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.
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.
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.
Abstract In this work, our prime objective is to study the phenomena of quantum chaos and complexity in the machine learning dynamics of Quantum Neural Network (QNN). A Parameterized Quantum Circuits (PQCs) in the hybrid quantum-classical framework is introduced as a universal function approximator to perform optimization with Stochastic Gradient Descent (SGD). We employ a statistical and differential geometric approach to study the learning theory of QNN. The evolution of parametrized unitary operators is correlated with the trajectory of parameters in the Diffusion metric. We establish the parametrized version of Quantum Complexity and Quantum Chaos in terms of physically relevant quantities, which are not only essential in determining the stability, but also essential in providing a very significant lower bound to the generalization capability of QNN. We explicitly prove that when the system executes limit cycles or oscillations in the phase space, the generalization capability of QNN is maximized. Finally, we have determined the generalization capability bound on the variance of parameters of the QNN in a steady state condition using Cauchy Schwartz Inequality.
The convergence of machine learning (ML) and quantum computing is quickly becoming a game-changing new frontier in computational science. The most intriguing area of overlap may be quantum optimization, which seeks to reduce the exponentially increasing complexity and computational cost of ML algorithms. As they grow larger and more complicated, traditional optimization methods, particularly in high-dimensional and non-convex space, will inevitably hit performance limits. Quantum optimization is the whole new paradigm that leverages quantum mechanical effects like superposition, entanglement, and tunnelling to explore solution spaces more efficiently. The chapter outlines the theory, practice, and potential of hybrid quantum-classical optimization algorithms for machine learning. The most brilliant research direction in the current NISQ (noisy intermediate-scale quantum) era is mixed systems leveraging classical computing capability augmented by quantum capacity. In such designs, computationally intensive subroutines like optimization for global optima or escape from local minima are left to quantum processors and data preprocessing, iterative refinement, and model testing are performed by classical processors. We start by defining the fundamental principles of quantum computing and acquainting readers with the fundamentals like qubits, quantum gates, and quantum circuits. These are the building blocks for quantum algorithms, which hopefully can be utilized for optimization, for example, the QAOA (quantum approximate optimization algorithm), the VQE (variational quantum Eigensolver), and quantum annealing. They all build to optimize the problems in some way differently: QAOA through parameterized quantum circuits, VQE through approximations of ground states of the Hamiltonians, and quantum annealing through quantum evolution to low-energy solutions. We then form a loop between these quantum algorithms and regular machine learning tasks. Optimization impacts virtually all ML phases, from training neural networks, where one reduces loss functions; feature selection, where one aims to discover the most relevant subset of input variables; and hyperparameter optimization, which often involves expensive search procedures over extremely large combinatorial spaces. Quantum optimization techniques can therefore, in principle, accelerate such procedures through computational complexity reduction and enhancing solutions over challenging landscapes. Realistic simulations and case studies emphasize the degree to which hybrid quantum-classical algorithms have been employed in solving ML problems, that is, QSVMs, quantum-enhanced clustering, and quantum reinforcement learning. Despite their experimental nature, these applications already show encouraging results in terms of improved performance metrics as well as reduction in time-to-solution compared to classical-only solutions. Notwithstanding that, this new field is not bereft of real challenges. Quantum hardware suffers from hard limits on qubit fidelity, decoherence, and scalability in the current times. Furthermore, quantum circuit integration into broad ML pipelines requires novel architectures and software platforms. As a counter to those issues, this chapter surveys recent advances in quantum programming languages, simulators, and hybrid platforms like PennyLane, Qiskit, and TensorFlow Quantum. We conclude by looking to the future possibility that quantum optimization will revolutionize the machine learning landscape. As hardware improves and algorithmic complexity increases, the possibility of quantum-powered learning systems solving hard problems becomes increasingly accessible. From speeding up deep network training to enabling new types of unsupervised learning, quantum optimization holds the potential to recast computational tractability in machine learning. This chapter is an all-encompassing introduction and state-of-the-art work for students, researchers, and practitioners interested in using quantum optimization for practical ML applications. It sets the foundation for understanding how to pursue hybrid approaches (how they may be applied today in the real world and bracing oneself for disruption in the future).
