Theory of Computing ------------------- Title : Tight Hardness of the Non-Commutative Grothendieck Problem Authors : Jop Briet, Oded Regev, and Rishi Saket Volume : 13 Number : 15 Pages : 1-24 URL : https://theoryofcomputing.org/articles/v013a015 Abstract -------- We prove that for any $\eps > 0$ it is NP-hard to approximate the non- commutative Grothendieck problem to within a factor $1/2 + \eps$, which matches the approximation ratio of the algorithm of Naor, Regev, and Vidick (STOC'13). Our proof uses an embedding of $\ell_2$ into the space of matrices endowed with the trace norm with the property that the image of standard basis vectors is longer than that of unit vectors with no large coordinates. We also observe that one can obtain a tight NP-hardness result for the _commutative_ Little Grothendieck problem; previously, this was only known based on the Unique Games Conjecture (Khot and Naor, Mathematika 2009). A conference version of this paper appeared in the Proceedings of the 56th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2015).