[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-117417-en":3,"doc-seo-117417-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},117417,7971461740909,"Levi","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Quantum Dynamics of Machine Learning","The paper derives a quantum dynamic equation (QDE) for machine learning by leveraging the Schrödinger equation and a potential-energy equivalence relationship. A Wick rotation is then used to connect quantum dynamics with thermodynamics. The resulting formulation converts the iterative learning process into a time-dependent partial differential equation with a clear mathematical structure. Using this framework, the diffusion model and activation functions such as Softmax and Sigmoid are analyzed to validate the proposed equations, supporting a rigorous theoretical basis and potential quantum-computer implementation.","QUANTUM DYNAMICS OF MACHINE LEARNING  \narXiv :2407 . 19890v1 [ quant-ph] 7 Jul 2024  \nPeng Wang B  \nChengdu Institution of Computer Application Chinese Academy of Sciences Chengdu, 610213, Sichuan, China.  \n[B](B wp002005@163.com)[ wp002005@163.com](B wp002005@163.com)  \nMaimaitiniyazi Maimaitiabudula  \nSouthwest Minzu University Chengdu, 610213, Sichuan, China. [mai1232021@163.com](mai1232021@163.com)  \nABSTRACT  \nThe quantum dynamic equation (QDE) of machine learning is obtained based on Schrödinger equation and potential energy equivalence relationship. Through Wick rotation, the relationship between quantum dynamics and thermodynamics is also established in this paper. This equation reformulates the iterative process of machine learning into a time-dependent partial differential equation with a clear mathematical structure, offering a theoretical framework for investigating machine learning iterations through quantum and mathematical theories. Within this framework, the fundamental iterative process, the diffusion model, and the Softmax and Sigmoid functions are examined, validating the proposed quantum dynamics equations. This approach not only presentsa rigorous theoretical foundation for machine learning but also holds promise for supporting the implementation of machine learning algorithms on quantum computers.  \nKeywords Quantum Dynamics · Machine Learning · Diffusion Model · Quantum Dynamics Framework · Schrödinger Equation · Quantum Dynamics Equation  \n1 Introduction  \nMachine learning is a typical optimisation problem, and its learning process is an iterative optimisation process in the parameter space. It is a natural way of thinking to consider the iterative motion process of this algorithm as a kinetic process. The theory of dynamics has been developed over a long period of time and is very complete, with quantum dynamics, Newtonian dynamics, thermodynamics, electrodynamics and molecular dynamics, which theoretically describes the laws of motion by establishing a set of kinetic equations. The establishment of the dynamics theory of machine learning is expected to address the lack of theoretical models in this field, thus advancing the development of machine learning theory and applications.  \nThe theoretical modeling of optimization problems using a dynamics approach was pioneered by Metropolis in 1953, drawing from thermodynamics [1] . This concept was later adapted by Kirkpatrick et al. for the simulated annealing algorithm, an early attempt to apply thermodynamics to optimization [2] . In 1994, Finnila et al. introduced a groundbreaking method that treated the objective function as a potential within the Schrödinger equation, transforming the optimization problem into one of finding a bound-state quantum ground state wave function. This marked the first application of quantum dynamics theory to optimization. In 2013, our research began exploring the potential of a quantum dynamics framework for optimization problems. Our results demonstrated that the Schrödinger equation can effectively describe the fundamental iterative process of optimization algorithms [3, 4, 5, 6] . Further evidence of the successful application of quantum dynamics in artificial intelligence includes the D-Wave quantum computer, which utilized quantum annealing to solve an intelligent optimization problem, marking a milestone in commercial quantum computing [7] .  \nIn 2015, Jascha Sohl-Dickstein and colleagues introduced the Diffusion Probabilistic Model, inspired by non-equilibrium thermodynamic principles[8] . This model has since undergone rapid development and found extensive applications. The Diffusion Model exemplifies the successful application of kinetic theory to machine learning. Recently, dynamics theory has shown growing theoretical and practical value in this field. New methods based on dynamics, such as those utilizing stochastic differential equations, have been developed and consistently validated through experimental resu","cbCaigxVv7BTcGdM","https://ap.wps.com/l/cbCaigxVv7BTcGdM","pdf",15717388,1,10,"English","en",105,"# Introduction\n## Dynamics-based modeling for machine learning optimization\n## From thermodynamics and simulated annealing to quantum dynamics\n# Quantum Dynamics Framework for Optimization Problems\n## Multi-scale Quantum Harmonic Oscillator Algorithm (MQHOA)\n## Schrödinger equation and potential-energy equivalence\n## Quantum Dynamic Framework figure interpretation","[{\"question\":\"How does the paper derive the quantum dynamic equation for machine learning?\",\"answer\":\"It starts from the Schrödinger equation and establishes a potential energy equivalence relationship, then formulates the quantum dynamic equation (QDE) for the learning process.\"},{\"question\":\"What role does Wick rotation play in the framework?\",\"answer\":\"Wick rotation is used to establish a relationship between quantum dynamics and thermodynamics, linking the dynamics formulation to thermodynamic descriptions.\"},{\"question\":\"Which machine learning components are examined to validate the proposed equations?\",\"answer\":\"The diffusion model and activation functions including Softmax and Sigmoid are examined within the quantum dynamics framework to validate the derived quantum dynamics equations.\"}]","Quantum Dynamics of Machine Learning | PDF",1785675770,25,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"quantum-dynamics-of-machine-learning","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/quantum-dynamics-of-machine-learning/117417/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-02",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"How does the paper derive the quantum dynamic equation for machine learning?","Question",{"text":75,"@type":76},"It starts from the Schrödinger equation and establishes a potential energy equivalence relationship, then formulates the quantum dynamic equation (QDE) for the learning process.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What role does Wick rotation play in the framework?",{"text":80,"@type":76},"Wick rotation is used to establish a relationship between quantum dynamics and thermodynamics, linking the dynamics formulation to thermodynamic descriptions.",{"name":82,"@type":73,"acceptedAnswer":83},"Which machine learning components are examined to validate the proposed equations?",{"text":84,"@type":76},"The diffusion model and activation functions including Softmax and Sigmoid are examined within the quantum dynamics framework to validate the derived quantum dynamics equations.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,134],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":21,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":21,"slug":133},"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]