$\newcommand{\eps}{\varepsilon} $We prove that for any $\eps > 0$ it is $\textsf{NP}$-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 $\textsf{NP}$-hardness result for the commutative Little Grothendieck problem; previously, this was only known based on the Unique Games Conjecture (Khot and Naor, Mathematika 2009).
翻译：$\ n newcommants69-pts- varepsilon} $We 证明,对于任何 $ps > 0$ 来说, 将非混合的Grothendieck 问题接近于 $1.2 +\ eps$, 与 Naor、 Regev 和 Vidick 算法(STOC'13) 的近似比率相匹配的因数中, 很难将非混合的 Grothendieck 问题大致地推到 $\ extsf{NP} $- 美元。 我们的证明使用将 $\ ell_ 2 美元嵌入带有跟踪规范的矩阵空间, 其属性是标准基矢量的图像比无大坐标单位矢量的单位矢量长。 我们还观察到, 对于小Grothendiecteck 问题, 人们可以获得一个紧凑合的 $\ textsf{NPNP} $- hardness $- $ $- dislate poiltypection production on the the Uniquen the the Uniquen the the Uniquen the the Uniquencienter Conjecture (Kt and Natort and Naor, Mathemat, Mathemattime.
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