Efficient algorithms for solving finite Sherrington–Kirkpatrick instances

Develop efficient algorithms to compute the ground-state configuration of the Sherrington–Kirkpatrick Ising spin-glass Hamiltonian E({m}) = -(1/√n) Σ_{i<j} J_{ij} m_i m_j, with m_i ∈ {−1,1} and couplings J_{ij} drawn from a standard normal distribution, for given finite n. The objective is to resolve the challenge of efficiently solving individual instances beyond the large-n asymptotic results that are known.

Background

The Sherrington–Kirkpatrick (SK) model is a canonical mean-field spin glass with all-to-all random couplings J_{ij} ∼ N(0,1), where the energy function is E({m}) = -(1/√n) Σ{i<j} J{ij} m_i m_j for binary spins m_i ∈ {−1,1}. While the ground-state energy per spin is known in the large-n limit via Parisi’s solution, efficiently determining the ground state for specific finite instances remains challenging.

In the paper, the authors benchmark their classical Probabilistic Approximate Optimization Algorithm (PAOA) against QAOA on n=26 SK instances and discuss performance and scalability. Within this context, they point out that, despite asymptotic knowledge of the ground-state energy, the existence of efficient algorithms to solve individual finite SK instances is unresolved.