# Optimization of the Sherrington-Kirkpatrick Hamiltonian

## TL;DR

This paper introduces a new message-passing algorithm that efficiently approximates the maximum of the Sherrington-Kirkpatrick Hamiltonian, closely matching the optimal value with high probability for large systems.

## Contribution

It presents a novel message-passing algorithm with quadratic time complexity that achieves near-optimal solutions for the SK model, extending its applicability to low-temperature regimes.

## Key findings

- Algorithm achieves (1-ε) approximation of the optimum
- Time complexity is quadratic in system size
- Constructs approximate solutions to TAP equations at low temperature

## Abstract

Let ${\boldsymbol A}\in{\mathbb R}^{n\times n}$ be a symmetric random matrix with independent and identically distributed Gaussian entries above the diagonal. We consider the problem of maximizing $\langle{\boldsymbol \sigma},{\boldsymbol A}{\boldsymbol \sigma}\rangle$ over binary vectors ${\boldsymbol \sigma}\in\{+1,-1\}^n$. In the language of statistical physics, this amounts to finding the ground state of the Sherrington-Kirkpatrick model of spin glasses. The asymptotic value of this optimization problem was characterized by Parisi via a celebrated variational principle, subsequently proved by Talagrand. We give an algorithm that, for any $\varepsilon>0$, outputs ${\boldsymbol \sigma}_*\in\{-1,+1\}^n$ such that $\langle{\boldsymbol \sigma}_*,{\boldsymbol A}{\boldsymbol \sigma}_*\rangle$ is at least $(1-\varepsilon)$ of the optimum value, with probability converging to one as $n\to\infty$. The algorithm's time complexity is $C(\varepsilon)\, n^2$. It is a message-passing algorithm, but the specific structure of its update rules is new. As a side result, we prove that, at (low) non-zero temperature, the algorithm constructs approximate solutions of the Thouless-Anderson-Palmer equations.