On the existence of 0/1 polytopes with high semidefinite extension complexity

In Rothvoß (Math Program 142(1–2):255–268, 2013) it was shown that there exists a 0/1 polytope (a polytope whose vertices are in $$\{0,1\}^{n}$${0,1}n) such that any higher-dimensional polytope projecting to it must have $$2^{\varOmega (n)}$$2Ω(n) facets, i.e., its linear extension complexity is exponential. The question whether there exists a 0/1 polytope with high positive semidefinite extension complexity was left open. We answer this question in the affirmative by showing that there is a 0/1 polytope such that any spectrahedron projecting to it must be the intersection of a semidefinite cone of dimension $$2^{\varOmega (n)}$$2Ω(n) and an affine space. Our proof relies on a new technique to rescale semidefinite factorizations.