Many-Sorted Algebras for Deep Learning and Quantum Technology

1st Edition - February 3, 2024
Author: Charles R. Giardina
Language: English
Paperback ISBN:
9 7 8 - 0 - 4 4 3 - 1 3 6 9 7 - 9
eBook ISBN:
9 7 8 - 0 - 4 4 3 - 1 3 6 9 8 - 6

Many-Sorted Algebras for Deep Learning and Quantum Technology presents a precise and rigorousdescription of basic concepts in quantum technologies and how they relate to deep learn… Read more

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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.

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

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Charles R. Giardina