This chapter offers a clear and accessible introduction to the field of quantum-inspired neural computing. This area combines ideas from quantum mechanics with traditional neural network models to create new possibilities in machine learning and optimization metaheuristic algorithms. Unlike actual quantum computing, quantum-inspired approaches work on classical computers but borrow key concepts like superposition, entanglement, and probabilistic behaviour to improve how neural networks learn and process information. The study begins by explaining why researchers are interested in using quantum principles in neural networks. It then introduces the basic mathematical ideas behind these concepts, such as quantum states and transformations, and shows how they can be used to design more powerful and flexible learning systems. It focuses on discussing current models, outlining the distinctions between quantum-inspired and true quantum systems, and evaluating the potential benefits and challenges of this approach. In addition, Quantum-inspired neural computing introduces a novel framework that bridges classical artificial intelligence with principles derived from quantum mechanics. By simulating quantum behaviours on classical hardware, it opens new avenues for developing more adaptive and efficient learning models. This emerging approach offers promising advantages in areas such as optimization, parallel processing, and high-dimensional data representation. Finally, despite its potential, the field faces notable limitations and challenges. These include the lack of limited theoretical formalization, and difficulties in scaling models for complex real-world applications. Moreover, distinguishing genuine quantum-inspired advantages from conservative algorithmic improvements remains an current research apprehension.
Machine learning is a branch of Artificial Intelligence that seeks to make machines learn from data. It is being applied for solving real world problems with huge amount of data. Though, Machine Learning is receiving wide acceptance, however, execution time is one of the major concerns in practical implementations of Machine Learning techniques. It largely comprises of a set of techniques that trains a model by reducing the error between the desired or actual outcome and an estimated or predicted outcome, which is often called as loss function. Thus, training in machine learning techniques often requires solving a difficult optimization problem, which is the most expensive step in the entire model-building process and its applications. One of the possible solutions in near future for reducing execution time of training process in Machine learning techniques is to implement them on quantum computers instead of classical computers. It is conjectured that quantum computers may be exponentially faster than classical computers for solving problems which involve matrix operations. Some of the machine learning techniques like support vector machines make extensive use of matrices, which can be made faster by implementing them on quantum computers. However, their efficient implementation is non-trivial and requires existence of quantum memories. Thus, another possible solution in near term is to use a hybrid of Classical Quantum approach, where a machine learning model is implemented in classical computer but the optimization of loss function during training is performed on quantum computer instead of classical computer. Several Quantum optimization algorithms have been proposed in recent years, which can be classified as gradient based and gradient free optimization techniques. Gradient based techniques require the nature of optimization problem being solved to be convex, continuous and differentiable otherwise if the problem is non-convex then they can find local optima only whereas gradient free optimization techniques work well even with non-continuous, non-linear and nonconvex optimization problems. This chapter discusses a global optimization technique based on Adiabatic Quantum Computation (AQC) to solve minimization of loss function without any restriction on its structure and the underlying model, which is being learned. Further, it is also shown that in the proposed framework, AQC based approach would be superior to circuit-based approach in solving global optimization problems.
With the rapid development of artificial intelligence technology, some shortcomings in the traditional machine learning model have begun to show. With the growth of the quantity of data, too many parameters have become a problem and traditional silicon-based computers are near their physical limits. Quantum machine learning is a new way that will be able to advance the development of artificial intelligence significantly in the future. This paper will introduce and investigate the promoting effect of quantum machine learning on the overall development process of artificial intelligence. By learning about the differences between traditional neural networks and quantum-enhanced algorithms, this paper will show that superposition and entanglement in quantum mechanics can solve computational problems that have been very difficult to solve for a long time. Also, based on the above, it will be explored how quantum feature spaces and quantum neural networks can significantly boost the speed of data processing and pattern recognition. Based on the above results, it is hoped that quantum machine learning will be able to provide a new computing system with enhanced capabilities compared with the current model and thus advance the continuous development of artificial intelligence.