  We give a randomized algorithm that approximates the number of independent
sets in a dense, regular bipartite graph -- in the language of approximate
counting, we give an FPRAS for #BIS on the class of dense, regular bipartite
graphs. Efficient counting algorithms typically apply to ``high-temperature''
problems on bounded-degree graphs, and our contribution is a notable exception
as it applies to dense graphs in a low-temperature setting. Our methods give a
counting-focused complement to the long line of work in combinatorial
optimization showing that CSPs such as Max-Cut and Unique Games are easy on
dense graphs via spectral arguments. The proof exploits the fact that dense,
regular graphs exhibit a kind of small-set expansion (i.e. bounded threshold
rank), which via subspace enumeration lets us enumerate small cuts efficiently.
