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Winter term 2021/22

Quantum computing

Quantum computing

Lecture: Quantum information theory (20110401)

  • Exam: The exam will be held on Wednesday, February 16, 8am - 11am, as a presence exam. It will take place in Hörsaal B.

  • Re-take Exam: There will also be a re-take exam.

  • Topic of the lecture:

    This course provides an overview of an exciting emerging field of research, that of quantum information theory. The field is concerned with the observation that single quantum systems used as elementary carriers of information allows for entirely new modes of quantum information processing and communication, quite radically different from their classical counterparts. Quantum key distribution suggests to communicate in a fashion, secure from any eavesdropping by illegitimate users. Quantum simulators can outperform classical supercomputers in simulation tasks. The anticipated - but now rapidly developing - devices of quantum computers can solve not all, but some delicate computational problems that are intractable on classical supercomputers. This course will give a comprehensive overview over these developments. At the heart of the course will be method development, setting the foundations in the field, building upon basic quantum theory. We will also make the point that quantum information is not only about information processing, but a mindset that can be used to tackle problems in other fields, most importantly in consensed matter research, with which quantum information is much intertwined for good reasons.

  • Content:

    1. Introduction
    1.1 Some introductory words
    1.2 Quantum information: A new kind of information?

    2. Elements of quantum (information) theory
    2.1 Quantum states and observables
    2.2 Unitary time evolution
    2.3 Composite quantum systems

    3. Two possible machines
    3.1 Quantum teleportation
    3.2 Dense coding

    4. Quantum channels and operations
    4.1 Complete positivity
    4.2 Kraus theorem
    4.3 Local operations and classical communication

    5. Entanglement theory
    5.1 Pure state entanglement
    5.2 Mixed state entanglement

    6. Quantum Shannon theory
    6.1 Capacities as optimal rates
    6.2 A glimpse at quantum Shannon theory

    7. Quantum key distribution
    7.1 BB84 scheme
    7.2 Entanglement-based schemes
    7.3 Words on quantum technologies

    8. Elements of quantum computing
    8.1 Why quantum computing?
    8.2 From classical to quantum computing
    8.3 Gottesman-Knill and Solovay-Kitaev theorems

    9. Quantum algorithms
    9.1 Deutsch and Deutsch-Jozsa algorithm
    9.2 Grover’s database search algorithm
    9.3 Exponential speed-up in Shor’s factoring algorithm
    9.4 Some thoughts on quantum algorithmic primitives

    10. Quantum computational models
    10.1 Adiabatic quantum computing
    10.2 Measurement-based quantum computing
    10.3 Further models of quantum computing

    11. Notions of computational complexity
    11.1 Cost functions
    11.2 Complexity classes

    12. Non-universal quantum computers
    12.1 Quantum simulators
    12.2 Variational quantum computers

    13. Quantum error correction
    13.1 Peres Code
    13.2 Shor code
    13.3 Elements of a theory of quantum error correction
    13.4 Stabilizer codes

  • Literature: See the script.
Group seminar: Recent advances in quantum many-body theory (20010416)