Revised: January 20, 2017

Published: December 2, 2017

**Keywords:**hardness of approximation, semidefinite programming, Grothendieck inequality

**Categories:**complexity theory, approximation algorithms, inapproximability, semidefinite programming, Grothendieck inequality

**ACM Classification:**G.1.6,

**AMS Classification:**68Q17, 15A60, 32A70, 03D15

**Abstract:**
[Plain Text Version]

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).