Studies of Quantum Dots using Machine Learning by Even Marius Nordhagen THESIS for the degree of MASTER OF SCIENCE Faculty of Mathematics and Natural Sciences University of Oslo December 2019 2 All illustrations in this thesis are created using the TikZ package1 if nothing else is specified. The plots are produced using a controversial combination of Matplotlib2 and PGFPlots1. The LATEX document preparation system was applied for typesetting. Abstract With the ability to solve the many-body Schrodinger¨ equation accurately, in principle all physics and chemistry could be derived from first principles. However, exact wave functions of re- alistic and interesting systems are in general unavailable because they are non-deterministic polynomial-hard to compute3. This implies that we need to rely on approximations. The varia- tional Monte Carlo (VMC) method is widely used for ground state studies, but requires a trial wave function ansatz which must trade off between efficiency and accuracy. The method also has many common features with machine learning algorithms, and as neural networks have shown impressive power as function approximators, the idea is to use a neural network as the trial wave function guess. For fermionic systems, like electronic structure systems, the wave function needs to obey Fermi-Dirac statistics. This is typically achieved using a Slater deter- minant. As a neural network hardly can model this feature, our approach is to replace the single-particle functions in the Slater determinant with restricted Boltzmann machines, getting the RBM ansatz. In addition, we add further correlations via so-called Jastrow factors4. The de facto standard trial wave function ansatz for electronic structure calculations is the Slater-Jastrow ansatz, which was implemented as a reference. Our primary focus is on closed-shell circular quantum dots, where we compute the ground state energy and electron density of two-dimensional systems with up to N = 90 electrons and three-dimensional systems with up to N = 70 electrons. The energy obtained by the RBM ansatz was reasonably close to experimental results, and it gradually became closer as we added more complex correlation factors. For our most complicated Jastrow factor, the energy was found to be lower than the energy provided by the Slater-Jastrow ansatz for small dots, but for larger dots it was slightly higher. However, the one-body density profile reveals that the RBM ansatz gives more distinctly located electrons compared to the Slater-Jastrow ansatz. This can be explained by the way the RBM ansatz models the correlations. From the two-body density profile, we also observe that the repulsive interactions get more significant as we add a Jastrow factor. Based on the electron densities and the energy distribution between kinetic and potential energy, it is certain that the various methods provide different electron configurations. For low-frequency dots, the electron density becomes more localized with an additional radial peak compared to high-frequency dots. This is reminiscent of what is known as Wigner localization5. The computational time consumption was found to be favorable for the RBM ansatz for small systems and the Slater-Jastrow ansatz for large systems. This can be explained by the exploding number of variational parameters in the RBM ansatz as the system sizes increase. The RBM ansatz with Jastrow factors were notably more computationally intensive than the other ansatze,¨ and evidently, there is no reason to use the simplest Jastrow factor when more complicated Jastrow factors are available. Acknowledgements After five exciting years at Blindern, I would sincerely like to acknowledge some people who have been important throughout my studies. First, I would like to thank my excellent supervisor, Morten Hjorth-Jensen, whom I luckily got to know three years ago. From day one, you took me under your wing and enthralled me with your eager, work ethic and massive knowledge. Few things make me more motivated than a conversation with you, be it in real-life at Blindern or through video conversations from whatever place you happen to be at. I also very much appreciate the support from mum, dad and my sister. Even though the schedule has been full and I have not spent as much time with you as I wanted, you were always understanding. Thanks to my friends (you know who you are), who are always ready for a beer (or ten) whenever I need to disconnect from the studies. These moments, filled with lively discussions and dark humor, make me able to focus on the studies when needed. The computational physics group is a funny composition of different people with a shared predilection for physics and programming. There are so many talented guys in the group, and I really enjoy spending early mornings and late nights with you. I would also like to thank my high school teacher, Jens Otto Opaker, who with his enthusiasm and dedication got me hooked on science in the first place. Last, but not least, thanks to Sebastian Gregorious Winther-Larsen, Robert Solli, Marius Jon- sson and Kaitlin Rose Preusser for proofreading and constructive feedback. Even Marius Nordhagen Contents 1 Introduction 1 1.1 The many-body problem..................................1 1.2 Machine learning......................................2 1.3 Quantum dots........................................2 1.4 Computer experiments...................................3 1.5 Ethics in science.......................................4 1.6 Goals and contributions..................................4 1.7 The developed code.....................................5 1.8 Structure of the thesis....................................5 I Quantum Theory7 2 Preliminary Quantum Mechanics9 2.1 The postulates of quantum mechanics.......................... 11 2.2 The Schrodinger¨ equation................................. 11 2.2.1 The hydrogen atom................................. 13 2.3 The variational principle.................................. 13 2.4 Quantum numbers..................................... 14 2.5 The virial theorem...................................... 14 3 Many-body Quantum Mechanics 17 3.1 The electronic Hamiltonian................................ 18 3.2 The many-body wave function.............................. 18 3.2.1 Anti-symmetry and the Pauli principle..................... 19 3.2.2 The Slater determinant............................... 19 3.2.3 Basis set....................................... 20 3.2.4 Modeling the correlations............................. 21 3.3 Electron density....................................... 21 3.3.1 Wigner crystals................................... 23 4 Systems 25 4.1 Quantum dots........................................ 26 4.2 Quantum double dots................................... 27 4.3 Atoms............................................ 27 Contents II Machine Learning Theory 31 5 Supervised Learning 33 5.1 Polynomial regression................................... 35 5.1.1 An example on polynomial regression...................... 36 5.2 Bias-variance tradeoff.................................... 38 5.3 Linear regression...................................... 39 5.3.1 Singular value decomposition.......................... 39 5.3.2 Ridge regression.................................. 40 5.3.3 LASSO regression................................. 40 5.4 Logistic regression..................................... 41 5.5 Neural networks...................................... 43 5.5.1 Forward phase................................... 44 5.5.2 Activation function................................. 44 5.5.3 Backward propagation............................... 45 5.6 Optimization algorithms.................................. 47 5.6.1 Gradient descent.................................. 47 5.6.2 Stochastic gradient descent............................ 48 5.6.3 Adding momentum................................ 48 5.6.4 ADAM........................................ 49 6 Boltzmann Machines 51 6.1 Statistical foundation.................................... 52 6.1.1 Marginal distributions............................... 53 6.1.2 Conditional distributions............................. 53 6.1.3 Maximum log-likelihood estimate........................ 54 6.2 Unrestricted Boltzmann machines............................ 54 6.3 Restricted Boltzmann machines.............................. 55 6.3.1 Gaussian-binary units............................... 56 6.4 Partly restricted Boltzmann machines.......................... 58 6.5 Deep Boltzmann machines................................. 59 III Methods 61 7 Quantum Monte Carlo Methods 63 7.1 Variational Monte Carlo.................................. 64 7.1.1 The trial wave function.............................. 64 7.1.2 The Jastrow factor................................. 66 7.1.3 The local energy.................................. 67 7.1.4 Parameter update.................................. 68 7.1.5 The electron density................................ 69 7.1.6 Common extensions................................ 69 7.2 Unifying Quantum Mechanics and Machine Learning................. 69 7.2.1 Restricted Boltzmann machine without Jastrow factor (RBM)........ 70 7.2.2 RBM with a simple Jastrow factor (RBM+SJ).................. 70 7.2.3 RBM with a Pade-Jastrow´ factor (RBM+PJ)................... 70 7.3 The Metropolis Algorithm................................. 71 7.3.1 Brute-force sampling................................ 71 7.3.2 Importance sampling................................ 72 Contents 7.3.3 Gibbs sampling..................................