Fast Numerical Solver of Ising Optimization Problems via Pruning and Domain Selection

TL;DR

The paper tackles Ising optimization by introducing a two-stage solver that first prunes spins using graph information to drastically reduce problem size, then solves the reduced problem via a domain-selection step that relaxes the discrete domain to a continuous space through an $L_1$-norm formulation and gradient descent. The approach leverages an augmented Lagrangian with two coupled variables and a continuous relaxation using $\sin(\bm{\theta})$, enabling efficient search for near-optimal domains. Empirically, it achieves up to an order-of-magnitude speedup over strong classical solvers and outperforms quantum-inspired annealers on benchmark problems, with high pruning rates and favorable scaling in problem size. The work presents a practical, hardware-light solver for near-term optimization challenges and offers a robust benchmark for assessing quantum devices.

Abstract

Quantum annealers, coherent Ising machines and digital Ising machines for solving quantum-inspired optimization problems have been developing rapidly due to their near-term applications. The numerical solvers of the digital Ising machines are based on traditional computing devices. In this work, we propose a fast and efficient solver for the Ising optimization problems. The algorithm consists of a pruning method that exploits the graph information of the Ising model to reduce the computational complexity, and a domain selection method which introduces significant acceleration by relaxing the discrete feasible domain into a continuous one to incorporate the efficient gradient descent method. The experiment results show that our solver can be an order of magnitude faster than the classical solver, and at least two times faster than the quantum-inspired annealers including the simulated quantum annealing on the benchmark problems. With more relaxed requirements on hardware and lower cost than quantum annealing, the proposed solver has the potential for near-term application in solving challenging optimization problems as well as serving as a benchmark for evaluating the advantage of quantum devices.

Figures (5)

• Figure 1: Illustration of our solver. (a) Our solver first applies pruning to simplify the optimization problem by utilizing the graph information of the Ising model. (b) Then the domain selection method is applied to find the approximate optimal solutions. As illustrated in (b1)-(b3), the original problem is firstly transformed into an $L_1$-norm maximization problem with discrete feasible domains. Then we relax the feasible domain to a continuous one and use the gradient descent optimizer to find approximate optimal domain. (c) Finally, the solution of the primal problem is retrieved by combining the results of domain selection and pruning.
• Figure 2: A lattice Ising model without external field.
• Figure 3: Comparisons of different solvers on the randomly generated problems. We sample $5000$ complete Ising problems of $1000$ spins. Except for Gurobi, the number of sweeps is set to be $5$. For Gurobi, we set the gap of mixed-integer programming as $\texttt{MIPGap}=0.8$. Running time is measured in seconds.
• Figure 4: Comparisons of different solvers on the typical fundamental optimization problems.
• Figure 5: Comparisons of different solvers on scalability. Here we show the variations of $T$ with the number of spins $d$. Apparently, the logarithmic horizontal and vertical axes can be related by polynomial functions. The slope of the line is the highest power of the terms of the polynomial, whose value is labeled on the right of the figure.