Unique Games hardness of Quantum Max-Cut, and a vector-valued Borell's inequality

Abstract

The Gaussian noise stability of a function f : R n → {−1, 1} is the expected value of f (x) • f (y) over ρ-correlated Gaussian random variables x and y. Borell's inequality states that for −1 ≤ ρ ≤ 0, this is minimized by the mean-zero halfspace f (x) = sign(x 1 ). In this work, we generalize this result to hold for functions f : R n → S k−1 which output k-dimensional unit vectors. Our main result shows that the expectation E x∼ρy f (x), f (y) is minimized by the function f (x) = x ≤k / x ≤k , where x ≤k = (x 1 , . . . , x k ). Our techniques extend the recent calculus of variations approach for proving Borell's inequality due to Heilman and Tarter [HT20] to output dimensions k larger than 1.As an application of this result, we show several hardness of approximation results for a special case of the local Hamiltonian problem related to the anti-ferromagnetic Heisenberg model known as Quantum Max-Cut. This can be viewed as a natural quantum analogue of the classical Max-Cut problem and has been proposed as a useful testbed for developing algorithms. We show the following:1. There exists an integrality gap of 0.498 for the basic SDP, matching the rounding algorithm of Gharibian and Parekh [GP19]. Combined with the work of Anshu, Gosset, and Morenz [AGM20], this shows that the basic SDP does not achieve the optimal approximation ratio.