Lecture and tutorial classes

Characterization and verification of quantum simulations

(Summer term 2019)

Quantum simulations and quantum computing are among the most exciting applications of quantum mechanics. More generally, in the quantum technology research field one aims to develop new devices using quantum superposition and entanglement. In a popular wording, these anticipated developments will lead to the second quantum revolution.

A main milestone is the use of quantum capabilities to solve a (computational) problem that cannot practically be solved otherwise. Theoretical proposals include integer factoring (Shor's algorithm), speed-ups for optimization and machine learning algorithms, the simulation of complex quantum systems, and certain sampling experiments specifically tailored to that milestone.

But if one cannot obtain the output of a quantum simulation or computation by conventional means how can one make sure that the outcome is correct? The output of integer factorization can efficiently be checked but, for instance, for the estimation of energies in quantum many-body systems, or outcomes of dynamical simulations, the situation is much less clear. Hence, for the development of trusted quantum technologies special characterization and verification techniques are urgently required.

This course gives an introduction to the research field, to the problems of characterization, validation, and verification, and first ways to solve them. More specifically, quantum state tomography, quantum states certification, quantum process tomography, and randomized benchmarking will be covered. In particular, the course provides an overview of the latest developments in this still young and very active research field. The approaches of the course are mainly of conceptual and mathematical nature.


  • Fidelity estimation and certification of quantum state preparations

  • Randomized benchmarking for quantum dynamics

  • Quantum state and process tomography (based on compressed sensing)

Exercises and other files will be uploaded here.

Lecture notes can be found here.

Formal things

Lecture and tutorial class

  • Monday 8:30am, room (lecture)

  • Tuesday 10:30am, room (tutorial class)

  • Prerequisites for attending are basic knowledge

    • Linear algebra,

    • Calculus, and

    • Quantum mechanics.

  • The course consists of

    • 2 hours lecture (L) by Martin Kliesch (teaching@mkliesch.eu) and

    • 2 hours tutorial (T) class (Übung) by Raphael Brieger (raphael.brieger@hhu.de)

    • There will be student presentations in the last part of the course (if the number of students admits it).

  • Assignments will be uploaded every two weeks. The solutions to the assignment sheets need to be handed in. At least 70% of the sheets need to be finished as a prerequisite for the exam. There will be no corrections but the solutions will be discussed in the tutorial classes.
    Collaboration policy: Collaborations are only allowed, if they are disclosed on the exercise submissions.

  • There will be oral exams after the term has ended.

Preliminary schedule


01 L Motivation, course outline, formalities Exercise 1 handed out
02 T Introduction, POVMs, SVD
08 L QST, informational completeness
09 L Informational completeness 2
15 L Linear inversion, frame theory Due date for Ex. 1, Ex. 2 handed out
16 T Student questions & discussion
22 Public holiday (Easter Monday)
23 L k-designs
29 L Symmetric measurements Due date for Ex. 2, Ex. 3 handed out
30 T Discussion of Ex. 2


06 L Symmetric measurements
07 T
13 L Certification Due date for Ex. 3, Ex. 4 handed out
14 T Tail bounds, direct fidelity estimation
20 L Direct fidelity estimation
21 T Student questions & discussion
27 L Direct fidelity estimation Due date for Ex. 4, Ex. 5 handed out
28 T Discussion Ex. 4


03 L Quantum state certification
04 L Quantum state certification
10 Public Holiday (Whit Monday)
11 T Discussion Ex. 5 Due date for Ex. 5, Ex. 6 posted on June 12
17 L Q. state tomography II
18 L Q. state tomography II
24 L Quantum processes Due date for Ex. 6, Ex. 7 handed out
25 T Discussion Ex. 6


01 L Randomized benchmarking (RB)
02 L Interleaved RB and process tomography
08 L Projected least squares estimation Due date for Ex. 7
09 T Discussion Ex. 7