Lecture "Quantum Information, Quantum Computing, and Quantum Algorithms" (winter term 2022/23)

Lecturer: Norbert Schuch


Overview

Quantum Information Theory is concerned with how we can process information and perform computations in a world which is governed by the laws of quantum mechanics, like the world we live in.  The lecture will provide an in-depth introduction to the field of Quantum Information Theory, with a special focus on Quantum Computing and Quantum Algorithms, taught from a mathematical perspective. In particular, no prior knowledge of quantum mechanics or any other physics will be required; solid foundations in linear algebra will be both necessary and sufficient to attend the lecture. The focus of the lecture will be on the underlying concepts and the key mathematical questions of Quantum Information and Computation, but depending on the interest, a birds-eye view on the main challenges and approaches towards building a real quantum computer can be included.

Planned topics include:

  • The basic formalism: States, evolution, and measurement
  • Mixed states, completely positive maps, and POVM measurements
  • Entanglement theory
  • Quantum cryptography
  • Quantum computation
  • Quantum algorithms
  • Quantum error correction
  • Quantum Shannon theory
  • Quantum complexity theory
  • Topological quantum computing


The lecture course consists of a four-hour lecture (250078 VO), and an associated two-hour tutorial/exercise session (250042 PS).

Prerequisites

No prior knowledge of quantum mechanics or any other physics will be required. Solid foundations in linear algebra will be both necessary and sufficient to attend the lecture.

Course material

Lecture notes

  1. Introduction
  2. The formalism: States, measurements, and evolution
    1. The formalism of Quantum Theory
    2. Mixed states
    3. The Schmidt decomposition & purifications
    4. POVM measurements
    5. General evolution: Completely positive maps
    6. Axioms of quantum theory (mixed version)
  3. Entanglement
    1. Introduction
    2. Bell inequalities
    3. Applications of entanglement: teleportation and dense coding
    4. Entanglement conversion and quantification
    5. Entanglement of mixed states
  4. Quantum Computing and Quantum Algorithms
    1. The circuit model
    2. Oracle-based algorithms
    3. The quantum Fourier transform, period finding, and Shor's factoring algorithm
    4. Grover's algorithm
  5. Quantum Error Correction
    1. Introduction
    2. The 9-qubit Shor code
    3. The Quantum Error Correction Conditions
    4. Basic propertes of Quantum Error Correction Codes
    5. Stabilizer codes

Exercise sheets

  • Sheet 1 (posted 14.10., discussed 19./24.10.)
  • Sheet 2 (posted 28.10., discussed 7./9.11.)
  • Sheet 3 (posted 9.11., discussed 14./16.11.)
  • Sheet 4 (posted 16.11., discussed 21./23.11.)
  • Sheet 5 (posted 23.11., discussed 28./30.11.)
  • Sheet 6 (posted 30.11., discussed 5./7.12.)
  • Sheet 7 (posted 7.12., discussed 12./14.12.)
  • Sheet 8 (posted 19.12., discussed 9./11.1.)
  • Sheet 9 (posted 11.1., discussed 16./18.1.)
  • Sheet 10 (posted 18.1., discussed 23./25.1.)

Literature

Main texts


Additional texts and lecture notes


Further reading

 

Organisatorial issues

The lecture takes place Monday 13:15-14:45 in Hörsaal 13, and Friday, 9:45-11:15 in Hörsaal 11 (Oskar-Morgenstern-Platz 1, 2nd floor). The first lecture is on 3.10..

For the tutorial/proseminar, there will be two groups, one on Monday 8:00-9:30, and one on Wednesday 8:00-9:30 (see ufind for locations).   If you are interested in participating in the lecture/tutorial but cannot register formally, please get in touch with me.

The examination/grading modalities for both the lecture and the tutorial will be communicated on moodle.