The course will start with a basic introduction of some experimental observations which justify quantum mechanics principles and in particular the probabilistic description of a physical system. We will then introduce the concepts of quantum bits and describe how to manipulate them through quantum circuits. The advantages of quantum computing will be illustrated through some specific examples and algorithms including quantum search, quantum teleportation, and quantum random walks. Two lectures will be devoted to quantum communications and one will provide additional information about the physics of quantum systems and the devices used for technological applications.
Charles H. Bennett and Gilles Brassard, Quantum cryptography: Public key distribution and coin tossing, Theoretical Computer Science, vol. 560, 1984, pp. 7–11.
Artur K. Ekert, Quantum cryptography based on Bell’s theorem, Physical Review Letters 67 (6), pp. 661-663, 1991.
C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen Channels, Physical Review Letters, vol. 70, pp. 1895-1899, 1993.
Ambainis, A., Bach, E., Nayak, A., Vishwanath, A., Watrous, J.: One-dimensional quantum walks. In: Proceedings of the 33th STOC, pp. 60–69. ACM, New York (2001).
Kempe, J.: Quantum random walks - an introductory overview. Contemp. Phys. 44(4), 302–327 (2003)
Szegedy, M.: Quantum speed-up of Markov chain based algorithms. In: Proceedings of FOCS 2004.
Portugal, R., Santos, R.A.M., Fernandes, T.D., Gonçalves, D.N.: The staggered quantum walk model. Quantum Inf. Process. 15(1), 85–101 (2016)
Avrachenkov, K., Jacquet, P., Sreedharan, J.K.: Distributed spectral decomposition in networks by complex diffusion and quantum random walk. In Proceedings of IEEE INFOCOM 2016.
Ali Anwar, Chithrabhanu Perumangatt, Fabian Steinlechner, Thomas Jennewein, and Alexander Ling, Entangled Photon-Pair Sources based on three-wave mixing in bulk crystals, arXiv:2007.15364v2, 2021.
Jay M. Gambetta, Jerry M. Chow and Matthias Steffen, Building logical qubits in a superconducting quantum computing system, Quantum Information, 2017
Evaluation
40% classwork (a 10-minute test at every lesson, only 5 best marks will be considered), 60% final exam.
Lessons
Lessons will be from 9.00 to 12.15 in room O+107, Templiers 1.
First lesson (October 14, G. Neglia): refresher on linear algebras and complex numbers,
an illustrative example (spin component measurements) to introduce some quantum mechanics findings (quantization, probabilistic description, measurements interact with the system), qubits, computation basis, bra-ket/Dirac notation, orthogonal kets, deriving a computational basis for spin component measurements,
principles of quantum mechanics (observables and Hermitian operators, measurements and eigenvalues, observation probabilities and orthonormal basis). See [Susskind, chapters 1-3].
Second lesson (October 21, G. Neglia): derive the operators for the 3 spin components, multi qubit systems (product states and Bell states, entanglement), unitary property of quantum gates, single qubit gates (Pauli matrices, rotations, Hadamard, quantum wires), multi-qubit gates (CNOT, Toffoli), quantum circuits, how to recover all classic logic circuits, a quantum computer to simulate quantum systems or to speed-up algorithms. See [Susskind, chapter 3], [Nielsen, Sec. 1.3], [Aaronson08].
Third lesson (October 28, P. Nain): BB84, E91, dense coding, and teleportation. See [Mermin, chapter 6] and the original papers for BB84 [Bennett84], E91 [Ekert91], and teleportation [Bennett93].
Fourth lesson (November 18, K. Avrachenkov): review of background material on discrete-time and continuous-time random walks on graphs, discrete-time quantum walks, in particular, the coined model [Ambainis01,Kempe03], Szegedy's model [Szegedy04] and the staggered model [Portugal16], continuous-time random walks [Kempe03], application of complex diffusion models and quantum walks to the distributed estimation of graph spectrum [Avrachenkov16].
Fifth lesson (November 25, G. Neglia): no-cloning algorithm, reversible circuits and Landauer's principle, ancilla bits and uncomputation, Grover's search algorithm (geometric description, number of Grover iterates and success probability, how to perform rotations). See [Nielsen, Sec. 3.2.5, Sec. 6.1] and the corresponding part of Quantum Country.
Decoherence, density matrix, fidelity. See [Susskind, Sec. 7.2-7.5], [Van Meter, Sec. 8.2.2].
Sixth lesson (December 2, G. Neglia): Purification, quantum repeaters, entanglement swapping [Van Meter, Sec. 1.3, Ch. 9, Ch. 10], quantum error correction [Nielsen, Sec. 10.1-10.2], [Mermin, Sec. 5.1-5.2].
Seventh lesson (December 9, V. D'Auria): Quantum state decoherence, Superconducting Qubit, Non-linear optics as source of quantum entanglement for quantum communication [Anwar21,Gambetta17].
Exam
The exam will be on December 16th in room O+107, Templiers 1
