Quantum Neural Computation

1 Introduction 1.1 Neurodynamics 1.2 Quantum Computation 1.3 Discrete Quantum Computers 1.4 Topological Quantum Computers 1.5 Computation at the Edge of Chaos and Quantum Neural Networks 1.6 Adaptive Path Integral: An 1-Dimensional QNN 1.6.1 Computational Partition Function 1.6.2 From Thermodynamics to Quantum Field Theory 1.6.3 1-Dimensional QNNs 1.7 Brain Topology vs. Small--World Topology 1.8 Quantum Brain and Mind 1.8.1 Connectionism, Control Theory and Brain Theory 1.8.2 Neocortical Biophysics 1.8.3 Quantum Neurodynamics 1.8.4 Bi-Stable Perception and Consciousness 1.9 Notational Conventions 2 Brain and Classical Neural Networks 2.1 Human Brain 2.1.1 Basics of Brain Physiology 2.1.2 Modern 3D Brain Imaging 2.2 Biological versus Artificial Neural Networks 2.2.1 Common Discrete ANNs 2.2.2 Common Continuous ANNs 2.3 Synchronization in Neurodynamics 2.3.1 Phase Synchronization in Coupled Chaotic Oscillators 2.3.2 Oscillatory Phase Neurodynamics 2.3.3 Kuramoto Synchronization Model 2.3.4 Lyapunov Chaotic Synchronization 2.4 Spike Neural Networks and Wavelet Resonance 2.4.1 Ensemble Neuron Model 2.4.2 Wavelet Neurodynamics 2.4.3 Wavelets of Epileptic Spikes 2.5 Human Motor Control and Learning 2.5.1 Motor Control 2.5.2 Human Memory 2.5.3 Human Learning 2.5.4 Spinal Musculo--Skeletal Control 2.5.5 Cerebellum and Muscular Synergy 3 Quantum Theory Basics 3.1 Basics of Non-Relativistic Quantum Mechanics 3.1.1 Soft Introduction to Quantum Mechanics 3.1.2 Quantum States and Operators 3.1.3 The Tree Standard Quantum Pictures 3.1.4 Dirac's Probability Amplitude and Perturbation 3.1.5 State--Space for n Non-Relativistic Quantum Particles 3.2 Introduction to Quantum Fields 3.2.1 Amplitude, Relativistic Invariance and Causality 3.2.2 Gauge Theories 3.2.3 Free and Interacting Field Theories 3.2.4 Dirac's Quantum Electrodynamics (QED) 3.2.5 Abelian Higgs Model 3.2.6 Topological Quantum Computation 3.3 The Feynman Path Integral 3.3.1 The Action--Amplitude Formalism 3.3.2 Correlation Functions and Generating Functional 3.3.3 Quantization of the Electromagnetic Field 3.3.4 Wavelet--Based QFT 3.4 The Path--Integral TQFT 3.4.1 Schwarz--Type and Witten--Type Theories 3.4.2 Hodge Decomposition Theorem 3.4.3 Hodge Decomposition and Chern--Simons Theory 3.5 Non-Abelian Gauge Theories 3.5.1 Introduction to Non-Abelian Theories 3.5.2 Yang--Mills Theory 3.5.3 Quantization of Yang--Mills theory 3.5.4 Basics of Conformal Field Theory (CFT) 4 Spatio-Temporal Chaos, Solitons and NLS 4.1 Reaction--Diffusion Processes and Ricci Flow 4.1.1 Bio--Reaction--Diffusion Systems 4.1.2 Reactive Neurodynamics 4.1.3 Dissipative Evolution Under the Ricci Flow 4.2 Turbulence and Chaos in PDEs 4.3 Quantum Chaos and Its Control 4.3.1 Quantum Chaos vs. Classical Chaos 4.3.2 Optimal Control of Quantum Chaos 4.4 Solitions 4.4.1 Short History of Solitons 4.4.2 Lie--Poisson Bracket 4.4.3 Solitons and Muscular Contraction 4.5 Dispersive Wave Equations and Stability of Solitons 4.5.1 KdV Solitons 4.5.2 The Inverse Scattering Approach 4.6 Nonlinear Schrodinger Equation (NLS) 4.6.1 Cubic NLS 4.6.2 