book

Many-Sorted Algebras for Deep Learning and Quantum Technology

Name: Many-Sorted Algebras for Deep Learning and Quantum Technology
Author: Charles R. Giardina
ISBN: 9780443136986

by Charles R. Giardina

February 2024

Intermediate to advanced

350 pages

14h 46m

English

Morgan Kaufmann

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Cover image
Title page
Table of Contents
Copyright
Dedication
List of figures
Preface
Acknowledgments
Chapter 1. Introduction to quantum many-sorted algebras
Abstract1.1 Introduction to quantum many-sorted algebrasReferences
Chapter 2. Basics of deep learning
Abstract2.1 Machine learning and data mining2.2 Deep learning2.3 Deep learning and relationship to quantum2.4 Affine transformations for nodes within neural net2.5 Global structure of neural net2.6 Activation functions and cost functions for neural net2.7 Classification with a single-node neural net2.8 Backpropagation for neural net learning2.9 Many-sorted algebra description of affine space2.10 Overview of convolutional neural networks2.11 Brief introduction to recurrent neural networksReferences

Chapter 3. Basic algebras underlying quantum and NN mechanisms
Abstract3.1 From a vector space to an algebra3.2 An algebra of time-limited signals3.3 The commutant in an algebra3.4 Algebra homomorphism3.5 Hilbert space of wraparound digital signals3.6 Many-sorted algebra description of a Banach space3.7 Banach algebra as a many-sorted algebra3.8 Many-sorted algebra for Banach* and C* algebra3.9 Banach* algebra of wraparound digital signals3.10 Complex-valued wraparound digital signalsReferences
Chapter 4. Quantum Hilbert spaces and their creation
Abstract4.1 Explicit Hilbert spaces underlying quantum technology4.2 Complexification4.3 Dual space used in quantum4.4 Double dual Hilbert space4.5 Outer product4.6 Multilinear forms, wedge, and interior products4.7 Many-sorted algebra for tensor vector spaces4.8 The determinant4.9 Tensor algebra4.10 Many-sorted algebra for tensor product of Hilbert spaces4.11 Hilbert space of rays4.12 Projective spaceReferences
Chapter 5. Quantum and machine learning applications involving matrices
Abstract5.1 Matrix operations5.2 Qubits and their matrix representations5.3 Complex representation for the Bloch sphere5.4 Interior, exterior, and Lie derivatives5.5 Spectra for matrices and Frobenius covariant matrices5.6 Principal component analysis5.7 Kernel principal component analysis5.8 Singular value decompositionReferences
Chapter 6. Quantum annealing and adiabatic quantum computing
Abstract6.1 Schrödinger’s characterization of quantum6.2 Quantum basics of annealing and adiabatic quantum computing6.3 Delta function potential well and tunneling6.4 Quantum memory and the no-cloning theorem6.5 Basic structure of atoms and ions6.6 Overview of qubit fabrication6.7 Trapped ions6.8 Super-conductance and the Josephson junction6.9 Quantum dots6.10 D-wave adiabatic quantum computers and computing6.11 Adiabatic theoremReferenceFurther reading
Chapter 7. Operators on Hilbert space
Abstract7.1 Linear operators, a MSA view7.2 Closed operators in Hilbert spaces7.3 Bounded operators7.4 Pure tensors versus pure state operators7.5 Trace class operators7.6 Hilbert-Schmidt operators7.7 Compact operatorsReferences
Chapter 8. Spaces and algebras for quantum operators
Abstract8.1 Banach and Hilbert space rank, boundedness, and Schauder bases8.2 Commutative and noncommutative Banach algebras8.3 Subgroup in a Banach algebra8.4 Bounded operators on a Hilbert space8.5 Invertible operator algebra criteria on a Hilbert space8.6 Spectrum in a Banach algebra8.7 Ideals in a Banach algebra8.8 Gelfand-Naimark-Segal construction8.9 Generating a C* algebra8.10 The Gelfand formulaReferences
Chapter 9. Von Neumann algebra
Abstract9.1 Operator topologies9.2 Two basic von Neumann algebras9.3 Commutant in a von Neumann algebra9.4 The Gelfand transformReferences
Chapter 10. Fiber bundles
Abstract10.1 MSA for the algebraic quotient spaces10.2 The topological quotient space10.3 Basic topological and manifold concepts10.4 Fiber bundles from manifolds10.5 Sections in a fiber bundle10.6 Line and vector bundles10.7 Analytic vector bundles10.8 Elliptic curves over C10.9 The quaternions10.10 Hopf fibrations10.11 Hopf fibration with bloch sphere S2, the one-qubit base10.12 Hopf fibration with sphere S4, the two-qubit baseReferences
Chapter 11. Lie algebras and Lie groups
Abstract11.1 Algebraic structure11.2 MSA view of a Lie algebra11.3 Dimension of a Lie algebra11.4 Ideals in a Lie algebra11.5 Representations and MSA of a Lie group of a Lie algebra11.6 Briefing on topological manifold properties of a Lie group11.7 Formal description of matrix Lie groups11.8 Mappings between Lie groups and Lie algebras11.9 Complexification of Lie algebrasReferences
Chapter 12. Fundamental and universal covering groups
Abstract12.1 Homotopy a graphical view12.2 Initial point equivalence for loops12.3 MSA description of the fundamental group12.4 Illustrating the fundamental group12.5 Homotopic equivalence for topological spaces12.6 The universal covering group12.7 The Cornwell mappingReferences
Chapter 13. Spectra for operators
