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 the need for a source that gives readers insights 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; hence, this thread is exposed using many-sorted algebras. This book includes hundreds of well-designed examples that 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 book, all laws or equational identities needed in specifying an algebraic structure are precisely described.
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Table of Contents
- 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
