Optimizing Sphere Valued Gaussian Noise Stability

. 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.

我们证明了一个关于高斯噪声稳定性的向量值不等式（即我们证明了一个向量值的博雷尔不等式），针对取值于二维球面的欧几里得函数，对于所有绝对值至多为\(1/10\)的相关参数。这个不等式是由黄（Hwang）、尼曼（Neeman）、帕雷克（Parekh）、汤普森（Thompson）和赖特（Wright）所猜想的（针对所有绝对值至多为\(1\)的相关参数）。在假定唯一博弈猜想成立的情况下，要证明积态量子最大割问题的精确计算难度，就需要这样一个不等式。