Sphere Valued Noise Stability and Quantum MAX-CUT Hardness
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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….
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