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Tensor Network and Neural Network Methods in Physical Systems

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Tensor Network and Neural Network Methods in Physical Systems Tensor network and neural network methods in physical systems Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Peiyuan Teng Graduate Program in Physics The Ohio State University 2018 Dissertation Committee: Dr. Yuan-Ming Lu, Advisor Dr. Ciriyam Jayaprakash Dr. Jay Gupta Dr. Comert Kural c Copyright by Peiyuan Teng 2018 Abstract In this dissertation, new ideas and methods from tensor network theory and neu- ral network theory are discussed. Firstly, common computational methods, such as the exact diagonalization method, the Density Matrix Renormalization Group ap- proach, and the tensor network theory are reviewed. Following this direction, a way of generalizing the tensor renormalization group (TRG) to all spatial dimensions is proposed. Mathematically, the connection between patterns of tensor renormalization group and the concept of truncation sequence in polytope geometry is discovered. A theoretical contraction framework is proposed. Furthermore, the canonical polyadic decomposition is introduced to tensor network theory. A numerical verification of this method on the 3-D Ising model is carried out. Secondly, this dissertation includes an efficient way of calculating the geomet- ric measure of entanglement using tensor decomposition methods. The connection between these two concepts is explored using the tensor representation of the wave- function. Numerical examples are benchmarked and compared. Furthermore, highly entangled qubit states are searched for to show the applicability of this method. Finally, machine learning approaches are reviewed. Machine learning methods are applied to quantum mechanics. The radial basis function network in a discrete basis is used as the variational wavefunction for the ground state of a quantum system. ii Variational Monte Carlo(VMC) calculations are carried out for some simple Hamil- tonians. The results are in good agreements with theoretical values. The smallest eigenvalue of a Hermitian matrix can also be acquired using VMC calculations. These results demonstrate that machine learning techniques are capable of solving quantum mechanical problems. iii This is dedicated to my parents, Mr. Yun Teng and Mrs. Min Xu. iv Acknowledgments I am sincerely thankful to my advisor Dr. Yuan-Ming Lu for the enlightening discussions, helpful suggestions, and careful comments. I'm amazed at his sharpness towards concepts, which clarifies a lot of my research ideas. His capability of digging deep into physics questions is also a good example for me to follow and to learn from. His love and dedication to physics influence me to move forward with my Ph.D. study. Learning from his good traits not only advances my research but will also help my future life. I'm also indebted to the Department of Physics of The Ohio State University for providing such a good environment for my study and for providing kindness support. I'm very happy to work in such a great place with distinguished professors and friendly classmates. Especially, I'd like to thank Dr. Jonathan Pelz for his wisdom and guidance about my study. I'm also extremely grateful to my committee members, Dr. Ciriyam Jayaprakash, Dr. Jay Gupta and Dr. Comert Kural for all the helpful talks and great suggestions. Their suggestions and helps guided through my study and will also be beneficial to my future career. I also want to express my appreciation to all the referees in the peer review pro- cesses of my papers. Their insightful comments and suggestions improved my work a lot. v I'm should also thank all the wonderful friends that I met during my life as a Ph.D. student in Columbus and in the United States, with whom I can explore many wonderful places and cultures in North America. Life would never be so amazing without them. Finally, I'd like to express my deepest gratitude to my parents, for their unrelenting love and selfless support. My thankfulness to them for everything is beyond any words. vi Vita January 2, 1990 . Born, Dandong, Liaoning, China 2008 - 2012 . B.S., Physics, Nankai University, China 2012 - 2015 . M.S., Physics, The Ohio State Univer- sity, USA 2015 - present(2018) . Ph.D. Candidate, Physics, The Ohio State University, USA Publications Research Publications Peiyuan Teng, "Generalization of the tensor renormalization group approach to 3-D or higher dimensions." Physica A: Statistical Mechanics and its Applications 472 (2017): 117-135. Peiyuan Teng, "Accurate calculation of the geometric measure of entanglement for multipartite quantum states." Quantum Information Processing 16.7 (2017): 181. Fields of Study Major Field: Physics vii Table of Contents Page Abstract . ii Dedication . iv Acknowledgments . v Vita . vii List of Tables . xii List of Figures . xiii 1. Introduction . 