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.
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.
In this letter, by establishing the Schrödinger equation of the optimization problem, the optimization problem is transformed into a constrained state quantum problem with the objective function as the potential energy. The mathematical relationship between the objective function and the wave function is established, and the quantum interpretation of the optimization problem is realized. Under the black box model, the Schrödinger equation of the optimization problem is used to establish the kinetic equation, i.e., the Fokker-Planck equation of the time evolution of the optimization algorithm, and the basic iterative structure of the optimization algorithm is given according to the interpretation of the Fokker-Planck equation. The establishment of the Fokker-Planck equation allows optimization algorithms to be studied using dynamic methods and is expected to become an important theoretical basis for algorithm dynamics.
In recent decades, with the emergence of numerous novel intelligent optimization algorithms, many optimization researchers have begun to look for a basic search mechanism for their schemes that provides a more essential explanation of their studies. This paper aims to study the basic mechanism of an algorithm for black-box optimization with quantum theory. To achieve this goal, the Schroedinger equation is employed to establish the relationship between the optimization problem and the quantum system, which makes it possible to study the dynamic search behaviors in the evolution process with quantum theory. Moreover, to explore the basic behavior of the optimization system, the optimization problem is assumed to be decomposed and approximated. Then, a multilevel approximation quantum dynamics model of the optimization algorithm is established, which provides a mathematical and physical framework for the analysis of the optimization algorithm. Correspondingly, the basic search behavior based on this model is derived, which is governed by quantum theory. Comparison experiments and analysis between different bare-bones algorithms confirm the existence of the quantum mechanic based basic search mechanism of the algorithm on black-box optimization.
We provide several quantum algorithms for continuous optimization that do not require gradient estimation. Instead, we encode the optimization problem into the dynamics of a physical system and coherently simulate the time evolution. We focus on the setting where the objective function can be accessed via a phase oracle. Our first two algorithms can find local optima of a differentiable function f : R N → R by simulating either classical or quantum dynamics with friction via a time-dependent Hamiltonian. We show that for the benchmark problem of optimizing a locally quadratic objective function, these methods require a total of O ( N 2 κ 2 / h x 2 ϵ ) queries to a phase oracle to find an ϵ -approximate local optimum, where κ is the condition number of the Hessian matrix and h x is the discretization spacing. In contrast, we show that methods based on gradient descent require O ( N 3 / 2 ( 1 / ϵ ) κ log ( 3 ) / 4 ) queries. This corresponds to an exponential separation between the query upper bounds for the benchmark problem. Our third algorithm can find the global optimum of f by preparing a classical low-temperature thermal state via simulation of the classical Liouvillian operator associated with the Nosé Hamiltonian. We use results from the quantum thermodynamics literature to bound the thermalization time for the discrete system. Additionally, we analyze barren plateau effects that commonly plague quantum optimization algorithms and observe that our approach is vastly less sensitive to this problem than standard gradient-based optimization. Our results suggest that these dynamical optimization approaches may be far more scalable for future quantum machine learning, optimization and variational experiments than was widely believed.
The rapid growth of machine learning (ML) has been accompanied by a significant challenge: the scaling of optimization algorithms to handle the immense parameter spaces of modern models, such as deep neural networks. Traditional optimization methods, including gradient descent and its variants, often encounter difficulties in efficiently exploring these spaces, leading to slow training times and potential stagnation. This paper investigates the application of quantum-inspired optimization techniques to address this scaling problem. We explore algorithms inspired by quantum annealing and other quantum phenomena, adapting their core principles to classical computation. Specifically, we propose a novel optimization framework utilizing a stochastic search strategy based on quantum superposition and measurement, designed to navigate high-dimensional parameter spaces more effectively. The efficacy of this approach is evaluated through theoretical analysis and conceptual simulations, demonstrating its potential to accelerate the training of large-scale machine learning models. We demonstrate how these methods can improve convergence speed and potentially escape local minima more effectively than conventional techniques. The core contribution lies in providing a theoretical foundation and a practical framework for applying quantum-inspired concepts to optimization within the context of machine learning.
This study presents an in-depth and focused review of Quantum-based conceptual optimization Strategies and their impact in enhancing classical machine learning training. Quantum-inspired methods simulate key fundamental concepts of quantum mechanics, including superposition, entanglement, and quantum tunneling within classical computing environments to improve optimization efficiency. These techniques enable better exploration of complex search spaces, faster convergence, and improved capability to avoid local minima. The review examines major quantum-inspired algorithms, including evolutionary models, annealing-based methods, and hybrid quantum–classical frameworks, highlighting their strengths, limitations, and suitability for modern ML tasks. The study also discusses their relevance to sustainable, energy-efficient digital transformation in education and research. The goal is to present clear insights into how quantum-inspired optimization can strengthen traditional ML training without requiring quantum hardware.