Fundamentals of quantum algorithms and their implementation

Quantum teleportation circuit with derived individual quantum states. As can be seen, only 2 classical bits are needed between the Sender and the Receiver, and also that they both own one qubit from the so-called EPR pair that was created in advance. The last equation says that 1 of 4 possible quantum states can occur at the output, and it is the value of these 2 measured classical bits that decides which one it will be. It is therefore possible to teleport the quantum state ψt over a theoretically unlimited distance using only a message about which of the 4 possible combinations occurred at the Sender. In addition, the security of the transmission is also a great advantage, because only 2 classical bits flow between the Sender and Receiver, from whose values alone the teleported quantum state cannot be reconstructed. Albert Einstein once described this possibility of quantum teleportation as "spooky action at a distance", but physicists have since shown that it is indeed possible, and even over very long distances.

Project reference: 2212

Quantum computers are based on a completely different principle than classical computers. The aim of this project is to explain this difference. Thanks to this insight, potential students should then understand, for example, why quantum computers are able to solve the problem of exponential complexity in less than exponential time, what is the difference between quantum natural parallelism and parallel programming on HPC, or what the principle of quantum teleportation is based on.
As this field has been undergoing hectic progress in recent times, new research results are constantly being published. It is therefore not possible to cover all of these new developments in a few weeks of teaching, so this project will focus mainly on the theoretical foundations, mathematical description and practical testing of the resulting quantum circuits on real quantum computers and their simulators.

Quantum teleportation circuit with derived individual quantum states. As can be seen, only 2 classical bits are needed between the Sender and the Receiver, and also that they both own one qubit from the so-called EPR pair that was created in advance. The last equation says that 1 of 4 possible quantum states can occur at the output, and it is the value of these 2 measured classical bits that decides which one it will be. It is therefore possible to teleport the quantum state ψt over a theoretically unlimited distance using only a message about which of the 4 possible combinations occurred at the Sender. In addition, the security of the transmission is also a great advantage, because only 2 classical bits flow between the Sender and Receiver, from whose values alone the teleported quantum state cannot be reconstructed. Albert Einstein once described this possibility of quantum teleportation as “spooky action at a distance”, but physicists have since shown that it is indeed possible, and even over very long distances.

Project Mentor: Jiří Tomčala

Project Co-mentor: /

Site Co-ordinator: Karina Pešatová

Learning Outcomes:
Students should be introduced to a completely different principle of quantum computers and their programming. They should also be able to design basic quantum circuits.

Student Prerequisites (compulsory):
Knowledge of basic linear algebra.

Student Prerequisites (desirable):
Imagination and experience with Python programming.

Training Materials:
“Quantum Computation and Quantum Information” by Isaac Chuang and Michael Nielsen.
“Quantum Algorithm Implementations for Beginners” by various authors, which can be downloaded from here: https://arxiv.org/pdf/1804.03719.pdf
Etc.

Workplan:

1st week: Quantum bits
2nd week: Single qubit operations
3rd week: Quantum gates
4th week: Quantum teleportation
5th week: Deutsch–Jozsa algorithm
6th week: Grover’s algorithm
7th week: Quantum Fourier transform
8th week: Shor’s algorithm

Final Product Description:
Students test their own quantum circuits on a simulator and, if possible, on a real quantum computer. This includes the correct evaluation of their measured results.

Adapting the Project: Increasing the Difficulty:
Attempting to create interesting modifications of the discussed algorithms.

Adapting the Project: Decreasing the Difficulty:
Skipping overly complicated algorithms.

Resources:

Organisation:
IT4Innovations National Supercomputing Center at VSB – Technical University of Ostrava

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