardness
for $K_{8}$-minor-free graphs by a simple and self-contained argument.
