Tight hardness of the non-commutative Grothendieck problem

Abstract

We prove that for any $\\varepsilon > 0$ it is NP-hard to approximate the \nnon-commutative Grothendieck problem to within a factor $1/2 + \\varepsilon$, \nwhich matches the approximation ratio of the algorithm of Naor, Regev, and \nVidick (STOC\'13). Our proof uses an embedding of $\\ell_2$ into the space of \nmatrices endowed with the trace norm with the property that the image of \nstandard basis vectors is longer than that of unit vectors with no large \ncoordinates