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IEEE International Symposium on Parallel & Distributed Processing, Workshops and Phd Forum

Computational Speedup of Simulated Annealing with Nested Monte Carlo Loop

K. Murashima

Published Jun 3, 2025• 1 citation
Publisher

Abstract

An Ising spin model interacting with neighboring spins has been eagerly studied along with the progress in Quantum Computing (QC). Quantum Monte Carlo (QMC) is often used in simulation, but it is such a heuristic approach that it has great difficulty in that it takes a long time to solve the problem. It mainly depends on the existence of a trotter layer derived from Suzuki-Trotter decomposition. In my previous work, I proposed a new approach to compute it much faster by changing the order in which all energies over the trotter layer are summed and minimized. In the algorithm, a plural of trotter layers could be treated in parallel and Time to Solution (TTS) could be reduced as well as that of Simulated Annealing (SA), although it requires a lot of computational resources.In this paper, in order to balance them, a new approach is proposed, where the number of inner loops is not changed but the number of the trotter layers is smaller. With such a small trotter loop repeated in several times, the proposed method works as so-called partial optimization and it leads to very fast convergence. Moreover, it doesn’t affect the result whether the updating spins are the same or not during the partial optimization. This fact also indicates that the new method would open another door to parallel computation. These results are verified by solving MaxCut problem, GSET.

DOI
10.1109/ipdpsw66978.2025.00185

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An Ising spin model interacting with neighboring spins has been eagerly studied along with the progress in Quantum Computing (QC). Quantum Monte Carlo (QMC) is often used in simulation, but it is such a heuristic approach that it has great difficulty in that it takes a long time to solve the problem. It mainly depends on the existence of a trotter layer derived from Suzuki-Trotter decomposition. In my previous work, I proposed a new approach to compute it much faster by changing the order in which all energies over the trotter layer are summed and minimized. In the algorithm, a plural of trotter layers could be treated in parallel and Time to Solution (TTS) could be reduced as well as that of Simulated Annealing (SA), although it requires a lot of computational resources.In this paper, in order to balance them, a new approach is proposed, where the number of inner loops is not changed but the number of the trotter layers is smaller. With such a small trotter loop repeated in several times, the proposed method works as so-called partial optimization and it leads to very fast convergence. Moreover, it doesn’t affect the result whether the updating spins are the same or not during the partial optimization. This fact also indicates that the new method would open another door to parallel computation. These results are verified by solving MaxCut problem, GSET.
Quantum Monte Carlo (QMC) is commonly used in simulations for Quantum Annealing (QA), but QMC as a heuristic approach has great difficulty in that it takes much time to find minimum energy. It mainly depends on the existence of a trotter layer derived from Suzuki-Trotter decomposition. In this paper, I propose a new approach to calculate it in short time, although it isn't rigorous mathematically. Its validity and advantageous points are also discussed, in comparison with conventional QMC methods.
Quantum Annealing (QA) has already put into practical use and considered useful for solving many social issues, such as reduction of traffic congestion and delivery optimization. But in QA, when the energy difference between the ground state and the first excited state is small, the transition probability between them increases. Therefore, in order to decrease the transition, QA has to be performed at extremely low temperatures. To simulate the situation, it has the problem in that it takes much time. Many studies are underway to accelerate QA theoretically, one of which is to incorporate non-stoquastic Hamiltonian. On the other hand, I proposed Nested Simulated Annealing (NSA) inspired by Quantum Monte Carlo (QMC). I showed the computational speedup could be achieved dramatically, by considering the idea that the effect of flipping a spin preferentially influenced the spins directly interacting with it. Although NSA was based on such classical concept of causality, the hybrid computation both quantum and classical approach worked well. In this paper, in order to discuss the relationship between NSA and non-stoquastic Hamiltonian like XX-interaction, the spins are treated as continuous variables. I derive the formula to calculate the total energy in both with a problem Hamiltonian and the perturbation Hamiltonian induced by a transverse electromagnetic field. And I will show a clear relationship between local-maxima and the convergence speed when calculating it with the binary spin. More precisely, the annealing path with the smallest local-maxima converges fast, even though it consumes fewer computational resources. Therefore, it is possible to induce an optimal adiabatic phase transition by selecting NSA parameters appropriately. This paper shows an effective method to choose the parameters when simulating QMC on a classical computer.
One of the vital roles of computing is to solve large-scale combinatorial optimization problems in a short time. In recent years, methods have been proposed that map optimization problems to ones of searching for the ground state of an Ising model by using a stochastic process. Simulated annealing (SA) is a representative algorithm. However, it is inherently difficult to perform a parallel search. Here we propose an algorithm called momentum annealing (MA), which, unlike SA, updates all spins of fully connected Ising models simultaneously and can be implemented on GPUs that are widely used for scientific computing. MA running in parallel on GPUs is 250 times faster than SA running on a modern CPU at solving problems involving 100 000 spin Ising models.
A numerical technique called Simulated Ising Annealing (SIA) uses digital computers to obtain approximations of the ground states of Ising models. The quadratic unconstrained binary optimization (QUBO) problem in combinatorial optimization, which is challenging to solve through exhaustive search, has significant linkages to the Ising model. Therefore, SIA's main objective is to identify a better solution in finite iterations. In order to enhance the performance of the simulated annealing procedure, we adopt a Gaussian augmented Hamiltonian Monte Carlo (GAHMC) method in this study. We contrasted our algorithm with Gibbs sampling and local quantum annealing method. Finally, we discuss its application on QUBO problem.
Abstract Quantum annealing aims to provide a faster method than classical computing for finding the minima of complicated functions, and it has created increasing interest in the relaxation dynamics of quantum spin systems. Moreover, problems in quantum annealing caused by first-order phase transitions can be reduced via appropriate temporal adjustment of control parameters, and in order to do this optimally, it is helpful to predict the evolution of the system at the level of macroscopic observables. Solving the dynamics of quantum ensembles is nontrivial, requiring modeling of both the quantum spin system and its interaction with the environment with which it exchanges energy. An alternative approach to the dynamics of quantum spin systems was proposed about a decade ago. It involves creating stochastic proxy dynamics via the Suzuki-Trotter mapping of the quantum ensemble to a classical one (the quantum Monte Carlo method), and deriving from this new dynamics closed macroscopic equations for macroscopic observables using the dynamical replica method. In this chapter, we give an introduction to this approach, focusing on the ideas and assumptions behind the derivations, and on its potential and limitations.
We introduce quantum fluctuations into the simulated annealing process of optimization problems, aiming at faster convergence to the optimal state. The idea is tested by the two models, the transverse Ising model and the traveling salesman problem (TSP). Adding the transverse field to the Ising model is a simple way to introduce quantum fluctuations. The strength of the transverse field is controlled as a function of time similarly to the temperature in the conventional method. TSP can be described by the Ising spin, so that we also apply the same technique to TSP. We directory solve the time-dependent Schrödinger equation for small-size systems and perform the quantum Monte Carlo simulation for large-size systems. Comparison with the results of the corresponding classical (thermal) method reveals that the quantum method leads to the ground state with much larger probability in almost all cases if we use the same annealing schedule of the control parameters. We also find that the relaxation time is quite short for quantum systems by numerical simulations. We consider this is one of the reasons why the annealing in quantum systems have a better performance of finding the optimal state in comparison with classical systems.