From Linear Algebra to Machine Learning: Ten Central Algorithms for Studying Quantum Mechanical Many-Particle Problems
Morten Hjorth-Jensen, Department of Physics and Astronomy and Facility for Rare Isotope Beams, Michigan State University, East Lansing, Michigan, USA and Department of Physics and Center for Computing in Science Education, Unversity of Oslo, Norway
Email: hjensen@msu.edu
In this contribution we outline central algorithms for studying quantum mechanical systems, with an emphasis on both computational and pedagogical aspects. Using simple systems that allow for analytical solutions, we show how one can move from linear algebra and eigenvalue algorithms using for example full configuration interaction theory, to stochastic methods like variational and Diffusion Monte Carlo approaches and finally, how we can use Monte Carlo methods together with deep learning algorithms. Along this journey we will present ten central algorithms which have changed considerably the way we study interacting many-particle systems. These algorithms span from Householder’s famous transformation of matrices, via iterative eigenvalue solvers to neural networks and automatic differentiation for optimizing multidimensional functions. Codes and jupyter-notebooks are provided, allowing thereby the reader to experiment and practice the various methods.
- Linear algebra and eigenvalue problems
- Houselholder's transformation
- Jacobi/Givens rotations
- Iterative methods, Lanczsos' method
- Monte Carlo methods
- Variational Monte Carlo
- Metropolis-Hastings algorithm
- Optimization problems
- Gradient descent and steepest gradient descent
- Adaptive methods
- Automatic differentiation
- Deep learning
- Neural Networks
- Reduced Boltzmann machines