Quantum Computing for the Brain

Contents
Preface
About the Authors
List of Figures
List of Tables
Chapter 1 Introduction to Quantum Neuroscience
1.1 The Brain Is the “Killer Application” of Quantum Computing
1.1.1 The complexity of the brain
1.2 The Brain and Quantum Computing
1.3 Status of Neuroscience
1.3.1 Whole-brain simulation
1.4 Status of Quantum Computing
1.4.1 2n scalability
1.4.2 Three-dimensional format
1.4.3 Quantum advantage over classical computing
1.4.4 Supercomputing versus quantum computing
1.4.5 Quantum finance and AdS/Finance
1.4.5.1 Classical-digital-quantum finance progression
1.5 What This Book Does Not Cover
1.6 Quantum Neuroscience and AdS/Brain
References
Part 1 Foundations
Chapter 2 Neural Signaling Basics
2.1 Scale Levels in the Brain
2.1.1 Relative size of neural entities
2.2 Neural Signaling Overview
2.2.1 Electrical-to-chemical interconnects
2.2.2 Neural signaling energy budget
2.3 Sending Neuron (Presynaptic Terminal)
2.4 Receiving Neuron (Postsynaptic Density)
2.5 Synaptic (Dendritic) Spike Integration
2.5.1 Excitatory and inhibitory postsynaptic potentials
2.5.2 Dendritic pathologies
2.5.3 Dendritic integration filtering
2.5.4 Computational neuroscience and biophysical modeling
2.6 Neural Signaling and Quantum Computing
References
Chapter 3 The AdS/Brain Correspondence
3.1 The AdS/CFT Correspondence
3.1.1 Stating the AdS/CFT correspondence
3.2 AdS/CFT Correspondence Studies
3.2.1 AdS/CFT hybrid approaches
3.2.2 Duality lens
3.3 Applied AdS/CFT
3.3.1 AdS/QCD (quantum chromodynamics)
3.3.2 AdS/CMT (condensed matter theory)
3.3.3 AdS/SYK (SYK model)
3.3.4 AdS/Chaos (thermal systems)
3.3.5 AdS/QIT (quantum information theory)
3.3.6 AdS/TN (tensor networks)
3.3.7 AdS/ML (machine learning)
3.4 AdS/DIY
3.4.1 The AdS/CFT equations
3.4.1.1 The AdS/CFT formalism
3.4.1.2 Correspondence formulation example
3.4.1.3 Listing of AdS/CFT correspondence formulations
References
Chapter 4 Tabletop Experiments
4.1 Black Holes and Quantum Gravity in the Lab
4.2 Particle Accelerator on a Chip
4.3 Quantum Gravity in the Lab
4.3.1 Quantum gravity
4.3.1.1 Justification for quantum gravity in the lab
4.3.2 Wormholes and holographic teleportation
4.3.2.1 Operator size and size winding distribution
4.3.2.2 Holographic teleportation protocols
4.3.3 Preparing the thermofield double state
4.3.3.1 Obtain wormhole-like physics with size winding
4.3.4 Rydberg atoms and trapped ions
4.3.4.1 Rydberg atom arrays
4.3.4.2 Trapped ions
4.4 Black Hole on a Chip
4.4.1 Fast scramblers
4.5 QSims: The SYK Model and Beyond
4.5.1 The SYK model
4.5.2 Tabletop platforms for quantum simulation
4.5.3 Simulation with ultracold gases
4.5.3.1 Large-scale and small-scale physics
4.5.3.2 Ultracold atoms in optical lattices
4.5.3.3 Black hole in a tabletop gas
4.5.4 Simulation with quantum computing
4.5.4.1 SYK simulation with ion traps and circuits
4.5.4.2 Demonstrations with NMR QSims
4.5.4.3 Ryu–Takayanagi entanglement entropy simulation
4.5.4.4 Scrambling Hamiltonian
References
Chapter 5 Neuronal Gauge Theory
5.1 Concept of the Neuronal Gauge Theory
5.1.1 Gauge theory
5.1.1.1 The principle of variational free energy
5.1.1.2 Symmetry, gauge invariance, and the Lagrangian
5.2 Details of the Neuronal Gauge Theory
5.2.1 Rebalancing global symmetry
5.2.1.1 Information-theoretic interpretation
5.2.1.2 Implications
5.2.2 Diffeomorphism invariance
5.2.3 Symmetry and Yang–Mills theory
