In recent years, quantum annealing has triggered active research on annealing methods for solving various combinatorial optimization problems (COPs) by mapping them to the Ising model based on spin glass theory. In particular, parallel annealing algorithms (PAAs) that can update all variables simultaneously attract attention due to fast optimization using parallel computers, either as an extension of Simulated Annealing rooted in classical thermodynamics or as a quantum-inspired algorithm. However, both types of PAAs face their own challenges. The classical thermodynamics-based PAAs (c-PAAs) perform inferior to the quantum-inspired PAAs (q-PAAs), whereas the q-PAAs require more parameters to be tuned than the c-PAAs. This paper proposes a new c-PAA based on Mean Field Annealing, which has the unique feature of updating analog variables deterministically. The proposed PAA achieves high speed and robustness despite fewer parameters than the q-PAAs, which means the proposed PAA breaks through the challenges of conventional PAAs. We demonstrate its performance through experiments on four types of COPs: Maximum Cut Problem, Graph Coloring Problem, Maximum Independent Set Problem, and Traveling Salesman Problem. These results imply that unless a real physical phenomenon is used, quantum-inspired algorithms cannot be considered superior to classical thermodynamics-based algorithms.
In recent years, quantum annealing has triggered active research on annealing methods for solving various combinatorial optimization problems (COPs) by mapping them to the Ising model based on spin glass theory. In particular, parallel annealing algorithms (PAAs) that can update all variables simultaneously attract attention due to fast optimization using parallel computers, either as an extension of Simulated Annealing rooted in classical thermodynamics or as a quantum-inspired algorithm. However, both types of PAAs face their own challenges. The classical thermodynamics-based PAAs (c-PAAs) perform inferior to the quantum-inspired PAAs (q-PAAs), whereas the q-PAAs require more parameters to be tuned than the c-PAAs. This paper proposes a new c-PAA based on Mean Field Annealing, which has the unique feature of updating analog variables deterministically. The proposed PAA achieves high speed and robustness despite fewer parameters than the q-PAAs, which means the proposed PAA breaks through the challenges of conventional PAAs. We demonstrate its performance through experiments on four types of COPs: Maximum Cut Problem, Graph Coloring Problem, Maximum Independent Set Problem, and Traveling Salesman Problem. These results imply that unless a real physical phenomenon is used, quantum-inspired algorithms cannot be considered superior to classical thermodynamics-based algorithms.
Abstract Combinatorial optimization problems and ground state problems of spin glasses are crucial in various fields of science and technology. However, they often belong to the computational class of NP-hard, presenting significant computational challenges. Traditional algorithms inspired by statistical physics like simulated annealing have been widely adopted. Recently, advancements in Ising machines, such as quantum annealers and coherent Ising machines, offer new paradigms for solving these problems efficiently by embedding them into the analog evolution of nonlinear dynamical systems. However, existing dynamics-based algorithms often suffer from low convergence rates and local minima traps. In this work, we introduce the dual mean-field dynamics into Ising machines. The approach integrates the gradient force and the transverse force into the dynamics of Ising machines in solving combinatorial optimization problems, making it easier for the system to jump out of the local minimums and allowing the dynamics to explore wider in configuration space. We conduct extensive numerical experiments using the Sherrington–Kirkpatrick spin glass up to 10 000 spins and the maximum cut problems with the standard G-set benchmarks. The numerical results demonstrate that our dual mean-field dynamics approach enhances the performance of base Ising machines, providing a more effective solution for large-scale combinatorial optimization problems.
