# Computational quantum physics

## Summary

The numerical simulation of quantum systems plays a central role in modern physics. This course gives an introduction to key simulation approaches, through lectures and practical programming exercises. Simulation methods based both on classical and quantum computers will be presented.

## Content

1.** Single-particle Problems:** Numerical solutions of the Schroedinger equation, Numerov's integration, the split operator method

2. **Quantum Spin Models**: Choice and representations of basis sets for the many-body problem, the Trotter decompososition for real and imaginary-time evolution

3. **Electronic Structure**: Second Quantization, Full Configuration Interaction, Hartree-Fock, Density Functional Theory

4. **Variational Methods**: Variational Monte Carlo. Machine Learning Based Techniques, Time-dependent Variational Approaches

5. **Quantum Monte Carlo Methods**: Path Integral Monte Carlo at finite and zero temperature

6. **Quantum Computing**: Quantum simulation on a quantum computer, Adiabatic State preparation, Variational Quantum Eigensolver

## Keywords

Quantum simulation, Variational Monte Carlo, Machine Learning in Physics, Density Functional Theory, Lanczos, Path Integral Monte Carlo, Quantum Computing, Second Quantization, Time-Dependent Variational Principle

## Learning Prerequisites

## Required courses

A solid understanding of quantum mechanics (I and II) is required.

Students should have a good working knowledge of at least one common programming language (Python, C, C++, Fortran, Julia...). Knowledge of Matlab is typically sufficient, but it is strongly advised to be familiar with Python, since the exercises will be typically presented and discussed in Python.

## Recommended courses

The following courses are recommended but not compulsory

**PHYS-403 - C**

**omputer simulation of physical systems I,**highly recommended to get an introduction to simulation paradigms for physical systems

**PHYS-467 - ****Machine learning for physicists, **highly recommended to get an introduction to modern machine learning, since part of the course will make us of machine learning to study quantum systems

**PHYS-641 - ****Quantum Information and Quantum Computing, **also highly recommended since part of the course will cover quantum algorithms

To have a broader view of the importance of the problems attacked during the course, it is also suggested to attend the following courses

**PHYS-419 - Solid State Physics III**

**PHYS-425 - Quantum Physics III**

**PHYS-502 - Interacting quantum matter**

## Learning Outcomes

By the end of the course, the student must be able to:

- Model a quantum problem through numerical tools
- Identify suitable algorithms to solve or approximately solve a certain quantum problem
- Discuss the limitations of a given algorithm
- Carry out computer simulations

## Teaching methods

Ex cathedra with exercises

## Expected student activities

Practical assignments will be given every week.

Solutions to the assignements will be handed out and the homework will not be graded.

It is strongly advised however to make the effort to do the homework weekly, since the final exam will also evaluate the understanding of the practical implementation aspects of the computational methods.

## Assessment methods

The course is graded through an oral exam.

The oral exam will assess both the general theory as well as the understanding of the practical implementation of the algorithms, as presented during the practical weekly exercises.

## Resources

## Bibliography

Suggested books to acquire a broader view on the topics discussed in the lecture notes

"Quantum Monte Carlo Approaches for Correlated Systems", F. Becca & S. Sorella, (Cambridge University Press, 2017)

"Computational Physics", J. M. Thijssen, (Cambridge University Press)

"Statistical Mechanics: Algorithms and Computations", W. Krauth, (Oxford Master Series in Physics)

## In the programs

**Semester:**Spring**Exam form:**Oral (summer session)**Subject examined:**Computational quantum physics**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Oral (summer session)**Subject examined:**Computational quantum physics**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Oral (summer session)**Subject examined:**Computational quantum physics**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Oral (summer session)**Subject examined:**Computational quantum physics**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Oral (summer session)**Subject examined:**Computational quantum physics**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Oral (summer session)**Subject examined:**Computational quantum physics**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks