# A Tight Degree 4 Sum-of-Squares Lower Bound for the Sherrington-Kirkpatrick Hamiltonian

## TL;DR

This paper proves a tight degree 4 sum-of-squares lower bound for certifying upper bounds on the Sherrington-Kirkpatrick Hamiltonian, showing it cannot do better than the spectral maximum, and proposes a conjecture for higher degrees.

## Contribution

It establishes a tight degree 4 SOS lower bound for the SK Hamiltonian and introduces a conjecture for extending lower bounds to higher degrees using pseudomoment constructions.

## Key findings

- Degree 4 SOS cannot certify bounds below the spectral maximum.
- With high probability, the lower bound matches the maximum eigenvalue asymptotically.
- Proposes a conjecture for lower bounds at any fixed degree as N grows.

## Abstract

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.