We prove a vector-valued inequality for the Gaussian noise stability (i.e. we prove a vector-valued Borell inequality) for Euclidean functions taking values in the two-dimensional sphere, for all correlation parameters at most $1/10$ in absolute value. This inequality was conjectured (for all correlation parameters at most $1$ in absolute value) by Hwang, Neeman, Parekh, Thompson and Wright. Such an inequality is needed to prove sharp computational hardness of the product state Quantum MAX-CUT problem, assuming the Unique Games Conjecture. In fact, assuming the Unique Games Conjecture, we show that the product state of Quantum MAX-CUT is NP-hard to approximate within a multiplicative factor of $.9859$. In contrast, a polynomial time algorithm is known with approximation factor $.956\ldots$.

翻译：我们证明了取值于二维球面的欧几里得函数在高斯噪声稳定性下的向量值不等式（即向量值Borell不等式），该不等式对所有绝对值不超过$1/10$的相关参数成立。该不等式由Hwang、Neeman、Parekh、Thompson和Wright提出猜想（对所有绝对值不超过$1$的相关参数成立）。此类不等式是证明乘积态量子MAX-CUT问题在唯一博弈猜想假设下具有尖锐计算复杂度的必要条件。事实上，在唯一博弈猜想成立的前提下，我们证明乘积态量子MAX-CUT问题在乘法近似因子$.9859$内为NP难。相比之下，已知存在多项式时间算法能达到$.956\ldots$的近似因子。