Many-Sorted Algebras for Deep Learning and Quantum Technology

Many-Sorted Algebras for Deep Learning and Quantum Technology

by Charles R. Giardina
Released February 2024
Publisher(s): Morgan Kaufmann
ISBN: 9780443136986

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Book description

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.
  • 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

Table of contents

  1. Cover image
  2. Title page
  3. Table of Contents
  4. Copyright
  5. Dedication
  6. List of figures
  7. Preface
  8. Acknowledgments
  9. Chapter 1. Introduction to quantum many-sorted algebras
    1. Abstract
    2. 1.1 Introduction to quantum many-sorted algebras
    3. References
  10. Chapter 2. Basics of deep learning
    1. Abstract
    2. 2.1 Machine learning and data mining
    3. 2.2 Deep learning
    4. 2.3 Deep learning and relationship to quantum
    5. 2.4 Affine transformations for nodes within neural net
    6. 2.5 Global structure of neural net
    7. 2.6 Activation functions and cost functions for neural net
    8. 2.7 Classification with a single-node neural net
    9. 2.8 Backpropagation for neural net learning
    10. 2.9 Many-sorted algebra description of affine space
    11. 2.10 Overview of convolutional neural networks
    12. 2.11 Brief introduction to recurrent neural networks
    13. References
  11. Chapter 3. Basic algebras underlying quantum and NN mechanisms
    1. Abstract
    2. 3.1 From a vector space to an algebra
    3. 3.2 An algebra of time-limited signals
    4. 3.3 The commutant in an algebra
    5. 3.4 Algebra homomorphism
    6. 3.5 Hilbert space of wraparound digital signals
    7. 3.6 Many-sorted algebra description of a Banach space
    8. 3.7 Banach algebra as a many-sorted algebra
    9. 3.8 Many-sorted algebra for Banach* and C* algebra
    10. 3.9 Banach* algebra of wraparound digital signals
    11. 3.10 Complex-valued wraparound digital signals
    12. References
  12. Chapter 4. Quantum Hilbert spaces and their creation
    1. Abstract
    2. 4.1 Explicit Hilbert spaces underlying quantum technology
    3. 4.2 Complexification
    4. 4.3 Dual space used in quantum
    5. 4.4 Double dual Hilbert space
    6. 4.5 Outer product
    7. 4.6 Multilinear forms, wedge, and interior products
    8. 4.7 Many-sorted algebra for tensor vector spaces
    9. 4.8 The determinant
    10. 4.9 Tensor algebra
    11. 4.10 Many-sorted algebra for tensor product of Hilbert spaces
    12. 4.11 Hilbert space of rays
    13. 4.12 Projective space
    14. References
  13. Chapter 5. Quantum and machine learning applications involving matrices
    1. Abstract
    2. 5.1 Matrix operations
    3. 5.2 Qubits and their matrix representations
    4. 5.3 Complex representation for the Bloch sphere
    5. 5.4 Interior, exterior, and Lie derivatives
    6. 5.5 Spectra for matrices and Frobenius covariant matrices
    7. 5.6 Principal component analysis
    8. 5.7 Kernel principal component analysis
    9. 5.8 Singular value decomposition
    10. References
  14. Chapter 6. Quantum annealing and adiabatic quantum computing
    1. Abstract
    2. 6.1 Schrödinger’s characterization of quantum
    3. 6.2 Quantum basics of annealing and adiabatic quantum computing
    4. 6.3 Delta function potential well and tunneling
    5. 6.4 Quantum memory and the no-cloning theorem
    6. 6.5 Basic structure of atoms and ions
    7. 6.6 Overview of qubit fabrication
    8. 6.7 Trapped ions
    9. 6.8 Super-conductance and the Josephson junction
    10. 6.9 Quantum dots
    11. 6.10 D-wave adiabatic quantum computers and computing
    12. 6.11 Adiabatic theorem
    13. Reference
    14. Further reading
  15. Chapter 7. Operators on Hilbert space
    1. Abstract
    2. 7.1 Linear operators, a MSA view
    3. 7.2 Closed operators in Hilbert spaces
    4. 7.3 Bounded operators
    5. 7.4 Pure tensors versus pure state operators
    6. 7.5 Trace class operators
