Generalized Mean-Field Ising Model as a Benchmark Family with Exact Solutions
Abstract
Exactly solvable models provide opportunities to understand complex optimization problems while offering controlled benchmarks for computational methods. However, nontrivial models that combine analytical tractability with challenging optimization structure remain scarce. Here, we introduce the Generalized Mean-Field Ising (GMFI) family, a densely connected and inhomogeneous class of Ising Hamiltonians whose ground states can nevertheless be obtained exactly. This exact solution reveals a hidden low-dimensional structure and enables a complete number-theoretic classification of different system sizes and degenerate ground states. Uniquely, the GMFI family can be used to benchmark both quantum and classical optimization methods over system sizes far beyond exact diagonalization. We investigate the GMFI family using several approaches, including theoretical analysis of quantum annealing, experiments on a D-Wave quantum processor, the quantum approximate optimization algorithm (QAOA), imaginary-time evolution (ITE), greedy algorithm, simulated annealing, and a quantum-inspired greedy algorithm. Despite the GMFI model's exact solvability, it exhibits nontrivial algorithmic behavior: although the quantum-annealing gap is expected to decrease with increasing system size, it shows pronounced and unexpected system-size-dependent oscillatory signatures. These results establish GMFI models as a rare bridge between exact many-body theory, number theory, optimization, and quantum hardware, providing a scalable benchmark for identifying and understanding algorithmic limitations in classical and quantum computing.
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BibTeX
@misc{willsch2026generalized,
title = {{Generalized Mean-Field Ising Model as a Benchmark Family with Exact Solutions}},
author = {Madita Willsch and Gopal Chandra Santra and Dennis Willsch and Samuel Morales and Lukas Mauth and Vrinda Mehta and Fengping Jin and Kristel Michielsen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38803},
eprint = {2610.38803},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38803},
}
RIS
TY - UNPB
TI - Generalized Mean-Field Ising Model as a Benchmark Family with Exact Solutions
AU - Madita Willsch
AU - Gopal Chandra Santra
AU - Dennis Willsch
AU - Samuel Morales
AU - Lukas Mauth
AU - Vrinda Mehta
AU - Fengping Jin
AU - Kristel Michielsen
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38803
UR - https://arxiv.org/abs/2610.38803
AB - Exactly solvable models provide opportunities to understand complex optimization problems while offering controlled benchmarks for computational methods. However, nontrivial models that combine analytical tractability with challenging optimization structure remain scarce. Here, we introduce the Generalized Mean-Field Ising (GMFI) family, a densely connected and inhomogeneous class of Ising Hamiltonians whose ground states can nevertheless be obtained exactly. This exact solution reveals a hidden low-dimensional structure and enables a complete number-theoretic classification of different system sizes and degenerate ground states. Uniquely, the GMFI family can be used to benchmark both quantum and classical optimization methods over system sizes far beyond exact diagonalization. We investigate the GMFI family using several approaches, including theoretical analysis of quantum annealing, experiments on a D-Wave quantum processor, the quantum approximate optimization algorithm (QAOA), imaginary-time evolution (ITE), greedy algorithm, simulated annealing, and a quantum-inspired greedy algorithm. Despite the GMFI model's exact solvability, it exhibits nontrivial algorithmic behavior: although the quantum-annealing gap is expected to decrease with increasing system size, it shows pronounced and unexpected system-size-dependent oscillatory signatures. These results establish GMFI models as a rare bridge between exact many-body theory, number theory, optimization, and quantum hardware, providing a scalable benchmark for identifying and understanding algorithmic limitations in classical and quantum computing.
ER -