  We consider the interchange process (IP) on the $d$-dimensional, discrete
hypercube of side-length $n$. Specifically, we compare the spectral gap of the
IP to the spectral gap of the random walk (RW) on the same graph. We prove that
the two spectral gaps are asymptotically equivalent, in the limit $n \to
\infty$. This result gives further supporting evidence for a conjecture of
Aldous, that the spectral gap of the IP equals the spectral gap of the RW on
all finite graphs. Our proof is based on an argument invented by Handjani and
Jungreis, who proved Aldous's conjecture for all trees. This also has
implications for the spectral gap of the quantum Heisenberg ferromagnet.