Abstract13.1 Spectral classification for bounded operators13.2 Spectra for operators on a Banach space13.3 Symmetric, self-adjoint, and unbounded operators13.4 Bounded operators and numerical range13.5 Self-adjoint operators13.6 Normal operators and nonbounded operators13.7 Spectral decomposition13.8 Spectra for self-adjoint, normal, and compact operators13.9 Pure states and density functions13.10 Spectrum and resolvent set13.11 Spectrum for nonbounded operators13.12 Brief descriptions of spectral measures and spectral theoremsReferences
Chapter 14. Canonical commutation relations
Abstract14.1 Isometries and unitary operations14.2 Canonical hypergroups—a multisorted algebra view14.3 Partial isometries14.4 Multisorted algebra for partial isometries14.5 Stone’s theorem14.6 Position and momentum14.7 The Weyl form of the canonical commutation relations and the Heisenberg group14.8 Stone-von Neumann and quantum mechanics equivalence14.9 Symplectic vector space—a multisorted algebra approach14.10 The Weyl canonical commutation relations C∗ algebraReferences
Chapter 15. Fock space
Abstract15.1 Particles within Fock spaces and Fock space structure15.2 The bosonic occupation numbers and the ladder operators15.3 The fermionic Fock space and the fermionic ladder operators15.4 The Slater determinant and the complex Clifford space15.5 Maya diagrams15.6 Maya diagram representation of fermionic Fock space15.7 Young diagrams representing quantum particles15.8 Bogoliubov transform15.9 Parafermionic and parabosonic spaces15.10 Segal–Bargmann–Fock operations15.11 Many-body systems and the Landau many-body expansion15.12 Single-body operations15.13 Two-body operationsReferences
Chapter 16. Underlying theory for quantum computing
Abstract16.1 Quantum computing and quantum circuits16.2 Single-qubit quantum gates16.3 Pauli rotational operators16.4 Multiple-qubit input gates16.5 The swapping operation16.6 Universal quantum gate set16.7 The Haar measure16.8 Solovay–Kitaev theorem16.9 Quantum Fourier transform and phase estimation16.10 Uniform superposition and amplitude amplification16.11 ReflectionsReferences
Chapter 17. Quantum computing applications
Abstract17.1 Deutsch problem description17.2 Oracle for Deutsch problem solution17.3 Quantum solution to Deutsch problem17.4 Deutsch-Jozsa problem description17.5 Quantum solution for the Deutsch-Jozsa problem17.6 Grover search problem17.7 Solution to the Grover search problem17.8 The Shor’s cryptography problem from an algebraic view17.9 Solution to the Shor’s problem17.10 Elliptic curve cryptography17.11 MSA of elliptic curve over a finite field17.12 Diffie–Hellman EEC key exchangeReferencesFurther reading
Chapter 18. Machine learning and data mining
Abstract18.1 Quantum machine learning applications18.2 Learning types and data structures18.3 Probably approximately correct learning and Vapnik-Chervonenkis dimension18.4 Regression18.5 K-nearest neighbor classification18.6 K-nearest neighbor regression18.7 Quantum K-means applications18.8 Support vector classifiers18.9 Kernel methods18.10 Radial basis function kernel18.11 Bound matrices18.12 Convolutional neural networks and quantum convolutional neural networksReferences
Chapter 19. Reproducing kernel and other Hilbert spaces
Abstract19.1 Algebraic solution to harmonic oscillator19.2 Reproducing kernel Hilbert space over C and the disk algebra19.3 Reproducing kernel Hilbert space over R19.4 Mercer’s theorem19.5 Spectral theorems19.6 The Riesz-Markov theorem19.7 Some nonseparable Hilbert spaces19.8 Separable Hilbert spaces are isometrically isomorphic to l2References
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.1Example B.2Example B.3Example B.4
Appendix C. Elliptic curves and Abelian group structure
Appendix D. Young diagrams
Example D.1References
Appendix E. Young diagrams and the symmetric group
Example E.1Example E.2References
Appendix F. Fundamental theorems in functional analysis
Example F.1Example F.2
Appendix G. Sturm–Liouville differential equations and consequences
Example G.1Example G.2Example G.3Example G.4Example G.5Example G.6Example G.7
Index

Content preview from Many-Sorted Algebras for Deep Learning and Quantum Technology

Appendix A

Hilbert space of wraparound digital signals

The set of all wraparound digital bound vectors and bound matrices both form Hilbert spaces. The inner product is evaluated on the intersection of their co-zero sets. For the empty intersection, this product is zero. Completeness follows using Cauchy sequences (CSs). In these cases, it was shown that they form CSs of bound vectors or bound matrices f_n. The objective is to show f_n converges, say to f, in the one norm; other norms follow in the same manner. The first thing to notice is that if f_n → f, each element v_n within f_n will converge to say v. The elements v_n are tuples within the bound vector, or matrix entries v_n, within the bound matrix f_n. Convergence in this manner follows since ...

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Many-Sorted Algebras for Deep Learning and Quantum Technology

by Charles R. Giardina

Hilbert space of wraparound digital signals

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