    7. 7.6 Hilbert-Schmidt operators
    8. 7.7 Compact operators
    9. References
  16. Chapter 8. Spaces and algebras for quantum operators
    1. Abstract
    2. 8.1 Banach and Hilbert space rank, boundedness, and Schauder bases
    3. 8.2 Commutative and noncommutative Banach algebras
    4. 8.3 Subgroup in a Banach algebra
    5. 8.4 Bounded operators on a Hilbert space
    6. 8.5 Invertible operator algebra criteria on a Hilbert space
    7. 8.6 Spectrum in a Banach algebra
    8. 8.7 Ideals in a Banach algebra
    9. 8.8 Gelfand-Naimark-Segal construction
    10. 8.9 Generating a C* algebra
    11. 8.10 The Gelfand formula
    12. References
  17. Chapter 9. Von Neumann algebra
    1. Abstract
    2. 9.1 Operator topologies
    3. 9.2 Two basic von Neumann algebras
    4. 9.3 Commutant in a von Neumann algebra
    5. 9.4 The Gelfand transform
    6. References
  18. Chapter 10. Fiber bundles
    1. Abstract
    2. 10.1 MSA for the algebraic quotient spaces
    3. 10.2 The topological quotient space
    4. 10.3 Basic topological and manifold concepts
    5. 10.4 Fiber bundles from manifolds
    6. 10.5 Sections in a fiber bundle
    7. 10.6 Line and vector bundles
    8. 10.7 Analytic vector bundles
    9. 10.8 Elliptic curves over C
    10. 10.9 The quaternions
    11. 10.10 Hopf fibrations
    12. 10.11 Hopf fibration with bloch sphere S2, the one-qubit base
    13. 10.12 Hopf fibration with sphere S4, the two-qubit base
    14. References
  19. Chapter 11. Lie algebras and Lie groups
    1. Abstract
    2. 11.1 Algebraic structure
    3. 11.2 MSA view of a Lie algebra
    4. 11.3 Dimension of a Lie algebra
    5. 11.4 Ideals in a Lie algebra
    6. 11.5 Representations and MSA of a Lie group of a Lie algebra
    7. 11.6 Briefing on topological manifold properties of a Lie group
    8. 11.7 Formal description of matrix Lie groups
    9. 11.8 Mappings between Lie groups and Lie algebras
    10. 11.9 Complexification of Lie algebras
    11. References
  20. Chapter 12. Fundamental and universal covering groups
    1. Abstract
    2. 12.1 Homotopy a graphical view
    3. 12.2 Initial point equivalence for loops
    4. 12.3 MSA description of the fundamental group
    5. 12.4 Illustrating the fundamental group
    6. 12.5 Homotopic equivalence for topological spaces
    7. 12.6 The universal covering group
    8. 12.7 The Cornwell mapping
    9. References
  21. Chapter 13. Spectra for operators
    1. Abstract
    2. 13.1 Spectral classification for bounded operators
    3. 13.2 Spectra for operators on a Banach space
    4. 13.3 Symmetric, self-adjoint, and unbounded operators
    5. 13.4 Bounded operators and numerical range
    6. 13.5 Self-adjoint operators
    7. 13.6 Normal operators and nonbounded operators
    8. 13.7 Spectral decomposition
    9. 13.8 Spectra for self-adjoint, normal, and compact operators
    10. 13.9 Pure states and density functions
    11. 13.10 Spectrum and resolvent set
    12. 13.11 Spectrum for nonbounded operators
    13. 13.12 Brief descriptions of spectral measures and spectral theorems
    14. References
  22. Chapter 14. Canonical commutation relations
    1. Abstract
    2. 14.1 Isometries and unitary operations
    3. 14.2 Canonical hypergroups—a multisorted algebra view
    4. 14.3 Partial isometries
    5. 14.4 Multisorted algebra for partial isometries
    6. 14.5 Stone’s theorem
    7. 14.6 Position and momentum
    8. 14.7 The Weyl form of the canonical commutation relations and the Heisenberg group
    9. 14.8 Stone-von Neumann and quantum mechanics equivalence
    10. 14.9 Symplectic vector space—a multisorted algebra approach
    11. 14.10 The Weyl canonical commutation relations C∗ algebra
    12. References
  23. Chapter 15. Fock space
    1. Abstract
    2. 15.1 Particles within Fock spaces and Fock space structure
    3. 15.2 The bosonic occupation numbers and the ladder operators
