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 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.
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.
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.
In order to solve the current lack of rigorous theoretical models in the machine learning process, in this paper the iterative motion process of machine learning is modeled by using quantum dynamic method based on the principles of first-principles thinking. This approach treats the iterative evolution of algorithms as a physical motion process, defines a generalized objective function in the parameter space of machine learning algorithms, and regards the iterative process of machine learning as the process of seeking the optimal value of this generalized objective function. In physical terms, this process corresponds to the system reaching its ground energy state. Since the dynamic equation of a quantum system is the Schrödinger equation, we can obtain the quantum dynamic equation that describes the iterative process of machine learning by treating the generalized objective function as the potential energy term in the Schrödinger equation. Therefore, machine learning is the process of seeking the ground energy state of the quantum system constrained by a generalized objective function. The quantum dynamic equation for machine learning transforms the iterative process into a time-dependent partial differential equation for precise mathematical representation, enabling the use of physical and mathematical theories to study the iterative process of machine learning. This provides theoretical support for implementing the iterative process of machine learning by using quantum computers. In order to further explain the iterative process of machine learning on classical computers by using quantum dynamic equation, the Wick rotation is used to transform the quantum dynamic equation into a thermodynamic equation, demonstrating the convergence of the time evolution process in machine learning. The system will be transformed into the ground energy state as time approaches infinity. Taylor expansion is used to approximate the generalized objective function, which has no analytical expression in the parameter space. Under the zero-order Taylor approximation of the generalized objective function, the quantum dynamic equation and thermodynamic equation for machine learning degrade into the free-particle equation and diffusion equation, respectively. This result indicates that the most basic dynamic processes during the iteration of machine learning on quantum computers and classical computers are wave packet dispersion and wave packet diffusion, respectively, thereby explaining, from a dynamic perspective, the basic principles of diffusion models that have been successfully utilized in the generative neural networks in recent years. Diffusion models indirectly realize the thermal diffusion process in the parameter space by adding Gaussian noise to and removing Gaussian noise from the image, thereby optimizing the generalized objective function in the parameter space. The diffusion process is the dynamic process in the zero-order approximation of the generalized objective function. Meanwhile, we also use the thermodynamic equation of machine learning to derive the Softmax function and Sigmoid function, which are commonly used in artificial intelligence. These results show that the quantum dynamic method is an effective theoretical approach to studying the iterative process of machine learning, which provides a rigorous mathematical and physical model for studying the iterative process of machine learning on both quantum computers and classical computers.
We explore whether quantum field theory can be understood as the statistical mechanics of a time-reversal-invariant stochastic generalization of Hamiltonian dynamics. The motivation for this project, started with this paper, is to assign sharp values to all observables and thereby avoid the quantum measurement problem. In classical mechanics, motion is deterministic and corresponds to an evolution of the phase space probability density according to Liouville's equation that is governed by first derivatives of the Hamiltonian in phase space. We derive a generalization of the Liouville equation with natural constraints -- namely, reduction to classical Hamiltonian dynamics as the stochasticity parameter ℏ↦0, Fokker-Planck form for the probability density evolution, local Hamiltonian dependence, time-reversal invariance, energy conservation, and minimality -- which turns out to be a Fokker-Planck equation with a generalized diffusion matrix that is symmetric, traceless, and constructed from the Hessian of the Hamiltonian. We then show that the Schrödinger equation in the coherent-state phase-space formulation of certain bosonic QFTs has precisely this form, with the Husimi function playing the role of the phase space probability density. The question to what extent this equation can be interpreted in terms of objective stochastic field theories is discussed in a companion paper.
In recent years, optimization algorithms have developed rapidly, especially those which introduce quantum ideas, which perform excellently. Inspired by quantum thought, this paper proposes a quantum dynamics framework (QDF) which converts optimization problems into the problem of the constrained ground state of the quantum system and analyzes optimization algorithms by simulating the dynamic evolution process of physical optimization systems in the ground state. Potential energy equivalence and Taylor expansion are performed on the objective function to obtain the basic iterative operations of optimization algorithms. Furthermore, a quantum dynamics framework based on the quantum tunneling effect (QDF-TE) is proposed which adopts dynamic multiple group collaborative sampling to improve the quantum tunneling effect of the QDF, thereby increasing the population diversity and improving algorithm performance. The probability distribution of solutions can be visually observed through the evolution of the wave function, which also indicates that the QDF-TE can strengthen the tunneling effect. The QDF-TE was evaluated on the CEC 2017 test suite and shown to be competitive with other heuristic optimization algorithms. The experimental results reveal the effectiveness of introducing a quantum mechanism to analyze and improve optimization algorithms.
We introduce a quantum approximate optimization algorithm (QAOA) for continuous optimization. The algorithm is based on the dynamics of a quantum system moving in an energy potential which encodes the objective function. By approximating the dynamics at finite time steps, the algorithm can be expressed as alternating evolution under two non-commuting Hamiltonians. We show that each step of the algorithm updates the wavefunction in the direction of its local gradient, with an additional momentum-dependent displacement. For initial states in a superposition over many points, this method can therefore be interpreted as a coherent version of gradient descent, i.e., 'gradient descent in superposition.' This approach can be used for both constrained and unconstrained optimization. In terms of computational complexity, we show how variants of the algorithm can recover continuous-variable Grover search, and how a single iteration can replicate continuous-variable instantaneous quantum polynomial circuits. We also discuss how the algorithm can be adapted to solve discrete optimization problems. Finally, we test the algorithm through numerical simulation in optimizing the Styblinski-Tang function.