References
Part 2 Substrate
Chapter 6 Quantum Information Theory
6.1 Quantum Information
6.1.1 Entropy and quantum information
6.1.1.1 Classical information theory
6.1.1.2 Entropy
6.1.2 Superposition, entanglement, and interference
6.1.2.1 Superposition and interference
6.1.2.2 Two-state qubit systems and quantum computing
6.1.2.3 Entanglement and Bell pairs
6.2 Quantum Toolbox
6.2.1 Quantum teleportation
6.2.2 Quantum error correction
6.2.2.1 AdS/QECC
6.2.2.2 QECC example: The three-qutrit code
6.2.3 Out-of-time-order correlators
6.2.4 Quantum walks and Hadamard coins
References
Chapter 7 Quantum Computing 101
7.1 Quantum Algorithms and Quantum Circuits
7.2 Qubit Encoding
7.2.1 Quantum circuit demonstrations
7.3 How Does Quantum Computing Work?
7.3.1 Input, processing, output, repeat
7.3.1.1 Quantum gate logic
7.3.1.2 Setting up a quantum computation
7.3.2 Step 1: Data encoding (embedding)
7.3.2.1 Classical data
7.3.2.2 Quantum data
7.3.3 Step 2: Data processing
7.3.3.1 Circuit architecture
7.3.3.2 Unitary parametrization
7.3.4 Steps 3 and 4: Results and repetition
7.3.4.1 Results measurement with complexity
7.4 Advances in Quantum Computing
7.5 Unitary Transformation
References
Chapter 8 Glia Neurotransmitter Synaptome
8.1 Glial Cells
8.1.1 Astrocyte calcium signaling
8.1.1.1 Tripartite synapse
8.1.1.2 Astrocyte tiling: Nonoverlapping territories
8.1.1.3 Astrocyte signaling: Calcium operations
8.1.1.4 Astrocytes and synapse formation
8.1.2 Glia and neuropathology
8.2 Neurotransmitters and Chemical Signaling
8.2.1 Glutamate (excitatory) and GABA (inhibitory)
8.2.2 Neurotransmitter transport
8.2.2.1 Transport into presynaptic vesicles: Proton gradient
8.2.2.2 Molecular economy
8.2.2.3 Transmitter uptake from cleft: Sodium gradient
8.3 Synaptome
8.3.1 Genome, connectome, and synaptome
8.3.1.1 Quantum computing-level complexity
8.3.1.2 Synapse proteome and neuropathology
8.3.2 Mouse synaptome: Aging pathologies
8.3.2.1 Three phases of development, stability, and decline
8.3.2.2 Synapse diversity and plasticity
8.3.2.3 EPSPs and fMRI data
8.3.3 Alzheimer’s disease synaptome
8.3.3.1 Synaptic plasticity and long-lived proteins in humans
8.3.3.2 Alzheimer’s disease synaptome and intervention
References
Chapter 9 Black Hole Information Theory
9.1 Black Holes
9.1.1 Black holes as a model system
9.1.1.1 Quantum gravity and black hole evaporation
9.1.1.2 Black hole in a box
9.1.2 Hologram decoding dictionaries
9.2 Practical Quantum Communications Protocols
9.2.1 UV–IR information compression
9.2.1.1 UV–IR correlations for interrogating bulk structure
References
Part 3 Connectivity
Chapter 10 Quantum Photonics and High-Dimensional Entanglement
10.1 Quantum Photonics
10.1.1 Technical benefits and qudits
10.1.1.1 Bosons and fermions
10.2 Boson Sampling
10.2.1 Gaussian boson sampling
10.2.1.1 Gaussian boson sampling demonstration
10.2.1.2 Gaussian boson sampling as a NISQ device
10.2.2 Gaussian boson sampling/graph theory
10.2.2.1 Application tools for boson sampling/graph theory
10.3 Space-Division Multiplexing Innovation
10.3.1 Information multiplexing
10.3.1.1 Spacetime states
10.3.2 Personal brain networks
10.4 Photonic Qubit Encoding
10.4.1 Physics: Angular momentum
10.4.1.1 Polarization: SAM
10.4.1.2 Spatial Modes: OAM
10.4.1.3 Polarization versus OAM encoding
10.4.2 Technology: Path and time-frequency bins
10.4.2.1 Propagation path waveguide encoding
