MATH 597 Topics Course on Quantum-related Computation (Spring 2023)

Time and Location: Spring 2023. TuTh 1:35PM – 2:50PM. 307 Boucke

Instructor: Prof. Xiantao Li.  Personal web page 

Brief Course Description:

Quantum mechanical models such as the modern density-functional theory and ab initio molecular dynamics have revolutionized the field of chemistry and material science. The implementation of these models also involves many interesting and challenging scientific computing problems, including eigenvalue problems, fixed-point iterations, geometric integrators, etc. This course will offer an introduction to the background of these problems, together with state-of-the-art computational techniques.

The second half of the course will provide an overview of quantum algorithms for large-scale scientific computing problems.

References:

  1. A Mathematical Introduction to Electronic Structure Theory, Lin and Lu, SIAM Spotlights, 2019.
  2. Lecture Notes on Quantum Algorithms, Andrew M. Childs, 2021, University of Maryland.

List of topics:

  • Structure-preserving time integration —  time-dependent Schrödinger equations
    • Time-ordered evolution operator
    • Magnus expansions
    • Operator-splitting
    • Krylov subspace projection for the matrix exponentials
    • Pseudo-spectral methods for the spatial discretization
    • Quantum dynamics in unbounded domains.
  • Nonlinear eigenvalue problems — the computation of electronic band structures
    • Davidson’s algorithm
    • Divide-and-conquer method
    • Conjugate-gradient method
    • Random algorithm for the computation of the density of states
  • Fixed-point problems — Self-consistent field (SCF) in density functional theory.
    • Andersen’s mixing
    • Optimization-based method
    • Machine-learning based algorithms
  • Dynamical Systems — Ion dynamics
    • Born-Oppenheimer dynamics (Ab Inito MD)
    • Ehrenfest dynamics and its geometric structures
    • The Lanvegin dynamics
    • Strong and weak convergence
    • Multi-level Monte Carlo Sampling
  • Basic quantum computing algorithms
    • Qubits and tensor-products of Hilbert spaces
    • Elementary quantum gates and the associated unitary operators
    • Quantum Fourier transform
    • Quantum phase estimation
    • Block encoding
  • Quantum Algorithms – Applications to Scientific Computing problems
    • Quantum algorithms for eigenvalue calculations.
    • Quantum computing algorithms for solving large-scale linear systems
    • Quantum computing algorithms for solving differential equations
    • Quantum machine learning