Probability for Deep Learning Quantum

A Many-Sorted Algebra View

Charles R. Giardina

• Publisher: Morgan Kaufmann
• Year: 2025
• eBook ISBN: 9780443248351
• Edition: 0
• Topic: Computer Science
• Subtopic: Programming
• Index: Computer Science

Probability for Deep Learning Quantum provides readers with the first book to address probabilistic methods in the deep learning environment and the quantum technological area simultaneously, by using a common platform: the Many-Sorted Algebra (MSA) view. While machine learning is created with a foundation of probability, probability is at the heart of quantum physics as well. It is the cornerstone in quantum applications. These applications include quantum measuring, quantum information theory, quantum communication theory, quantum sensing, quantum signal processing, quantum computing, quantum cryptography, and quantum machine learning. Although some of the probabilistic methods differ in machine learning disciplines from those in the quantum technologies, many techniques are very similar. Probability is introduced in the text rigorously, in Komogorov's vision. It is however, slightly modified by developing the theory in a Many-Sorted Algebra setting. This algebraic construct is also used in showing the shared structures underlying much of both machine learning and quantum theory. Both deep learning and quantum technologies have several probabilistic and stochastic methods in common. These methods are described and illustrated using numerous examples within the text. Concepts in entropy are provided from a Shannon as well as a von-Neumann view. Singular value decomposition is applied in machine learning as a basic tool and presented in the Schmidt decomposition. Besides the in-common methods, Born's rule as well as positive operator valued measures are described and illustrated, along with quasi-probabilities. Author Charles R. Giardina provides clear and concise explanations, accompanied by insightful and thought-provoking visualizations, to deepen your understanding and enable you to apply the concepts to real-world scenarios. - Provides readers with a resource that is loaded with hundreds of well-crafted examples illustrating the difficult concepts pertaining to quantum and stochastic processes - Addresses probabilistic methods in the deep learning environment and in the quantum technological area - Includes a rigorous and precise presentation of the algebraic underpinning of both quantum and deep learning

Table of contents

• Title of Book
• 1 Introduction to a many-sorted algebra view
• 2 Information geometry decision theory
• 3 Symplectic tomographic probability
• 4 The Born rule for quantum probability calculations
• 5 MSA view for a random variable algebra
• 6 Algebra illustrations using probability indicators
• 7 Algebras for complex and quaternion random variables
• 8 MSA for stochastic processes, and large deviation theory
• 9 Probability in Fock space
• 10 Applied probability in quantum
• 11 Entanglement
• 12 Quasiprobability
• 13 Noisy intermediate scale quantum NISQ computing
• 14 Machine learning meets quantum
• Appendix A1. MSA description for a ring, a field, a vector space and an inner product or Hilbert space
• Appendix A2. MSA description for an algebra, a normed vector space, and a Banach space
• Appendix A3. MSA description of a Banach algebra, a Banach* algebra, and a C* algebra
• Appendix A4. MSA description of the quaternion skew field
• Appendix A5. MSA description of Schwartz space and delta functions
• Appendix A6. MSA description of a lattice
• Appendix A7. Classical and quantum probability
• Index