Matrix Product States (2015)

Name of course:

Matrix Product States

ECTS credits: 5

Course parameters:

Language: English

Level of course: PhD Course

Semester/quarter: Q1 2015

Hours per week: 6

Capacity limits: No limit

Objectives of the course:

The course objectives are to give the students an introduction to numerical methods for simulating time-dependent dynamics of many-body systems with focus on the matrix product state method. A theoretical introduction to the concepts will be given, but a main goal of the course is also the actual numerical implementation.

The course will present the matrix product state method for deterministic dynamics and generalize to the case of measurement induced stochastic dynamics.

Professor Ulrich Schollwöck will be invited to give an introductory tutorial with applications from an expert on the subject. Furthermore, we will try to arrange talks by other relevant speakers throughout the quarter.

Learning outcomes and competences:

At the end of the course the student should be able to:

  • Describe the problem of simulating time-dependent dynamics of many-body systems.
  • Understand exact diagonalization procedures and their shortcomings.
  • Describe the theoretical concept of matrix product states.
  • Apply the matrix product states to calculate ground-states, time-dependent dynamics and measurement induced dynamics.
  • Evaluate the quality of matrix product state methods in current research.
  • Reflect upon the forces and weaknesses of matrix product states.


Compulsory programme:

Active participation in every part of the course. Completion of final exam project.

Course contents:

Exact diagonalization, matrix product states and operators, many-body groundstates, many-body time evolution, time-dependent simulation of many-body systems, many body quantum measurements and trajectories in the matrix product state formulation.


Advanced quantum mechanics, or similar course.

A rudimentary understand of programming outside MATLAB is an advantage.

Name of lecturer:

Professor Klaus Mølmer (Responsible), Department of Physics and Astronomy, Aarhus University

Type of course/teaching methods:

The course will take the form of a study group.

6 hours per week with a mixture of lectures prepared by the students, discussions among the students, and sessions with numerical exercises.

Guest lecturers will be invited to present relevant topics and applications.


The density-matrix renormalization group in the age of matrix product states by Ulrich Schollwöck

Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems by F. Verstraete, J.I. Cirac, V. Murg

Efficient parametric inference, estimation and simulation of open quantum systems by Søren Gammelmark (PhD thesis)

Supplementary notes.

Scientific papers.

Selected chapters from relevant books.

Course homepage:


Course assessment:

Approval of final project by Klaus Mølmer. (Passed/Not passed)


Department of Physics and Astronomy

Special comments on this course:



Q1 of 2015: 24 August to 28 October 2015


Department of Physics and Astronomy, Room TBA


For registration, contact Alexander Holm Kiilerich or Jens Jakob Sørensen, e-mail: or

Comments on content: 
Revised 20.06.2016