Annealing processors that can quickly solve combinatorial optimization problems (COPs) are attracting attention. They solve COPs by converting them into spin-spin interactions and external fields of the Ising model. Annealing processors that use fully-coupled spin coupling can solve a wide range of COPs. However, it is difficult to execute parallel annealing in fully-coupled spin coupling because one spin is updated by referring to all the other spins. In this paper, we proposed multi-state pseudo annealing (MSPA), which is an annealing method for changing the spin value from a conventional binary to a multi state that takes into account intermediate states. In a software validation, we solved the Maxcut problem and the traveling salesman problem and confirmed that the proposed method improves the solution quality in parallel annealing compared with simulated annealing and pseudo annealing. In addition, we implemented MSPA on a field-programmable gate array as a 1024 -spin 16 -parallel system using a scalable structure. With the proposed system, we solved the Max-cut problem and confirmed that it achieves a 9.40 times faster solving speed and yields a 1% larger Max-cut value compared with the conventional parallelized annealing processors.
Abstract Two classes of quantum-annealing-inspired-algorithms (QAIA), namely different variants of simulated coherent Ising machine and simulated bifurcation, have been proposed for efficiently solving combinatorial optimization problems recently. In order to certify the superiority of these algorithms, standardized comparisons among them and against other physics-based algorithms are necessary. In this work, for Max-Cut problems up to 20,000 nodes, we benchmark QAIA against quantum annealing and other physics-based algorithms. We found that ballistic simulated bifurcation excelled for chimera and small-scale graphs, achieving nearly a 50-fold reduction in time-to-solution compared to quantum annealing. For large-scale graphs, discrete simulated bifurcation achieves the lowest time-to-target and outperforms D-Wave Advantage system when tasked with finding the maximum cut value in pegasus graphs. Our results suggest that QAIA represents a promising means for solving combinatorial optimization problems in practice, and can act as a natural baseline for competing quantum algorithms.
Optimization algorithms for combinatorial problems play a critical role in a wide range of applications, from logistics and scheduling to cryptography and artificial intelligence. Traditional classical methods, such as simulated annealing, genetic algorithms, and branch-and-bound techniques, have been widely employed to solve these problems, but they often face limitations in terms of computational efficiency and scalability as problem complexity grows. In recent years, quantum computing has emerged as a promising alternative, with quantum annealing offering a novel approach to solving combinatorial optimization problems. This study provides a comparative analysis of quantum annealing and classical optimization methods, highlighting their respective strengths and limitations. We explore how quantum annealing, through leveraging quantum tunnelling and superposition, has the potential to outperform classical algorithms in specific problem domains and Noisy Mean Field Annealing (NMFA) is used to comparisons of all algorithms. Benchmarking various combinatorial problems, such as the Traveling Salesman Problem (TSP), Max-Cut, Knapsack, and Quadratic Assignment Problem (QAP), this paper discusses performance metrics, computational time, and scalability. The findings suggest that while classical methods remain robust and practical for many real-world problems, quantum annealing holds significant promise, especially as quantum hardware continues to mature. However, there remain challenges in terms of noise, coherence, and problem mapping, which currently limit the full realization of quantum annealing's potential. This study offers insights into the future directions of optimization techniques and the evolving role of quantum computing in solving complex combinatorial problems.
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.
Combinatorial optimization problems are frequently classified as NP-hard, which means that the time needed to find the optimal solution generally increases exponentially with the problem size. However, the existing research gap highlights the necessity to develop combinatorial optimization methods capable of scaling to large problems while maintaining low solution latency. In this paper, we introduce stochastic simulated quantum annealing (SSQA) for fast solution of combinatorial optimization problems. SSQA is designed based on stochastic computing, which can simulate quantum annealing (QA) by using multiple replicas of spins (probabilistic bits) in classical computing. The use of stochastic computing leads to an efficient parallel spin-state update algorithm, enabling quick search for a solution around the global minimum energy. Therefore, SSQA realizes quantum-like annealing for large-scale problems and can handle fully connected models in combinatorial optimization, unlike QA. The proposed method is evaluated in MATLAB on graph isomorphism problems, which are typical combinatorial optimization problems. It can handle a 100-times larger problem size compared to QA and a 25-times larger problem size compared to a traditional SA method, respectively, for similar convergence probabilities. The time to solution of SSQA is 11.5 times and 423 times smaller than that of the conventional stochastic simulated annealing method and traditional SA, respectively, when solving a 2,025-node combinatorial optimization problem.