Inhalt

•  
   Abstract
•  
   Zusammenfassung
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   List of Symbols
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   Contents
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   List of Figures
•  
   Introduction
•  
  Physical Framework
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   Magnetism - The Exchange Interaction
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   Heisenberg Model
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   J1-J2 Model
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   Hubbard Model
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   Marshall-Peierls Sign Rule and Stoquastic Hamiltonians
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   Phase Transitions and Critical Exponents
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   Quantum Geometry
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   Projective Hilbert Space and Tangent Space
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  Variational Manifold
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   Variational Manifold of the Projective Hilbert Space
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   Variational Manifold of the Hilbert Space
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   Quantum Natural Gradient (QNG)
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   Quantum Velocity
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  Variational Monte Carlo Method
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   Ground State Optimization
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   Stochastic Gradient Descent (SGD)
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   Quantum Natural Gradient (QNG)
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   Time Evolution
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   Imaginary-Time Evolution
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   Real-Time Evolution
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   Adaptive Heun Method
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   Imaginary-Time Evolution on the Manifold and Modification
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   Regularization
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   Ground State Optimization
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   Time Evolution
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  Neural Network (NN)
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   Hilbert's 13-th Problem
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   Kolmogorov-Arnold Representation Theorem
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   Single-Layer Feed-Forward Neural Network (SLFFNN)
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   Multi-Layer Feed-Forward Neural Network (MLFFNN)
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   Feed-Forward Neural Network (FFNN) over the Complex Field
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   Neural Network Quantum State (NNQS)
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   Sigmoidal Ansatz (SA)
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   Boltzmann Machine (BM)
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   Effective Update Scheme
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   Details of the Learning Algorithm
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   Curse of Dimensionality
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   Entanglement Entropy and Expressive Power of RBMs
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  Variational Wave Functions for Strongly Correlated Systems
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   H2-Molecule
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   Heitler-London Ansatz
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   Hartree-Fock Ansatz
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   Many-Body Ansatz Wave Function
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   Hartree-Fock Ansatz
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   Gutzwiller and Jastrow Factor
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  Pair-Product (PP) Wave Function
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   Bardeen-Cooper-Schrieffer (BCS) Wave Function
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   Resonating Valence Bond (RVB) Wave Function
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   Pair-Product (PP) Wave Function
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   Efficient Computation of the Pfaffian and its Derivative
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   Transformation from Fermions to Spins
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   PP-NN Ansatz
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   Ground State Optimization with Feed-Forward Neural Networks (FFNNs)
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   Barren Plateaus
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   Improvement by the PP-NN Ansatz
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   Exact Representations with Deep Boltzmann Machines (DBMs)
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   Wave Function for Finite-Temperature Computations
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   Finite-Temperature Results for the One-Dimensional Heisenberg Model
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  Purification Method
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   Purification
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   Macroscopic Thermodynamic Variables
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   Numerical Results
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  Sampling Method
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   Minimally Entangled Thermal States (METS)
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   Numerical Results
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  Thermal Pure Quantum State (TPQS)
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   Measure Concentration on Normed Spaces
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   Infinite-Temperature Thermal Pure Quantum State (TPQS)
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   Extension to Finite Temperatures
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   Canonical Thermal Pure Quantum State (CTPQS)
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   Variational Canonical Thermal Pure Quantum State (VCTPQS)
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   PP-NN Ansatz
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   The Infinite-Temperature VCTPQS
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   Numerical Results
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   Dynamical Correlation Functions for the One-Dimensional Heisenberg Model at Infinite Temperature
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   Spin-Spin Autocorrelation Function and Entanglement Entropy
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   Numerical Results
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  Finite-Temperature Results for the Two-Dimensional J1-J2 Model on the Square Lattice
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   Ground State Phase Diagram
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   Modifications of the Purification and the Sampling Algorithm
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  Finite-Temperature Results
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   Finite-Temperature Results for the Two-Dimensional Heisenberg Model
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   Finite-Temperature Results for the Two-Dimensional J1-J2 Model
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   Phase Transition at Finite Temperatures
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   Critical Slowing Down
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   Conclusion
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   Outlook
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   Bibliography
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   Appendix Schrödinger Equation on the Variational Manifold of the Projective Hilbert Space
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   Appendix Schrödinger Equation on the Variational Manifold of the Hilbert Space