Nonlinear Wave and Schrodinger Equations 4.6.3 Physical NLS--Derivation 4.6.4 A Compact Attractor for High--Dimensional NLS 4.6.5 Finite--Difference Scheme for NLS 4.6.6 Method of Lines for NLS 5 Quantum Brain and Cognition 5.1 Biochemistry of Microtubules 5.2 Kink Soliton Model of MT--Dynamics 5.3 Macro-- and Microscopic Neurodynamical Self--Similarity 5.3.1 Open Liouville Equation 5.4 Dissipative Quantum Brain Model 5.5 QED Brain Model 5.6 Stochastic NLS--Filtering and Robotic Eye Tracking 5.7 Field QNN Application: Cerebellum Motion Controller 5.7.1 Spinal Reflex Control of Biodynamics 5.7.2 Local Muscle--Joint Mechanics 5.7.3 Cerebellum: Adaptive Path--Integral Comparator 5.8 Nonlinear Quantum Psychodynamics 5.8.1 Classical versus Quantum Probability 5.8.2 The Life Space Foam 5.8.3 Geometric Chaos and Topological Phase Transitions 5.8.4 Joint Action of Several Agents 5.8.5 Chaos and Bernstein--Brooks Adaptation 5.9 Entropic Model for Crowd Behavior Dynamics 5.9.1 Cognition and Crowd Behavior 5.9.2 Generic Three--Step Crowd Behavioral Dynamics 5.9.3 Formal Individual, Aggregate and Crowd dynamics 5.9.4 Crowd Entropy, Chaos and Phase Transitions 5.9.5 Crowd Ricci Flow and Perelman Entropy 5.9.6 Chaotic Inter-Phase in Crowd Dynamics 5.9.7 Crowd Phase Transitions 6 Quantum Information, Games and Computation 6.1 Quantum Information and Computing 6.1.1 Entanglement, Teleportation and Information 6.1.2 The Circuit Model for Quantum Computers 6.1.3 Elementary Quantum Algorithms 6.2 Quantum Games 6.2.1 Quantum Strategies 6.2.2 Quantum Games 6.2.3 Two--Qubit Quantum Games 6.2.4 Quantum Cryptography and Quantum Gambling 6.2.5 Formal Quantum Games 6.3 Hardware for Quantum Computers 6.3.1 Josephson Effect and Pendulum Analog 6.3.2 Dissipative Josephson Junction 6.3.3 Josephson Junction Ladder (JJL) 6.3.4 Synchronization in Arrays of Josephson Junctions 6.4 Topological Machinery for Quantum Computation 6.4.1 Non-Abelian Anyons 6.4.2 Emergent Anyons 7 Appendix: Mathematical and Computational Tools 7.1 Meta-Language of Categories and Functors 7.1.1 Maps 7.1.2 Categories 7.1.3 Functors 7.1.4 Natural Transformations 7.1.5 Limits and Colimits 7.1.6 Adjunction 7.1.7 n-Categories and n-Functors 7.1.8 Topological Structure of n-Categories 7.2 Frequently Used Mathematical Concepts 7.2.1 Groups and Related Algebraic Structures 7.2.2 Manifolds, Bundles and Lie Groups 7.2.3 Unitary Matrix and Group 7.2.4 Differential Forms and Stokes Theorem 7.2.5 Symmetry Breaking and Partition Function 7.2.6 Basics of Kalman Filtering 7.2.7 Basics of Wavelet Transforms 7.2.8 Basic of Nonlinear Dynamics and Chaos Theory 7.2.9 Basics of Nash's Game Theory 7.3 Frequently Used Computational Tools 7.3.1 Basic Numerical Algorithms 7.3.2 Numerical Integration of Functions 7.3.3 Numerical Integration of ODEs 7.3.4 Vector--Field and Lyapunov Function 7.3.5 Basics of Qualitative Dynamics 7.3.6 Kuramoto's Neural Model 7.3.7 Boundary Value Problems and PDEs . 7.3.8 Evolution PDEs in MathematicaTM 7.3.9 Fourier Transforms 7.3.10 Sophisticated Random Algorithms 7.3.11 Sophisticated Integration Algorithms References Index