This thesis is devoted to the application of tensor-network methods to problems in quantum many-body physics in one and two dimensions and in machine learning. It contributes to the theory of infinite dimensional tensor networks giving rise to critical or fractional quantum Hall lattice models. The power of artificial neural networks at representing such states is investigated both analytically and numerically, and connections with tensor networks are uncovered, with applications both in the study of quantum states and in machine learning.This thesis is devoted to the application of tensor-network methods to problems in quantum many-body physics in one and two dimensions and in machine learning. It contributes to the theory of infinite dimensional tensor networks giving rise to critical or fractional quantum Hall lattice models. The power of artificial neural networks at representing such states is investigated both analytically and numerically, and connections with tensor networks are uncovered, with applications both in the study of quantum states and in machine learning.

Jan 1, 20180 citations

Machine Learning Quantum States — Extensions to Fermion–Boson Coupled Systems and Excited-State Calculations

Yusuke Nomura

To analyze quantum many-body Hamiltonians, recently, machine learning techniques have been shown to be quite useful and powerful. However, the applicability of such machine learning solvers is still limited. Here, we propose schemes that make it possible to apply machine learning techniques to analyze fermion-boson coupled Hamiltonians and to calculate excited states. As for the extension to fermion-boson coupled systems, we study the Holstein model as a representative of the fermion-boson coupled Hamiltonians. We show that the machine-learning solver achieves highly accurate ground-state energy, improving the accuracy substantially compared to that obtained by the variational Monte Carlo method. As for the calculations of excited states, we propose a different approach than that proposed in K. Choo et al., Phys. Rev. Lett. 121 (2018) 167204. We discuss the difference in detail and compare the accuracy of two methods using the one-dimensional $S=1/2$ Heisenberg chain. We also show the benchmark for the frustrated two-dimensional $S=1/2$ $J_1$ - $J_2$ Heisenberg model and show an excellent agreement with the results obtained by the exact diagonalization. The extensions shown here open a way to analyze general quantum many-body problems using machine learning techniques.To analyze quantum many-body Hamiltonians, recently, machine learning techniques have been shown to be quite useful and powerful. However, the applicability of such machine learning solvers is still limited. Here, we propose schemes that make it possible to apply machine learning techniques to analyze fermion-boson coupled Hamiltonians and to calculate excited states. As for the extension to fermion-boson coupled systems, we study the Holstein model as a representative of the fermion-boson coupled Hamiltonians. We show that the machine-learning solver achieves highly accurate ground-state energy, improving the accuracy substantially compared to that obtained by the variational Monte Carlo method. As for the calculations of excited states, we propose a different approach than that proposed in K. Choo et al., Phys. Rev. Lett. 121 (2018) 167204. We discuss the difference in detail and compare the accuracy of two methods using the one-dimensional $S=1/2$ Heisenberg chain. We also show the benchmark for the frustrated two-dimensional $S=1/2$ $J_1$-$J_2$ Heisenberg model and show an excellent agreement with the results obtained by the exact diagonalization. The extensions shown here open a way to analyze general quantum many-body problems using machine learning techniques.

arXiv:2001.02106Journal of the Physical Society of JapanApr 15, 202031 citations

Neural network representations of quantum many-body states

Ying Zhen Yang, Huaixin Cao, Zhanjun Zhang

Science China Physics Mechanics and AstronomyJun 20, 201913 citations

S=1/2

Heisenberg chain. We also show the benchmark for the frustrated two-dimensional

S=1/2

J_1

J_2

Heisenberg model and show an excellent agreement with the results obtained by the exact diagonalization. The extensions shown here open a way to analyze general quantum many-body problems using machine learning techniques.