We show that, if $\mathbf{W} \in \mathbb{R}^{N \times N}_{\mathsf{sym}}$ is drawn from the gaussian orthogonal ensemble, then with high probability the degree 4 sum-of-squares relaxation cannot certify an upper bound on the objective $N^{-1} \cdot \mathbf{x}^\top \mathbf{W} \mathbf{x}$ under the constraints $x_i^2 - 1 = 0$ (i.e. $\mathbf{x} \in \{ \pm 1 \}^N$) that is asymptotically smaller than $\lambda_{\max}(\mathbf{W}) \approx 2$. We also conjecture a proof technique for lower bounds against sum-of-squares relaxations of any degree held constant as $N \to \infty$, by proposing an approximate pseudomoment construction.

中文翻译：

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Sherrington-Kirkpatrick Hamiltonian 的紧度 4 平方和下界

我们证明，如果 $\mathbf{W} \in \mathbb{R}^{N \times N}_{\mathsf{sym}}$ 是从高斯正交系综中提取的，那么 4 次和的概率很高-of-squares 松弛不能证明目标 $N^{-1} \cdot \mathbf{x}^\top \mathbf{W} \mathbf{x}$ 在约束 $x_i^2 - 1 下的上限= 0$（即 $\mathbf{x} \in \{ \pm 1 \}^N$）渐近小于 $\lambda_{\max}(\mathbf{W}) \approx 2$。我们还通过提出近似的伪矩构造，推测了一种下界证明技术，以对抗任何保持常数为 $N \to \infty$ 的任何程度的平方和松弛。