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giardina c many sorted algebras deep learning quantum technology 2024

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Textbook in PDF format Many-Sorted Algebras for Deep Learning and Quantum Technology presents a precise and rigorous description of basic concepts in Quantum technologies and how they relate to Deep Learning and Quantum Theory. Current merging of Quantum Theory and Deep Learning techniques provides a need for a text that can give readers insight into the algebraic underpinnings of these disciplines. Although analytical, topological, probabilistic, as well as geometrical concepts are employed in many of these areas, algebra exhibits the principal thread. This thread is exposed using Many-Sorted Algebras (MSA). In almost every aspect of Quantum Theory as well as Deep Learning more than one sort or type of object is involved. For instance, in Quantum areas Hilbert spaces require two sorts, while in affine spaces, three sorts are needed. Both a global level and a local level of precise specification is described using MSA. At a local level operation involving neural nets may appear to be very algebraically different than those used in Quantum systems, but at a global level they may be identical. Again, MSA is well equipped to easily detail their equivalence through text as well as visual diagrams. Among the reasons for using MSA is in illustrating this sameness. Author Charles R. Giardina includes hundreds of well-designed examples in the text to illustrate the intriguing concepts in Quantum systems. Along with these examples are numerous visual displays. In particular, the Polyadic Graph shows the types or sorts of objects used in Quantum or Deep Learning. It also illustrates all the inter and intra sort operations needed in describing algebras. In brief, it provides the closure conditions. Throughout the text, all laws or equational identities needed in specifying an algebraic structure are precisely described. Specific quantum and Machine Learning fields are presented along with general Hilbert space conditions that underly all quantum methodology. Time-limited signals are developed under inner product space conditions. These signals are basic constructs for convolutional neural networks. Kernel methods, useful in both quantum and Machine Learning disciplines, are presented. In later chapters, kernel methods are shown to be a fundamental ingredient in support vector machines. This chapter ends with a description and application of R modules. These structures have an MSA description almost identical to a vector space structure. Includes hundreds of well-designed examples to illustrate the intriguing concepts in quantum systems Provides precise description of all laws or equational identities that are needed in specifying an algebraic structure Illustrates all the inter and intra sort operations needed in describing algebras Preface Introduction to quantum many-sorted algebra Basics of deep learning Basic algebras underlying quantum and neural net Quantum Hilbert spaces and their creation Quantum and machine learning applications involving matrices Quantum annealing and adiabatic quantum computing Operators on Hilbert space Spaces and algebras for quantum operators Von Neumann algebra Fiber bundles Lie algebras and Lie groups Fundamental and universal covering groups Spectra for operators Canonical commutation relations Fock space Underlying theory for quantum computing Quantum computing applications Machine learning and data mining Reproducing kernel and other Hilbert spaces

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Giardina C. Many-Sorted Algebras for Deep Learning & Quantum Technology 2024.pdf 14.8 MB

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giardina c many sorted algebras deep learning quantum technology 2024

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