10.4.2.2 Time-based encoding: Time, frequency, energy
10.5 High-dimensional Quantum Entanglement
10.5.1 Theoretical development
10.5.1.1 Qudits and optimal information content
10.5.1.2 Greenberger–Horne–Zeilinger (GHZ) state
10.5.2 Experimental implementation
10.5.2.1 Polarization: Spin angular momentum
10.5.2.2 Spatial modes: OAM
10.5.2.3 Propagation path waveguide encoding
10.5.2.4 Time-based encoding: Time, frequency, energy
10.5.2.5 Hybrid entanglement: Multiple degrees of freedom
10.5.2.6 Degrees of freedom conversion
10.5.2.7 Planck’s constant
References
Chapter 11 Optical Machine Learning and Quantum Networks
11.1 Quantum Optical Machine Learning
11.1.1 Optical quantum computing
11.1.2 Optical neural networks
11.1.2.1 All-optical waveguide platforms
11.1.2.2 All-optical reservoir computing
11.1.3 Quantum optical machine learning
11.1.3.1 GHZ state preparation and Hamiltonian simulation
11.1.3.2 New protocols
11.1.3.3 Quantum optical autoencoder
11.1.3.4 Quantum reinforcement learning
11.1.3.5 One-way quantum repeaters
11.2 Global Quantum Networks
11.2.1 End-to-end qubits
11.2.1.1 Quantum network stack with entanglement
11.2.2 Long-distance entanglement
11.2.2.1 Heralded (confirmed) entanglement
11.2.2.2 Use cases for quantum network entanglement
11.2.2.3 Smart routing and SL A certification
11.2.2.4 Status of long-distance quantum entanglement
11.3 Global Quantum Clock Network
11.3.1 GHZ state and optical oscillators
11.3.1.1 Cooperative quantum clock network
11.3.1.2 Step 1 initialization: Prepare network-wide GHZ states
11.3.1.3 Step 2: Interrogation of local nodes
11.3.1.4 Step 3: Feedback and local node clock updating
11.3.1.5 Time trust in the cooperative clock system
11.3.2 Paper clocks
References
Chapter 12 Connectome and Brain Imaging
12.1 Connectomics
12.2 Brain Imaging
12.2.1 Connectome parcellation
12.3 High-Throughput Connectome Imaging
12.3.1 Electron microscopy
12.3.2 Light sheet microscopy
12.3.3 Expansion light sheet microscopy
12.3.4 X-ray microtomography
12.4 High-Throughput Recording
12.4.1 Light field microscopy
12.4.1.1 Fruit fly grooming and walking
12.4.2 Calcium imaging
References
Chapter 13 Brain Networks
13.1 Brain Networks’ Approach
13.1.1 The brain as a communications network
13.2 Wiring and Circuit Layout
13.2.1 The brain is three-dimensional
13.2.1.1 Topographical projection
13.2.1.2 Optimal ratios of axonal to dendritic arbor volumes
13.3 Connectivity
13.3.1 Gray matter and white matter
13.3.1.1 Local gray matter and long-distance white matter
13.3.1.2 Sparse small-world connectivity
13.4 Energy Consumption
13.4.1 Imputing traffic volume from energy consumption
13.4.2 Bandwidth
13.5 Signal Processing
13.5.1 Signal conversion
13.5.1.1 Probabilistic signal transmission
13.6 Signal-to-Noise Ratio
13.6.1 Ion channels
13.6.1.1 Molecular channel noise is nonlinear
13.6.1.2 Volumetric connectome data
13.7 Network Rewiring: Synaptic Plasticity
13.7.1 Neural signaling path integral
13.7.1.1 Implications of brain networks’ approach
References
Part 4 System Evolution
Chapter 14 Quantum Dynamics
14.1 Dynamics of Quantum Systems
14.2 Operator Size and Distribution Growth
14.3 The Holographic SYK Model
14.3.1 The Heisenberg uncertainty principle
14.3.1.1 Size-momentum correspondence and holographic SYK
14.3.2 Out-of-time-order correlators
14.3.2.1 Quantum dynamics and temperature
