Unique Games hardness of Quantum Max-Cut, and a vector-valued Borell's inequality
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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 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 expected value of ⟨ f(x), f(y)⟩ over ρ-correlated Gaussians x and y is minimized by the function f(x) = x_≤ k / ‖ x_≤ k‖, where x_≤ k = (x_1, …, x_k). As an application, we show several hardness of approximation results for Quantum Max-Cut, a special case of the local Hamiltonian problem related to the anti-ferromagnetic Heisenberg model. Quantum Max-Cut is a natural quantum analogue of classical Max-Cut and has become testbed for designing quantum approximation algorithms. We show the following: (1) The integrality gap of the basic SDP is 0.498, matching an existing rounding algorithm. Combined with existing approximation results for Quantum Max-Cut, this shows that the basic SDP does not achieve the optimal approximation ratio. (2) It is Unique Games-hard (UG-hard) to compute a (0.956+ε)-approximation to the value of the best product state, matching an existing approximation algorithm. This result may be viewed as applying to a generalization of Max-Cut where one seeks to assign 3-dimensional unit vectors to each vertex; we also give tight hardness results for the analogous k-dimensional generalization of Max-Cut. (3) It is UG-hard to compute a (0.956+ε)-approximation to the value of the best (possibly entangled) state.
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