1 2. Numerical methods for many-body systems . 5 2.1 Exact Diagonalization . 5 2.1.1 Methodology . 5 2.1.2 Numerical implementation of exact diagonalization method 1 for spin- 2 models . 8 2.2 Density Matrix Renormalization Group . 9 2.2.1 Methodology . 9 2.2.2 Numerical simulations using Density Matrix Renormalization Group . 14 2.3 Tensor Network and Matrix Product States . 15 2.3.1 Tensor Network . 15 2.3.2 Matrix Product States . 17 2.4 Quantum Monte Carlo methods . 18 2.4.1 Introduction . 18 2.4.2 Variational Monte Carlo (VMC) . 19 viii 2.4.3 World Line Monte Carlo . 20 3. Generalization of the Tensor Renormalization Group method. 22 3.1 Introduction . 22 3.2 Tensor renormalization approach for a 2-D system . 25 3.2.1 Classical Ising model and tensor network . 25 3.2.2 Contraction of a 2-D honeycomb network . 27 3.2.3 Contraction of a 2-D square tensor network . 30 3.2.4 Free energy calculation for a 2d kagome Ising model . 31 3.3 Tensor renormalization approach for a 3-D system . 34 3.3.1 Dual polyhedron . 34 3.3.2 Tensor renormalization group (TRG) . 35 3.3.3 Canonical polyadic decomposition (CPD) . 37 3.3.4 TRG in detail: From cube to octahedron . 40 3.3.5 TRG in detail: From octahedron back to a cube . 47 3.4 Numerical results of the 3-D cubic tensor network model . 49 3.4.1 Ising model and tensor network . 49 3.4.2 Calculation framework . 50 3.4.3 Calculation steps: tensor size and cutoff . 50 3.4.4 Accuracy of CPD . 51 3.4.5 Rescaling of the tensor network and the dual tensor network 53 3.4.6 Derivation of free energy . 56 3.4.7 Numerical results . 57 3.4.8 Current restrictions of 3D-TRG method . 60 3.5 Tensor RG for higher dimensional tensor network . 61 3.6 Discussion . 64 3.6.1 Applications to quantum system . 64 3.6.2 Potential problems of TRG in Higher dimensions . 64 3.6.3 Is CPD the only choice? . 65 3.6.4 CPD and best rank-r approximation . 65 4. Tensor methods for Geometric Measure of Entanglement . 67 4.1 Introduction . 67 4.2 Geometric measure of entanglement and tensor decomposition . 70 4.2.1 Geometric measure of entanglement . 70 4.2.2 Tensor decomposition . 71 4.2.3 Numerical algorithm . 74 4.3 Numerical evaluation of the geometric measure of entanglement using Alternate Least Square algorithm . 76 ix 4.3.1 Geometric measure of entanglement for symmetric qubits pure states . 76 4.3.2 Geometric measure of entanglement for combinations of three qubits W states . 76 4.3.3 Geometric measure of entanglement for d-level system (qudits) 78 4.3.4 Hierarchies of Geometric measure of entanglement . 82 4.4 Discussions . 84 4.4.1 Geometric measure of entanglement for many-body systems 84 4.4.2 Several comments . 85 4.A Appendix: Searching for highly entangled states and maximally en- tangled states. 86 4.A.1 Bounds on the geometric measure of entanglement . 87 4.A.2 Maximally entangled four qubits states . 87 4.A.3 Highly entangled four qubits states . 89 4.A.4 Highly entangled five qubits states . 92 4.A.5 Highly entangled six and seven qubits states . 92 5. Theory of Machine Learning . 94 5.1 Introduction . 94 5.2 Regression . 95 5.3 K-nearest neighbors algorithm(KNN) . 97 5.4 Decision tree . 98 5.5 Support vector machine . 99 5.6 K-means clustering . 100 5.7 Principal component analysis . 102 5.8 Restricted Boltzmann Machines (RBM) as an artificial neural network102 5.9 Model evaluation . 103 6. Solving quantum mechanics problems using radial basis function network 107 6.1 Introduction . 107 6.2 Artificial neural network theory and the variational Monte Carlo method . 109 6.2.1 Artificial neural network theory . 109 6.2.2 Variational Monte Carlo method(VMC) . 115 6.3 Solving quantum mechanics problems using artificial neural network 116 6.3.1 Theoretical outline . 117 6.3.2 One dimensional quantum harmonic oscillator in electric field 120 6.3.3 Two dimensional quantum harmonic oscillator in electric field 125 6.3.4 Particle in a box . 128 6.3.5 Neural network as a Hermitian matrix lowest eigenvalue solver131 x 6.4 Discussion . 134 7. Conclusions . 136 Bibliography . 138 xi List of Tables Table Page 4.1 Overlaps for n-partite qubit systems . 77 4.2 Overlaps for n-partite qudit systems . 81 4.3 Hierarchies of 5-qubits W state . 83 6.1 The relation between nmax and the VMC energy at Ex = 4:0;Ey = 2:0.127 6.2 Comparison between exact values, perturbation results and numerical VMC energy at different a. 131 6.3 VMC results of the lowest eigenvalue of H(d). 133 xii List of Figures Figure Page 2.1 The infinite-system DMRG . 12 2.2 The finite-system DMRG .
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