Tensor Network and Neural Network Methods in Physical Systems
Author: Peiyuan Teng
Publisher:
Published: 2018
Total Pages: 145
ISBN-13:
DOWNLOAD EBOOKIn this dissertation, new ideas and methods from tensor network theory and neural network theory are discussed. Firstly, common computational methods, such as the exact diagonalization method, the Density Matrix Renormalization Group approach, 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 geometric measure of entanglement using tensor decomposition methods. The connection between these two concepts is explored using the tensor representation of the wavefunction. 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. Variational Monte Carlo(VMC) calculations are carried out for some simple Hamiltonians. 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.