On the Existence of 0/1 Polytopes with High Semidefinite Extension Complexity

In Rothvoÿ [2011] it was shown that there exists a 0/1 polytope (a polytope whose vertices are in {0, 1}) such that any higherdimensional polytope projecting to it must have 2 facets, i.e., its linear extension complexity is exponential. The question whether there exists a 0/1 polytope with high PSD extension complexity was left open. We answer this question in the a rmative by showing that there is a 0/1 polytope such that any spectrahedron projecting to it must be the intersection of a semide nite cone of dimension 2 and an a ne space. Our proof relies on a new technique to rescale semide nite factorizations.