Many-Sorted Algebras for Deep Learning and Quantum Technology / Charles R. Giardina.

Format:

Book

Author/Creator:

Giardina, Charles R., 1942- author.

Language:

English

Subjects (All):

Computer science--Mathematics.

Computer science.

Machine learning.

Quantum computing.

Physical Description:

1 online resource (423 pages)

Place of Publication:

Cambridge, MA : Mara Conner, [2024]

Summary:

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.

Contents:

Front Cover

Many-Sorted Algebras for Deep Learning and Quantum Technology

Copyright Page

Dedication

Contents

List of figures

Preface

Acknowledgments

1 Introduction to quantum many-sorted algebras

1.1 Introduction to quantum many-sorted algebras

1.1.1 Algebraic structures

1.1.2 Many-sorted algebra methodology

1.1.3 Global field structure

1.1.4 Global algebraic structures in quantum and in machine learning

1.1.5 Specific machine learning field structure

1.1.6 Specific quantum field structure

1.1.7 Vector space as many-sorted algebra

1.1.8 Fundamental illustration of MSA in quantum

1.1.9 Time-limited signals as an inner product space

1.1.10 Kernel methods in real Hilbert spaces

1.1.11 R-Modules

References

2 Basics of deep learning

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

3 Basic algebras underlying quantum and NN mechanisms

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.

4 Quantum Hilbert spaces and their creation

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

5 Quantum and machine learning applications involving matrices

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

6 Quantum annealing and adiabatic quantum computing

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

7 Operators on Hilbert space

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

8 Spaces and algebras for quantum operators.

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

9 Von Neumann algebra

9.1 Operator topologies

9.2 Two basic von Neumann algebras

9.3 Commutant in a von Neumann algebra

9.4 The Gelfand transform

10 Fiber bundles

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

11 Lie algebras and Lie groups

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

12 Fundamental and universal covering groups

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

13 Spectra for operators

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

14 Canonical commutation relations

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&amp

lowast

algebra

15 Fock space

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

16 Underlying theory for quantum computing

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

17 Quantum computing applications

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

18 Machine learning and data mining

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

19 Reproducing kernel and other Hilbert spaces

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.

Notes:

Includes bibliographical references and index.

Description based on print version record.

ISBN:

9780443136986

044313698X

OCLC:

1420007435