用QTT去探討在Efimov物理裡具有尺度不變性的哈密頓量

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研究生：	許哲嘉 Hsu, Che-Chia
論文名稱：	用QTT去探討在Efimov物理裡具有尺度不變性的哈密頓量 Explore Scale Invariant Hamiltonian in Efimov Physics Using Quantized Tensor Train Approach
指導教授：	黃一平 Huang, Yi-ping
口試委員:	鍾佳民 Chung, Chia-Min 陳柏中 Chen, Po-Chung
學位類別：	碩士 Master
系所名稱：	理學院 - 物理學系 Department of Physics
論文出版年：	2024
畢業學年度：	112
語文別：	英文
論文頁數：	62
中文關鍵詞：	張量網絡、量子張量訓練
外文關鍵詞：	QTT, Efimov physics, MPS
相關次數：	點閱：20 下載：0
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矩陣乘積狀態(MPS)作為一種資料結構，其糾纏結構使它善於捕捉系統內複雜的相關性。MPS最近進一步擴大了這種方法的效用，有效的應用在流體動力學方程，包括諸如渦輪等現象的複雜動力學，並且可以很好地壓縮資料。本研究努力採用相關方法去研究Efimov物理，其特點是其獨特的尺度不變性。在此背景下，透過量子張量訓練(QTT)，兩種不同的新方法被使用以產生與距離平方反比位能的MPS，從而能用密度矩陣重整化群(DMRG)在更高精度下計算並重現能量比例不變及離散縮放不變，並詳細探索了相關的數值挑戰。

Matrix product state (MPS) is a data structure used to represent the quantum states in many-body physics since it efficiently captures the low-entangled structure within the quantum many-body system with less memory. By broadening the scope of MPS as a data representation framework, it becomes more adept at capturing intricate correlations within the system. Recently, physicists found MPS can also used to solve different kinds of numerical problems, such as hydrodynamic equations, including the complex dynamics of phenomena like turbulence, and can compress data well. Moreover, the quantized tensor train (QTT) approach allows for accurate approximation of the data into MPS with less memory. This study endeavors to apply these methods to Efimov physics, which is characterized by scaling invariance. Two new distinct approaches for generating the potential with the inverse square of the distance into MPS using QTT have been used in this study, successfully reproducing discrete scaling invariance and energy ratio invariance features with high precision and less memory using density matrix renormalization group (DMRG), and a detailed exploration of associated numerical challenges.

Contents
論文發表聲明書 iii
Acknowledgements
摘要 i
Abstract ii
Introduction 1
1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 Overview of matrix product state (MPS), matrix product operator (MPO),
density matrix renormalization group (DMRG) in quantum many-body
systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.3 Overview of quantum-inspired classical algorithm . . . . . . . . . . . 4
2.4 Overview of Efimov states . . . . . . . . . . . . . . . . . . . . . . . . 5
Matrix product states (MPS), matrix product operators (MPO), density matrix
renormalization group (DMRG) and quantum many-body physics 7
1 What is a quantum many-body physics? . . . . . . . . . . . . . . . . . . . . . 7
2 What is matrix product state? Why is it helpful for solving quantum many-body
problems? Limitation? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1 Many body states states to matrix product states (MPS) . . . . . . . . . 8
2.2 Represent many-body states into MPS . . . . . . . . . . . . . . . . . . 11
2.3 General MPS to canonical form . . . . . . . . . . . . . . . . . . . . . 13
3 What is matrix product operator (MPO)? Why is it useful to develop the variational method for the density matrix renormalization group (DMRG)? . . . . . 14
3.1 Operators to matrix product operators (MPO) . . . . . . . . . . . . . . 14
3.2 Represent many-body Hamiltonian into MPO . . . . . . . . . . . . . . 15
4 Density matrix renormalization group (DMRG) . . . . . . . . . . . . . . . . . 17
4.1 Searching for ground state and low excited states . . . . . . . . . . . . 17
4.2 One-site and two-site DMRG . . . . . . . . . . . . . . . . . . . . . . 19
Quantum-inspired classical algorithm 21
1 Differential operator as MPO . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 Functions with exact MPS representation . . . . . . . . . . . . . . . . . . . . 23
3 Example of using QTT to solve Schrödinger equation . . . . . . . . . . . . . . 25
4 Review why most smooth functions can be represented into MPS successfully? 29
iii
Introduction to Efimov physics 33
1 Borromean binding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2 Scattering length and unitary limit . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Zero-range theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4 Derivation of the Efimov attraction in three identical bosons case . . . . . . . . 36
4.1 Jacobi coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.2 Hyper-spherical coordinates . . . . . . . . . . . . . . . . . . . . . . . 37
4.3 Hyper-angular part solution . . . . . . . . . . . . . . . . . . . . . . . 38
4.4 Hyper-radius part solution . . . . . . . . . . . . . . . . . . . . . . . . 39
5 Three non-identical bosons case . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.1 Equal masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.2 Unequal masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Quantum-inspired classical algorithm for Efimov physics 43
1 Method 1: searching for 1
f (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2 Method 2: changing to ex coordinate . . . . . . . . . . . . . . . . . . . . . . . 45
3 Features in Efimov physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Conclusion & discussion 53
1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
1.1 Compare different coordinates . . . . . . . . . . . . . . . . . . . . . . 53
1.2 Slow converge issue . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
1.3 The difficulty of DMRG to search for the function is function-dependent 53
2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
A Lanczos algorithm 59
                                

