The quantized electromagnetic field is a well-known example of a continuous variable system. The partially reflecting mirror, termed the beam-splitter, is the primitive for all passive transformations [1]. The beam-splitter transform has been used for the basis of generalizing the entropy power inequality to quantum mechanical setting [2,3]. The central formula is the map (ρ_X, ρ_Y) ↦ ρ_{X ⊞λ Y} = Tr₂(U_λ(ρ_X ⊗ ρ_Y)U^†_λ), where U_λ is the beam-splitter transform. To achieve the same for the discrete or finite dimensional case [4,5] we need a generalization of the beam-splitter for finite dimensions. A promising approach is to use the discrete phase space distribution introduced by Wootters [6].

The aim of this assigment is to understand the phase space formalism for continuous and discrete quantum systems, and to define a discrete version of the usual beam-splitter in a way that is consistent in the limit of large dimensionality with the continuous variable case.

The assigment is suitable both for a bachelors and masters thesis. In case of a bachelor thesis the emphasis will be on the study of the phase space formalism and the formulation of the problem of the discrete beam-splitter, with the possibility to continue the research as master's thesis.

For more about what other things our research group does please visit our web page.

1. Wolfgang P. Schleich: "Quantum Optics in Phase Space," Wiley-VCH Verlag, Berlin (2001)
2. Robert König and Graeme Smith: "The Entropy Power Inequality for Quantum Systems," IEEE Trans. Inf. Theory, 60, 1536-1548, (2014)
3. Stefan Huber, Robert König, and Anna Vershynina: "Geometric inequalities from phase space translations," J. Math. Phys. 58, 012206 (2017)
4. K. Audenaert, N. Datta, and M. Ozols: “Entropy power inequalities for qudits,” J. Math. Phys. 57, 052202 (2016)
5. S. Guha et al.: “Thinning, photonic beam splitting, and a general discrete entropy power inequality,” in 2016 IEEE International Symposium on Information Theory (ISIT), IEEE, Barcelona, 2016. pp. 705–709.
6. W. K. Wootters: "A Wigner-Function Formulation of Finite-State Quantum Mechanics," Ann. Phys. 176, 1-21 (1987)

Description: Quantum computers exploit genuine features of quantum mechanics to achieve a stunning speedup over classical computers in solving certain problems. While practical quantum computer hardware is not yet available, there exist a few tools for developing quantum programs. At present the most popular quantum programming toolkit is the open source Qiskit, initially developed to control the IBM-Q superconducting quantum computers, however, more platforms are being added with time.

The goal of the project: Learning and understanding basic quantum algorithms and implementing them to be run on a chosen quantum hardware. The project is meant to also provide an opportunity to understand in more detail the physics of the particular quantum hardware, as well as to establish the basis for more advanced applications of quantum computers or to extend the functionalities of the quantum SDK subsequently.

Expected skills and knowledge: solid knowledge of quantum mechanics (for masters'), and familiarity with programming, preferably python.

Language: The working language will be English. The preferred language for thesis is English, although Czech is also possible.

[1] M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Information: 10th Anniversary Edition. (Cambridge University Press, 2011).
[2] Eleanor Rieffel and Wolfgang Polak. Quantum Computing: A Gentle Introduction (MIT Press, 2014)
[3] http://www.qiskit.org
[4] Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics, A. Gilyén et al. STOC 2019: Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, 193–204 (2019)

Kvantové procházky popisují šíření kvantové částice na grafu nebo mřížce. Na rozdíl od klasické náhodné procházky, kde je pohyb částice náhodný, se kvantová procházka vyvíjí v koherentní superpozici možných stavů. Cílem práce bude zkoumat asymptotické vlastnosti kvantových procházek s jednou i více částicemi. Důraz bude kladen zejména na určení tvaru limitního pravděpodobnostního rozdělení v závislosti na dynamice kvantové procházky a počátečních podmínkách. Dále se bude zkoumat vliv interakce s okolím, dekoherence, statického nebo dynamického šumu a perkolace. V případě procházek s více částicemi se zaměříme rovněž na roli bosonové a fermionové statistiky a interakce mezi částicemi.

• D. Reitzner, D. Nagaj, V. Bužek, Quantum walks, Acta Physica Slovaca 61, 603-725 (2011)
• A. Ahlbrecht, V.B. Scholz, and A.H. Werner, Disordered quantum walks in one lattice dimension, J. Math. Phys. 52, 102201 (2011)
• B. Kollár, T. Kiss, J. Novotný, I. Jex, Asymptotic dynamics of coined quantum walks on percolation graphs, Phys. Rev. Lett. 108, 230505 (2012)