Quantum Mechanics and Bayesian Machines

The following sections are included:

• Preface
• About the Author
• Acknowledgments
• Contents

One of the most important challenges for modern computer science is how to emulate the remarkable cognitive capabilities of the mammalian brain; particularly in situations where there is a necessity for rapidly selecting among multiple possible interpretations for sensory data (cf. Fig. 1.1). At first sight the “machine learning” techniques that are currently used to emulate the information processing capabilities of the mammalian brain would appear to lack any coherent mathematical foundation. It has long been thought [1] that “deep” artificial neural networks (DNNs), which can be either deterministic or probabilistic, might provide a pathway to emulating the cognitive capabilities of the mammalian brain. However, it turns out that these techniques are better suited to interpolation than extrapolation. Moreover, the Herculean efforts typically required to train neural networks for complex reinforcement learning (RL) [2] or feedback control [3] problems have given pause to the assumption that deep neural networks (DNNs) are the best path forward. An alternative view is that probability theory [4–6], and Bayes’s theorem [5–9], provide the best framework for automating pattern recognition and decision-making…

The following sections are included:

• Bayes’s Probability Formula
• The Wiener and Kalman–Bucy Filters
• Bellman’s Dynamic Programming Approach to Optimal Control
• Feynman’s Path Integral Approach to Quantum Mechanics
• Quantum Solution of the Traveling Salesman Problem (TSP)

The following sections are included:

• Bayesian Searches
• A Tale of Two Costs
• Hidden Factors and the Helmholtz Machine

The following sections are included:

• The Hamilton–Jacobi–Bellman Equation
• Pontryagin Maximum Principle
• Lie–Poisson Dynamics
• H_∞ Control

The following sections are included:

• RH Solution of the Airy Equation
• The KdV Equation
• Segal–Wilson Construction
• The NLS Equation
• Galois Remembered

The following sections are included:

• Weyl Remembered
• Helstrom’s Theorem and Universal Hilbert Spaces
• Measurement-based Quantum Computation

The following sections are included:

• Pontryagin Control and Quantum Criticality
• Quantum Theory of Innovations
• Quantum Helmholtz Machine
• Ad Mammalian Intelligence

The following sections are included:

• Quantum Mechanics and 3D Geometry
• Cognitive Science and Quantum Physics

The following sections are included:

• Gaussian Processes
• Wiener–Hopf Methods
• Riemann Surfaces
• The Eightfold Way
• Quantum Theory of Brownian Motion

The following sections are included:

• References
• Index