[1] Mazen Ali and Anthony Nouy. “Approximation Theory of Tree Tensor Networks: Tensorized Univariate Functions”. In: Constructive Approximation 58 (2020), pp. 463–544.
[2] Eric Braaten and H.-W. Hammer. “Universality in few-body systems with large scattering length”. In: Physics Reports 428.5 (2006), pp. 259–390.
[3] Jielun Chen, E.M. Stoudenmire, and Steven R. White. “Quantum Fourier Transform Has Small Entanglement”. In: PRX Quantum 4 (Oct. 2023), p. 040318.
[4] Juan José García-Ripoll. “Quantum-inspired algorithms for multivariate analysis: from interpolation to partial differential equations”. In: Quantum 5 (Apr. 2021), p. 431.
[5] Nikita Gourianov et al. “A quantum-inspired approach to exploit turbulence structures”. In: Nature Computational Science 2 (2021), pp. 30–37.
[6] Nicolas Jolly, Yuriel Núñez Fernández, and Xavier Waintal. Tensorized orbitals for computational chemistry. 2023. arXiv: 2308.03508.
[7] David B. Kaplan et al. “Conformality lost”. In: Phys. Rev. D 80 (Dec. 2009), p. 125005.
[8] Boris N. Khoromskij. “O(dlog N)-Quantics Approximation of N-d Tensors in High-Dimensional Numerical Modeling”. In: Constructive Approximation 34 (2011), pp. 257–280.
[9] Cornelius Lanczos. “An iteration method for the solution of the eigenvalue problem of linear differential and integral operators”. In: J. Res. Natl. Bur. Stand. B 45 (1950), pp. 255–282.
[10] Simone Montangero. “Introduction to Tensor Network Methods: Numerical simulations of low-dimensional many-body quantum systems”. In: 2018.
[11] Pascal Naidon and Shimpei Endo. “Efimov physics: a review”. In: Reports on Progress in Physics 80.5 (Mar. 2017), p. 056001.
[12] I. Oseledets. “Constructive Representation of Functions in Low-Rank Tensor Formats”. In: Constructive Approximation 37 (2012), pp. 1–18.
[13] I. Oseledets. “Tensor-Train Decomposition”. In: SIAM J. Sci. Comput. 33 (2011), pp. 2295–2317.
[14] Raghavendra Dheeraj Peddinti et al. “Quantum-inspired framework for computational fluid dynamics”. In: Communications Physics 7.1, 135 (Apr. 2024), p. 135.
[15] Shi-Ju Ran and Gang Su. “Tensor Networks for Interpretable and Efficient Quantum-Inspired Machine Learning”. In: Intelligent Computing 2 (2023), p. 0061.
[16] Shi-Ju Ran et al. Tensor Network Contractions: Methods and Applications to Quantum Many-Body Systems. 1st. Springer Publishing Company, Incorporated, 2020.
[17] Marc K. Ritter et al. “Quantics Tensor Cross Interpolation for High-Resolution Parsimonious Representations of Multivariate Functions”. In: Phys. Rev. Lett. 132 (Jan. 2024), p. 056501.
[18] Ulrich Schollwöck. “The density-matrix renormalization group in the age of matrix product states”. In: Annals of Physics 326.1 (2011), pp. 96–192.
[19] Steven R. White. “Density matrix formulation for quantum renormalization groups”. In: Phys. Rev. Lett. 69 (19 Nov. 1992), pp. 2863–2866.
[20] Erika Ye and Nuno F. G. Loureiro. “Quantum-inspired method for solving the Vlasov-Poisson equations”. In: Phys. Rev. E 106 (3 Sept. 2022), p. 035208.

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