  In this paper, we investigate the validity of the Unique Games Conjecture
when the constraint graph is the boolean hypercube. We construct an almost
optimal integrality gap instance on the Hypercube for the Goemans-Williamson
semidefinite program (SDP) for Max-2-LIN$(\mathbb{Z}_2)$. We conjecture that
adding triangle inequalities to the SDP provides a polynomial time algorithm to
solve Unique Games on the hypercube.
