[Paper Review] Highly parallel algorithm for the Ising ground state searching problem

A. Yavorsky, L. A. Markovich+2 more|arXiv (Cornell University)|Jul 11, 2019

Quantum Computing Algorithms and Architecture50 references4 citations

TL;DR

This paper proposes MARS, a highly parallel algorithm for finding the ground state of the Ising model by combining mean-field descent with simulated annealing from random initial states. It achieves superior performance on large Ising systems and MAX-CUT benchmarks, outperforming classical simulated annealing, NMFA, and SimCIM in both solution quality and computational speed due to massive parallelization of low-cost single runs.

ABSTRACT

Finding an energy minimum in the Ising model is an exemplar objective, associated with many combinatorial optimization problems, that is computationally hard in general, but occurs in all areas of modern science. There are several numerical methods, providing solution for the medium size Ising spin systems. However, they are either computationally slow and badly parallelized, or do not give sufficiently good results for the large systems. In this paper, we present a highly parallel algorithm, called Mean-field Annealing from a Random State (MARS), incorporating the best features of the classical simulated annealing (SA) and Mean-Field Annealing (MFA) methods. The algorithm is based on the mean-field descent from a randomly selected configuration and temperature. Since a single run requires little computational effort, the effectiveness can be achieved by massive parallelisation. MARS shows excellent performance both on the large Ising spin systems and on the set of exemplary maximum cut benchmark instances in terms of both solution quality and computational time.

Motivation & Objective

• To address the computational hardness of finding the ground state in the Ising model, a canonical NP-hard problem with broad applications in physics, optimization, and machine learning.
• To overcome limitations of existing methods—particularly poor parallelization and slow convergence—on large-scale Ising spin systems and combinatorial optimization instances.
• To develop a scalable, massively parallel algorithm that achieves high-quality solutions with minimal per-run computational cost.
• To evaluate performance on standard benchmarks, including the G-set of MAX-CUT instances and large fully connected Ising systems.
• To analyze parameter sensitivity and compare with state-of-the-art algorithms in terms of solution quality, runtime, and success probability.

Proposed method

• MARS initiates search from a randomly selected spin configuration and applies mean-field descent at a fixed temperature, leveraging the mean-field approximation to accelerate convergence.
• The algorithm uses a temperature schedule that allows for efficient exploration of the energy landscape, with descent from high to low temperature to avoid local minima.
• Each run is computationally lightweight, enabling massive parallelization across many cores or processors, with results aggregated to improve solution quality.
• The method integrates features of classical simulated annealing (SA) and mean-field annealing (MFA), combining SA’s stochastic escape with MFA’s fast mean-field updates.
• Temperature bounds $T_{ ext{min}} = 0$ and $T_{ ext{max}} = 40$ are used, with sensitivity analysis showing strong dependence of performance on these parameters.
• The algorithm is tested on both fully connected Ising systems and the G-set of MAX-CUT graphs, using standard benchmark instances from the literature.

Experimental results

Research questions

• RQ1 Can a highly parallel algorithm significantly outperform classical and recent heuristic methods in solving the Ising ground state problem for large systems?
• RQ2 How does the performance of MARS compare to SA, NMFA, and SimCIM in terms of solution quality and computational time on large Ising spin systems?
• RQ3 To what extent does parameter selection—particularly $T_{ ext{min}}$ and $T_{ ext{max}}$—affect the quality and success probability of the solution?
• RQ4 Can MARS achieve or match the best-known results on standard MAX-CUT benchmarks, especially for random and toroidal graphs?
• RQ5 How does the algorithm scale with increasing number of parallel simulation blocks, and what is the optimal configuration for performance?

Key findings

• On a 100-spin Ising system, MARS achieved a ground state energy of $-33311$ in $14$ hours $15$ minutes with $50,000$ simulations, outperforming SA, NMFA, and SimCIM in both solution quality and speed.
• For the same system, the success probability was $P = 2 \times 10^{-5}$, with a hit count of 1, indicating consistent convergence to high-quality solutions.
• On the G-set of MAX-CUT instances, MARS matched or slightly exceeded the best-known results for unweighted random graphs and toroidal grids with random edge weights.
• For the "almost planar" family of graphs, MARS performed comparably to the best-known algorithms but with significantly reduced computational time.
• Parameter sensitivity analysis showed that optimal $T_{ ext{min}}$ and $T_{ ext{max}}$ values strongly influence solution quality and acceptance probability.
• MARS outperformed all baseline algorithms (SA, NMFA, SimCIM) in both best-cut estimation and computational time, even on small sample sizes, due to efficient parallelization of low-cost runs.