14.3.2.2 The Heisenberg equation of motion
14.3.3 Thermofield double state
14.3.3.1 Implications for the holographic SYK model
14.4 Superconductivity and Spacetime Superfluids
14.4.1 Time crystals
14.4.1.1 Spacetime superfluids and temperature
14.4.1.2 UV–IR correlations: Order-disorder phase transition
References
Chapter 15 Neural Dynamics
15.1 Multiscale Modeling
15.1.1 Centrality of wavefunction modeling
15.1.1.1 Practical perspective: Scale and model integration
15.1.1.2 Nonlinear dynamical systems approach
15.2 Approaches to Collective Neural Behavior
15.2.1 Nonlinear dynamical systems
15.2.1.1 Stochastic calculus and diffusion
15.2.2 Neural dynamics in large-scale models
15.3 Neural Ensemble Models
15.3.1 Fokker–Planck dynamics for normal distributions
15.3.1.1 Heavy tail distributions
15.3.2 Beyond linear Fokker–Planck equations
15.3.2.1 Recognized nonlinear probability distributions
15.3.2.2 Unknown probability distributions
15.3.3 Neural signaling: Orbits and bifurcation
15.3.3.1 Empirical data and oscillatory neural dynamics
15.4 Neural Mass Models
15.4.1 Brain networks approach
15.4.2 Technical aspects of neural mass methods
15.4.2.1 Oscillatory dynamics: Jansen–Rit neural mass model
15.4.2.2 Non-smooth dynamics and the Floquet model
15.5 Neural Field Models
15.5.1 Statistical theory of neuron dynamics
15.5.1.1 Oscillatory neural dynamics
15.5.2 Neural field theory in practice
15.5.2.1 Multiscale models
15.5.2.2 Synchrony: Simultaneous arrival of signals
15.5.2.3 Neural field theory simulation and filtering
15.5.2.4 Firing synchrony within populations of neurons
15.5.3 Statistical neural field theory
15.5.4 Quantum neural field theory
References
Part 5 Modeling Toolkit
Chapter 16 Quantum Machine Learning
16.1 Machine Learning-Physics Collaboration
16.1.1 Quantum machine learning overview
16.1.2 Structural similarities
16.1.3 Problems in quantum mechanics
16.1.3.1 Variational methods: The method of varying
16.2 Wavefunction Approximation
16.2.1 Quantum state neural networks
16.2.1.1 Traditional approaches to wavefunction modeling
16.2.1.2 Motivation for machine learning
16.2.1.3 Machine learning approach to wavefunction modeling
16.2.1.4 Encoding quantum states
16.2.1.5 Neural network mathematics
16.2.1.6 Demonstration: Quantum spin systems’ ground states
16.2.1.7 Neural network-tensor network comparison
16.2.1.8 Demonstration: Time-dependent quantum dynamics
16.2.1.9 Implications of quantum state neural networks
16.3 Quantum Transformer Neural Networks
16.3.1 Transformer attention mechanism
16.3.1.1 Transformer neural networks and quantum states
16.3.1.2 The attention mechanism
References
Chapter 17 Born Machine and Pixel = Qubit
17.1 The Born Machine
17.1.1 Boltzmann machine versus born machine
17.1.2 Supervised versus unsupervised learning
17.1.3 Unsupervised generative learning
17.1.3.1 Born machine implications for quantum computing
17.2 Probabilistic Methods: Reduced Density Matrix
17.2.1 Modeling classical data with quantum states
17.2.1.1 Quantum entanglement found in classical data
17.2.1.2 Practical implementation
17.2.1.3 Advantages of probabilistic quantum methods
17.2.2 Density matrices and density operators
17.3 Tensor Networks: Pixel = Spin (Qubit)
17.3.1 Decomposition of high-dimensional vectors
17.3.1.1 Machine learning algorithm: Weighted data features
17.3.1.2 Step 1: Encoding input data into tensor networks
