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International Conference on Information and Communication Technology Convergence (ICTC)

Tunneling Effect of quantization-based optimization

Jinwuk Seok, Chang-Sik Cho

Published Oct 14, 2025• 0 citations
Publisher

Abstract

In this paper, we present an analysis of the tunneling effect in the quantization-based optimization. We prove that, when the quantization step size is an exponential decay function with respect to an iteration, the quantized objective function admits a decomposition into longitudinal and transverse fields. From the perspective of quantum mechanics, since the transverse field gives the quantum tunneling effect, quantization-based optimization can outperform a conventional thermodynamics-based optimization scheme. Experimental results for a vanilla objective function and a typical combinatorial optimization problem, such as the traveling salesman problem, demonstrate that the presented analysis is valid.

DOI
10.1109/ictc66702.2025.11388700

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In this paper, we present an analysis of the tunneling effect in the quantization-based optimization. We prove that, when the quantization step size is an exponential decay function with respect to an iteration, the quantized objective function admits a decomposition into longitudinal and transverse fields. From the perspective of quantum mechanics, since the transverse field gives the quantum tunneling effect, quantization-based optimization can outperform a conventional thermodynamics-based optimization scheme. Experimental results for a vanilla objective function and a typical combinatorial optimization problem, such as the traveling salesman problem, demonstrate that the presented analysis is valid.
This work presents a quantum mechanical framework for analyzing quantization-based optimization algorithms. The sampling process of the quantization-based search is modeled as a gradient-flow dissipative system, leading to a Hamilton-Jacobi-Bellman (HJB) representation. Through a suitable transformation of the objective function, this formulation yields the Schroedinger equation, which reveals that quantum tunneling enables escape from local minima and guarantees access to the global optimum. By establishing the connection to the Fokker-Planck equation, the framework provides a thermodynamic interpretation of global convergence. Such an analysis between the thermodynamic and the quantum dynamic methodology unifies combinatorial and continuous optimization, and extends naturally to machine learning tasks such as image classification. Numerical experiments demonstrate that quantization-based optimization consistently outperforms conventional algorithms across both combinatorial problems and nonconvex continuous functions.
Statistical and stochastic analysis based on thermodynamics has been the main analysis framework for stochastic global optimization. Recently, with the appearance of quantum annealing or quantum tunneling algorithms for global optimization, we require a new research framework for global optimization algorithms. In this paper, we provide the analysis for quantization-based optimization based on the Schrödinger equation to reveal what property in quantum mechanics enables global optimization. We present that the tunneling effect derived by the Schrödinger equation in quantization-based optimization enables to escape of a local minimum. Additionally, we confirm that this tunneling effect is the same property included in quantum mechanics-based global optimization. Experiments with standard multi-modal benchmark functions represent that the proposed analysis is valid.
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.
Abstract We propose a number‐theory‐based quantized mathematical optimization scheme for various NP‐hard and similar problems. Conventional global optimization schemes, such as simulated and quantum annealing, assume stochastic properties that require multiple attempts. Although our quantization‐based optimization proposal also depends on stochastic features (i.e., the white‐noise hypothesis), it provides a more reliable optimization performance. Our numerical analysis equates quantization‐based optimization to quantum annealing, and its quantization property effectively provides global optimization by decreasing the measure of the level sets associated with the objective function. Consequently, the proposed combinatorial optimization method allows the removal of the acceptance probability used in conventional heuristic algorithms to provide a more effective optimization. Numerical experiments show that the proposed algorithm determines the global optimum in less operational time than conventional schemes.
This paper investigates the application of quantum annealing to solve combinatorial optimization problems. The core claim is that quantum annealing algorithms can effectively enhance the efficiency and accuracy of solving such problems. The underlying mechanism involves simulating the quantum annealing process, leveraging the quantum tunneling effect to explore the solution space and identify optimal solutions. This work presents a novel approach by applying quantum annealing to combinatorial optimization, offering a fresh perspective on problem-solving. We explore the theoretical foundations, model the annealing process, and discuss potential challenges and future research directions. The focus is on demonstrating the feasibility and potential benefits of this approach rather than presenting results from experimental implementations.
The Traveling Salesman Problem is a fundamental combinatorial optimization problem widely studied in operations research. Despite its simple formulation, it remains computationally challenging due to the exponential growth of the search space and the large number of constraints required to eliminate subtours. This paper introduces a preprocessing strategy that significantly reduces the size of the optimization model by restricting the set of candidate arcs and retaining only the lowest-cost neighbors for each vertex. Computational experiments on TSPLIB benchmark instances demonstrate that the proposed approach substantially reduces the number of decision variables. The method is evaluated using both classical and quantum optimization techniques, showing improvements in computational time and reductions in optimality gaps. Overall, the results indicate that the proposed preprocessing enhances the scalability of the formulations and makes them more suitable for both classical solvers and emerging quantum optimization frameworks.