# Optimization of the Sherrington-Kirkpatrick Hamiltonian

@article{Montanari2019OptimizationOT, title={Optimization of the Sherrington-Kirkpatrick Hamiltonian}, author={Andrea Montanari}, journal={2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)}, year={2019}, pages={1417-1433} }

Let A be a symmetric random matrix with independent and identically distributed Gaussian entries above the diagonal. We consider the problem of maximizing the quadratic form associated to A over binary vectors. 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…

## 77 Citations

Computing the Partition Function of the Sherrington-Kirkpatrick Model is Hard on Average

- Computer Science, Mathematics2020 IEEE International Symposium on Information Theory (ISIT)
- 2020

We establish the average-case hardness of the algorithmic problem of exactly computing the partition function of the Sherrington-Kirkpatrick model of spin glasses with Gaussian couplings. In…

The Quantum Approximate Optimization Algorithm and the Sherrington-Kirkpatrick Model at Infinite Size

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The first implementation of a new approximate message passing (AMP) algorithm is described and it is observed numerically that the intermediate states generated by the algorithm have the properties of ancestor states in the ultrametric tree.

Realizable solutions of the Thouless-Anderson-Palmer equations.

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The factors which determine the free-energy difference between a stationary solution corresponding to a saddle point and its associated minimum are investigated, which is the barrier which has to be surmounted to escape from the vicinity of a TAP minimum or pure state.

Optimizing strongly interacting fermionic Hamiltonians

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Among other results, this work gives an efficient classical certification algorithm for upper-bounding the largest eigenvalue in the q=4 SYK model, and an efficient quantum Certification algorithm for lower- bounding this largest eigensvalue; both algorithms achieve constant-factor approximations with high probability.

The $\ell^p$-Gaussian-Grothendieck problem with vector spins

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We study the vector spin generalization of the l-Gaussian-Grothendieck problem. In other words, given integer κ ≥ 1, we investigate the asymptotic behaviour of the ground state energy associated with…

Low-Degree Hardness of Random Optimization Problems

- Computer Science2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)
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This work considers the problem of finding nearly optimal solutions of optimization problems with random objective functions and considers the Langevin dynamics algorithm, a canonical Monte Carlo analogue of the gradient descent algorithm (applicable only for the spherical p-spin glass Hamiltonian).

Optimizing Mean Field Spin Glasses with External Field

- Computer Science
- 2021

This work gives a two-phase message pasing algorithm to approximately maximize HN when a no overlap-gap condition holds and gives a branching variant of the algorithm which constructs a full ultrametric tree of approximate maxima.

The realizable solutions of the TAP equations

- Physics
- 2019

We show that the only solutions of the TAP equations for the Sherrington-Kirkpatrick model of Ising spin glasses which can be found by iteration are those whose free energy lies on the border between…

On convergence of Bolthausen's TAP iteration to the local magnetization

- Mathematics
- 2020

The Thouless, Anderson, and Palmer (TAP) equations state that the local magnetization in the Sherrington-Kirkpatrick mean-field spin glass model satisfies a system of nonlinear equations. In the…

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