17.3.1.3 Step 2: Selecting network architecture
17.3.1.4 Step 3: Training the network
17.3.1.5 Step 4: Testing real-life data
17.3.1.6 Advantages of tensor networks for machine learning
17.4 Tensor Networks: Wavelet = Spin (Qubit)
17.4.1 Wavelet transform
17.4.1.1 Wavelet transform equates to MERA tensor network
17.4.1.2 Step 1: Data encoding: MERA tensor network
17.4.1.3 Steps 2 and 3: Network architecture and training
17.4.1.4 Step 4: Testing real-life datasets
References
Chapter 18 Quantum Kernel Learning and Entanglement Design
18.1 Quantum Kernel Methods
18.1.1 Machine learning approaches
18.1.2 Kernel methods
18.1.2.1 Kernel methods reduce dimensionality
18.1.2.2 Dimensionality reduction and squeezed light states
18.1.2.3 Quantum algorithm design
18.1.3 Quantum kernel methods
18.1.3.1 Quantum kernel methods: Feature map approach
18.1.4 Embedded data Hilbert spaces
18.1.5 Quantum finance
18.1.5.1 Reproducing kernel Hilbert space formalism
18.1.5.2 Time series analysis and AdS/RKHS
18.1.6 Squeezed states of light
18.1.6.1 Squeezed states: Quantum noise reduction technique
18.1.6.2 Global telecommunications networks
18.1.6.3 Continuous basis quantum systems
18.1.7 RHKS and machine learning
18.2 Entanglement as a Design Principle
18.2.1 Entanglement and tensor networks
18.2.1.1 Blocks as a renormalization method
18.2.1.2 Density matrix renormalization group
18.2.2 Classical data and quantum states
18.2.3 Entanglement entropy
References
Chapter 19 Brain Modeling and Machine Learning
19.1 Brain Modeling
19.1.1 Compartmental neuroscience models
19.1.1.1 Compartment model classes
19.1.1.2 Synaptic integration
19.1.1.3 The future of compartmental models
19.1.2 Theoretical neuroscience
19.1.2.1 Network neuroscience modeling
19.1.2.2 Empirical context: Brain–computer interfaces
19.2 Classical Machine Learning and Neuroscience
19.2.1 Machine learning and biomedicine
19.2.2 Machine learning and neuroscience
19.2.2.1 Machine learning and brain tumors
19.2.2.2 Machine learning and neuropathologies of aging
19.2.3 Machine learning and connectomics
19.2.3.1 Neuron reconstruction: TeraVR and DeepNeuron
19.2.3.2 Neural connectivity: Synapse detection
19.2.3.3 Brain atlas annotation and deep learning network
19.2.3.4 Generative machine learning for unlabeled data
19.2.4 Rapprochement
19.2.4.1 Machine learning applied to neuroscience
19.2.4.2 Machine learning and neuroscience at odds
19.2.4.3 Next-generation machine learning
19.3 Neuromorphics and Spiking Neural Networks
19.3.1 Neuromorphic computing
19.3.1.1 Neuromorphic computing chips and projects
19.3.2 Spiking neural networks
19.3.2.1 Spike-based activation
19.3.2.2 Backpropagation and the learning problem
19.3.2.3 Eligibility propagation synaptic plasticity
19.3.2.4 Wider application of spiking neural networks
19.4 Optical Spiking Neural Networks
References
Chapter 20 Conclusion: AdS/Brain Theory and Quantum Neuroscience
20.1 Quantum Computing for the Brain
20.2 AdS/Brain Theory
20.2.1 Quantum neural signaling
20.2.1.1 AdS/Brain theory of neural signaling
20.2.1.2 Information-theoretic black hole-like physics
20.2.1.3 Implementation of the AdS/Brain theory
20.2.2 Risks and limitations
20.3 Millennium Prize-Type Challenges
20.3.1 NISQ device neuroscience applications
20.3.1.1 Physics-based time technologies
20.3.1.2 Standard neuroscience quantum circuits
20.4 The Future of Quantum Neuroscience
References
Glossary
Index