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.
1. Chapter 1 Introduction to quantum many-sorted algebras - p. 1 2. Chapter 2 Basics of deep learning - p. 21 3. Chapter 3 Basic algebras underlying quantum and NN mechanisms - p. 41 4. Chapter 4 Quantum Hilbert spaces and their creation - p. 57 5. Chapter 5 Quantum and machine learning applications involving matrices - p. 83 6. Chapter 6 Quantum annealing and adiabatic quantum computing - p. 103 7. Chapter 7 Operators on Hilbert space - p. 131 8. Chapter 8 Spaces and algebras for quantum operators - p. 145 9. Chapter 9 Von Neumann algebra - p. 165 10. Chapter 10 Fiber bundles - p. 171 11. Chapter 11 Lie algebras and Lie groups - p. 191 12. Chapter 12 Fundamental and universal covering groups - p. 217 13. Chapter 13 Spectra for operators - p. 231 14. Chapter 14 Canonical commutation relations - p. 255 15. Chapter 15 Fock space - p. 271 16. Chapter 16 Underlying theory for quantum computing - p. 291 17. Chapter 17 Quantum computing applications - p. 307 18. Chapter 18 Machine learning and data mining - p. 327 19. Chapter 19 Reproducing kernel and other Hilbert spaces - p. 349 20. Appendix A Hilbert space of wraparound digital signals - p. 365 21. Appendix B Many-sorted algebra for the description of a measurable and measure spaces - p. 369 22. Appendix C Elliptic curves and Abelian group structure - p. 373 23. Appendix D Young diagrams - p. 377 24. Appendix E Young diagrams and the symmetric group - p. 379 25. Appendix F Fundamental theorems in functional analysis - p. 383 26. Appendix G Sturm–Liouville differential equations and consequences - p. 387 27. Index - p. 391
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.
1. Chapter 1 Introduction to quantum many-sorted algebras - p. 1 2. Chapter 2 Basics of deep learning - p. 21 3. Chapter 3 Basic algebras underlying quantum and NN mechanisms - p. 41 4. Chapter 4 Quantum Hilbert spaces and their creation - p. 57 5. Chapter 5 Quantum and machine learning applications involving matrices - p. 83 6. Chapter 6 Quantum annealing and adiabatic quantum computing - p. 103 7. Chapter 7 Operators on Hilbert space - p. 131 8. Chapter 8 Spaces and algebras for quantum operators - p. 145 9. Chapter 9 Von Neumann algebra - p. 165 10. Chapter 10 Fiber bundles - p. 171 11. Chapter 11 Lie algebras and Lie groups - p. 191 12. Chapter 12 Fundamental and universal covering groups - p. 217 13. Chapter 13 Spectra for operators - p. 231 14. Chapter 14 Canonical commutation relations - p. 255 15. Chapter 15 Fock space - p. 271 16. Chapter 16 Underlying theory for quantum computing - p. 291 17. Chapter 17 Quantum computing applications - p. 307 18. Chapter 18 Machine learning and data mining - p. 327 19. Chapter 19 Reproducing kernel and other Hilbert spaces - p. 349 20. Appendix A Hilbert space of wraparound digital signals - p. 365 21. Appendix B Many-sorted algebra for the description of a measurable and measure spaces - p. 369 22. Appendix C Elliptic curves and Abelian group structure - p. 373 23. Appendix D Young diagrams - p. 377 24. Appendix E Young diagrams and the symmetric group - p. 379 25. Appendix F Fundamental theorems in functional analysis - p. 383 26. Appendix G Sturm–Liouville differential equations and consequences - p. 387 27. Index - p. 391
