Compression of Tensor Train Cores via Tucker Decomposition, and Applications to the Density Matrix Renormalization Group Open Access

Abstract

The tensor train (TT), or matrix product state (MPS), is a tensor decomposition which aims to avoid the curse of dimensionality in high-dimensional problems by compressing a tensor of arbitrary order into a chain of contractions of order-3 tensors, which we call the TT cores. In this work, we further compress the TT cores by means of a Tucker decomposition, or higher-order singular value decomposition (HOSVD). We perform error analysis for putting a TT into this form, and provide operations analogous to those for the uncompressed form, and discuss several properties. We showcase these theoretical results by compressing solutions of the density matrix renormalization group (DMRG) for the one-dimensional Hubbard model.

Table of Contents

Contents

1 Introduction and Background 1

1.1 Mathematical Background . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Matrices and the Singular Value Decomposition . . . . . . . . 3

1.1.2 Tensor Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.3 Tensor Train . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 Physical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.1 Second Quantization . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.2 The Schrödinger Equation and the Hubbard Model . . . . . . 17

1.2.3 The Curse of Dimensionality . . . . . . . . . . . . . . . . . . . 18

1.3 Density Matrix Renormalization Group . . . . . . . . . . . . . . . . . 19

1.3.1 The DMRG Sweep Algorithm . . . . . . . . . . . . . . . . . . 20

1.3.2 Variational Rounding . . . . . . . . . . . . . . . . . . . . . . . 23

1.3.3 The Structure of the Hubbard Hamiltonian . . . . . . . . . . . 25

2 Compression of TT Cores 27

2.1 The Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.1.1 Algebraic Operations . . . . . . . . . . . . . . . . . . . . . . . 29

2.1.2 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2 Revisiting DMRG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2.1 Utilizing Nonuniqueness of the Tucker Decomposition . . . . . 33

2.2.2 Lanczos Iteration on Tucker Decompositions . . . . . . . . . . 34

3 Numerical Results 36

3.1 More on the Error Bound . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 Compression of DMRG Solutions . . . . . . . . . . . . . . . . . . . . 38

4 Concluding Remarks 41

5 Appendix 43

Bibliography 49

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Chemistry, Physical Physics, Condensed Matter Mathematics
Keyword	electronic structure DMRG condensed matter singular value decomposition Hubbard model matrix product state density matrix renormalization group tensor decomposition tensor train Tucker decomposition tensor
Committee Chair / Thesis Advisor	Elizabeth Newman, Emory University Francesco Evangelista, Emory University
Committee Members	Cosmin Pohoata, Emory University

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