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Tensor Contraction Schemes for 2D Problems in Quantum Simulation Technische Universit¨atM¨unchen Computational Science and Engineering (Int. Master's Program) Master's Thesis Tensor Contraction Schemes for 2D Problems in Quantum Simulation Ing. Javier Enciso First examiner: Univ.-Prof. Dr. Thomas Huckle Second examiner: Univ.-Prof. Dr. Hans-Joachim Bungartz Assistant advisor: Dipl.-Math. Konrad Waldherr Thesis handed in on 30.09.2010 Declaration I hereby declare that this thesis is entirely the result of my own work except where otherwise indicated. I have only used the resources given in the list of references. M¨unchen, 30.09.2010 Javier Enciso Contents 1 Introduction 1 1.1 Overview . 1 1.2 Basic Notions . 1 1.3 Problem Definition . 4 1.4 Challenges . 5 2 Quantum Simulation 7 2.1 Overview . 7 2.2 Quantum Dynamics . 7 2.3 Spin Interaction . 9 2.4 Ising Model . 11 2.5 Numerical Solution . 12 3 Ground State Calculation 19 3.1 Overview . 19 3.2 Data Representation . 19 3.3 Contraction Scheme . 23 3.4 Numerical Ansatz . 24 4 Numerical Experiments 29 4.1 Overview . 29 4.2 Experimental Setup . 29 4.3 Results . 35 4.4 Errors . 38 5 Conclusions and Outlook 41 5.1 Overview . 41 5.2 Conclusions . 41 5.3 Further Improvements . 42 A Computer Program 43 A.1 Function Specification . 43 A.2 Code Example . 43 A.3 Licence . 45 i B Configuration Management 47 B.1 Overview . 47 B.2 Hardware Configuration . 47 B.3 Software Configuration . 47 Bibliography 51 Index 53 ii List of Figures 1.1 Dimensional and time requirements . 5 2.1 1D lattice with n sites . 8 2.2 2D lattice with m × n sites . 9 2.3 Bloch sphere . 10 2.4 1D lattice with external field . 12 2.5 Sparsity pattern in Hamiltonians . 13 2.6 Sparsity pattern after exact diagonalisation . 14 2.7 Integration via Monte Carlo method . 16 2.8 Decomposition and sweep directions in DMRG . 18 4.1 Setup for numerical experiments . 30 4.2 Sparsity pattern in numerical experiments . 30 4.3 Tensor blocks . 31 4.4 Index set arrangement for six tensors . 33 4.5 Simulation results . 36 4.6 Energy level after 100 iterations . 37 B.1 Profiling sample output . 49 iii iv List of Tables 3.1 Mapping components of final state . 21 4.1 Tensor configurations . 34 4.2 Statistics for six tensors . 35 4.3 Statistics for four tensors . 36 4.4 Statistics for three tensors . 36 4.5 Statistics for two tensors . 37 4.6 Statistics for ground state calculation . 37 4.7 Condition number for selected Hamiltonians . 39 A.1 Function description . 44 B.1 Hardware configuration . 48 B.2 Software configuration . 48 v vi List of Notations C Complex number (2) ⊗ Kronecker product operator (2) H Hamiltonian (4) Ln 1D lattice with n sites (8) Lm×n 2D lattice with m × n sites (8) σ0; : : : ; σ3 Pauli Matrices (9) In n × n Identity Matrix (9) MC Monte Carlo (14) QMC Quantum Monte Carlo (14) DoF Degrees of Freedom (14) sinc x Sinc Function (14) D&C Divide and Conquer (17) DMRG Density Matrix Renormalisation Group (17) R(H; j i) Rayleigh Quotient (24) ALS Alternating Least Squares (25) CPU Central Process Unit (47) GPU Graphic Process Unit (47) RAM Random Access Memory (47) HD Hard Disk (47) GB Giga Bytes (47) vii viii Acknowledgements Thanks are due to the Colombian Government, Colfuturo and the German Academic Exchange Service (DAAD) for the financial support through the Scholarship-loan Programme 2008 and Scholarship A/08/72528 respectively. Thanks to the people at the education and Public Outreach Department (ePOD) at the European Southern Observatory (ESO) for their outstanding collaboration during the internship and the writing process. Thanks to Univ.-Prof. Dr. Thomas Huckle at the Institute for Scien- tific Computing for his continuous advice and extraordinary support along the two years of the Master Programme. Thanks to Dipl.-Math. Konrad Waldherr for his valuable discussions and sharp thoughts that smoothed the development of the scripts. Thanks to Wilmer Hern´andezfor the superb visual ideas, spooky sound effects, and mind-blowing animation presented during the talk at the Scientific Computing in Computer Science colloquium. I thank my parents, brother, and sister for their omniscient support and good-will. Last but not least, special thanks are due to my beautiful wife for her kind support, lovely understanding, and invaluable companion in the time being. Perhaps my memory is not good enough to remember all people that made this academic project possible, however, my most sincere gratitude is with all of them, after all you know how you are. This document was produced with LATEX, vector graphics were created with Inkscape, and the drawings were processed with GIMP. ix x Abstract Quantum simulation describes a great variety of process observed in labo- ratories [16] and daily life [5], however, as soon as the computer programs involve more particles, time and