    4. 15.3 The fermionic Fock space and the fermionic ladder operators
    5. 15.4 The Slater determinant and the complex Clifford space
    6. 15.5 Maya diagrams
    7. 15.6 Maya diagram representation of fermionic Fock space
    8. 15.7 Young diagrams representing quantum particles
    9. 15.8 Bogoliubov transform
    10. 15.9 Parafermionic and parabosonic spaces
    11. 15.10 Segal–Bargmann–Fock operations
    12. 15.11 Many-body systems and the Landau many-body expansion
    13. 15.12 Single-body operations
    14. 15.13 Two-body operations
    15. References
  24. Chapter 16. Underlying theory for quantum computing
    1. Abstract
    2. 16.1 Quantum computing and quantum circuits
    3. 16.2 Single-qubit quantum gates
    4. 16.3 Pauli rotational operators
    5. 16.4 Multiple-qubit input gates
    6. 16.5 The swapping operation
    7. 16.6 Universal quantum gate set
    8. 16.7 The Haar measure
    9. 16.8 Solovay–Kitaev theorem
    10. 16.9 Quantum Fourier transform and phase estimation
    11. 16.10 Uniform superposition and amplitude amplification
    12. 16.11 Reflections
    13. References
  25. Chapter 17. Quantum computing applications
    1. Abstract
    2. 17.1 Deutsch problem description
    3. 17.2 Oracle for Deutsch problem solution
    4. 17.3 Quantum solution to Deutsch problem
    5. 17.4 Deutsch-Jozsa problem description
    6. 17.5 Quantum solution for the Deutsch-Jozsa problem
    7. 17.6 Grover search problem
    8. 17.7 Solution to the Grover search problem
    9. 17.8 The Shor’s cryptography problem from an algebraic view
    10. 17.9 Solution to the Shor’s problem
    11. 17.10 Elliptic curve cryptography
    12. 17.11 MSA of elliptic curve over a finite field
    13. 17.12 Diffie–Hellman EEC key exchange
    14. References
    15. Further reading
  26. Chapter 18. Machine learning and data mining
    1. Abstract
    2. 18.1 Quantum machine learning applications
    3. 18.2 Learning types and data structures
    4. 18.3 Probably approximately correct learning and Vapnik-Chervonenkis dimension
    5. 18.4 Regression
    6. 18.5 K-nearest neighbor classification
    7. 18.6 K-nearest neighbor regression
    8. 18.7 Quantum K-means applications
    9. 18.8 Support vector classifiers
    10. 18.9 Kernel methods
    11. 18.10 Radial basis function kernel
    12. 18.11 Bound matrices
    13. 18.12 Convolutional neural networks and quantum convolutional neural networks
    14. References
  27. Chapter 19. Reproducing kernel and other Hilbert spaces
    1. Abstract
    2. 19.1 Algebraic solution to harmonic oscillator
    3. 19.2 Reproducing kernel Hilbert space over C and the disk algebra
    4. 19.3 Reproducing kernel Hilbert space over R
    5. 19.4 Mercer’s theorem
    6. 19.5 Spectral theorems
    7. 19.6 The Riesz-Markov theorem
    8. 19.7 Some nonseparable Hilbert spaces
    9. 19.8 Separable Hilbert spaces are isometrically isomorphic to l2
    10. References
  28. Appendix A. Hilbert space of wraparound digital signals
    1. References
  29. Appendix B. Many-sorted algebra for the description of a measurable and measure spaces
    1. Example B.1
    2. Example B.2
    3. Example B.3
    4. Example B.4
  30. Appendix C. Elliptic curves and Abelian group structure
  31. Appendix D. Young diagrams
    1. Example D.1
    2. References
  32. Appendix E. Young diagrams and the symmetric group
    1. Example E.1
    2. Example E.2
    3. References
  33. Appendix F. Fundamental theorems in functional analysis
    1. Example F.1
    2. Example F.2
  34. Appendix G. Sturm–Liouville differential equations and consequences
    1. Example G.1
    2. Example G.2
    3. Example G.3
    4. Example G.4
    5. Example G.5
    6. Example G.6
    7. Example G.7
  35. Index

Product information

  • Title: Many-Sorted Algebras for Deep Learning and Quantum Technology
  • Author(s): Charles R. Giardina
  • Release date: February 2024
  • Publisher(s): Morgan Kaufmann
  • ISBN: 9780443136986