storage turn out to be critical issues that constraint the execution of larger simulations. Tensor Contraction Schemes for 2D Problems in Quantum Simulation is based on the ideas of Huckle and Waldherr [11] and is a courageous attempt to describe many-body sys- tems accurately without the intrinsic dimensional problems. To achieve this goal, two well-established techniques are applied to represent large vectors implicitly and, through a variational-based numerical method, minimise the related optimisation problem. Remarkably, the contraction schemes demand linear storage and time resources only. As result, the proposed scheme was able to produce solutions up to the first significant digit for 12-particle sys- tem, taking no more than few iterations. Finally, there are still open issues regarding the accuracy in the vector representation and the local optimality of the iterative method that must to be addressed in future investigations. xi Chapter 1 Introduction \Number is the ruler of forms and ideas, and the cause of gods and daemons" Pythagoras 1.1 Overview From the scientific crisis that leaded the definition of a new set of ideas to explain really to the difficulties in the realisation of the quantum simula- tions, first chapter is intended to provide to the non-experienced reader a general introduction into the field of quantum simulation. In section 1.2, the basic introduction to the realm of quantum mechanics is presented, mak- ing emphasis in the clever notation used by physicists. In section 1.3, the definition of the problem that motivated the implementation of the contrac- tion schemes is described. Finally in section 1.4, the main challenges in the implementation of the numerical scheme are listed. 1.2 Basic Notions Newtonian mechanics provide reasonable explanation for the motion of bod- ies we experience in daily life [5]. From planes flying around to the tidal effects that govern the sees and the beautiful formation of the rainbow, nonetheless, the notions underlying classical mechanics straggle in the accu- rate description of the really fast/small systems, since then a new paradigm in science to reconcile reason with reality was needed. In 1907, the first problem was addressed by Albert Einstein [6, 21] ele- gantly. Thanks to the theoretical contributions done by the German physi- cist one hundred years ago, modern marvels such as the Global Positioning System[3] have been possible. Einstein's brilliant insight was able to explain the physics of objects subject to really intense gravitational fields, with ve- locities approaching the cosmic barrier of the speed of light. Certainly, his 1 moment of genius redefined mainstream science and paved the highway for the upcoming theories that would defy common sense. The second problem, related to the infinite small objects like electrons and photons, posed serious difficulties for the scientific community of the late nineteenth and early twentieth century. Among them Max Planck, who in 1900 settled the foundation of the latter know Quantum Mechanics, a new conceptual framework so powerful, that no other theory can describe at the exquisite level of detail and astonishing accuracy the phenomena at the atomic scale depicted by the quantum formalism. Quantum mechanics introduced several concepts far beyond common sense. For example, according to Heisenberg's uncertainty principle [10], it is not possible to know particle's position and momentum simultaneously without disturbing the other. Heisenberg's observations implied not only profound changes in physics but a revolution in the understanding of real- ity. Additionally, quantum superposition allows particles to be in all possible states at the same time, in other words, a quantum coin might be heads and tails at the same time and the real outcome will be known after measure- ment only. Althought no formal definition for a state of a system has been presented, it will be developed thoroughly in the following section. 1.2.1 Quantum State Contrary to the intuitive notion, a quantum state is a abstract mathematical construction. It does not describe any physical or measurable property of the system, instead it supports the formalism of quantum mechanics and allows the evolution of the system through the application of unitary operations. The foundation for the evolution of quantum systems will be presented in the next section. In mathematical terms, a quantum state j i is a complex row vector that lays on a finite Hilbert space H. The dimension d = dim (H) of the Hilbert space is also replicated by the number of components of the state j i, which grows exponentially with the number of particles in the system. A quantum state j i can be expressed as the linear combination of a set P P 2 of orthonormal basis states j i = i cijkii where ci 2 C and i jcij = 1.
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