• Sophie Laplante (Université Paris Cité, IRIF)
• Frédéric Magniez (CNRS, IRIF)

Each year computing machines become faster and faster, but they use still use at their base the same Newtonian physics. Feynman in 1982 already asked about the necessity of this restriction to classical physics. The idea behind quantum computation is to use quantum phenomena to solve tasks that conventional machines cannot achieve.

Historically the first result that showed the superiority of the quantum model was in cryptography. Bennett and Brassard in 1984 gave a first quantum protocol for perfectly secure key distribution. Such an unconditional security does not exist in the classical world.

At present many important concepts of theoretical computer science have been extended to quantum computation, from communication to algorithms and error correcting codes.

The aim of this course is to present the bases of several concepts about quantum computation. The emphasis will be on quantum algorithms and communication. We will describe the basics of Quantum Computation and its applications in algorithms, communication complexity and nonlocality.

Algorithms, basic notions in computational complexity, basic notions in linear algebra and probability.

Lectures take place Fridays 12:45, room 1002. Unless specified otherwise lectures are in the 3-hour format. Lectures will take place in French, English upon request. (In the past all lectures have been in English.) References in brackets refer to Ronald de Wolf's (RdW) and John Watrous' (JW) lecture notes.

Basic notions (order and contents subjects to change)

• Sept 20 SL: States, measurements, entanglement. Mermin-GHZ game, CHSH [RdW: Chapter 1.1-1.4 + 16.1-16.2] [JW: Lectures 1&2, Lecture 3 pp.6-8, Lecture 20]
• Sept 27 SL: Quantum non-locality. Local, quantum and non-signaling sets. Bell inequalities. Tsirelson inequalities. Gates and circuits, superdense coding.
• Oct 4 SL: density matrices, Entropy and Holevo's theorem. no-cloning theorem, teleportation. BPP in BQP. [RdW: 1.5] [JW Lecture 3, Lecture 4 pp. 1-3]
• Oct 10: NO CLASS
• Oct 18 SL: Quantum query model, Deutsch-Jozsa, Bernstein-Vazirani [RdW: 2.1, 2.4] [JW Lecture 4 pp. 3-6, Lecture 5 ]. Simon's algorithm. [RdW: 3.1-3.2] [JW: Lecture 6]

Algorithms and complexity (order and contents subjects to change, lecture notes will be provided)

• Oct 25 FM: Quantum Fourier Transform, Phase Estimation, Period Finding
• Nov 1st: NO CLASS → HOMEWORK
• Nov 8 FM: Factorization, Grover Search
• Nov 15 FM: Amplitude Amplification, Quantum Counting
• Nov 22 FM: Lower bounds for query complexity

This course is also part of the Quantum technologies Graduate School of Université Paris Cité. As a consequence, a joint QuanTech seminar is offered to MPRI students by IRIF and MPQ, which you are welcome to attend.

• Distributed by email

The final exam will take place on Dec 6, 9:00-12:00 (usual room 1002). Handwritten and printed lecture notes (from the class and the ones mentioned in this page) are allowed for the exam.

Please make sure to have your student ID with you for the exam.

Sample exam:

• Exam from 2013. Be aware that the content varies from year to year and in particular there is no crypto in the course this year.

We recommend the following lecture notes to use alongside the lectures:

• Ronald de Wolf Quantum Computing lecture notes
• John Watrous Quantum Computation Lecture notes
• John Preskill Lecture notes for PH219/CS219 mainly for information theory (10.1, 10.2.1) and Holevo's bound (10.6.2)

The following textbook is also suggested:

• Quantum Computation and Quantum Information. M. Nielsen et I. Chuang. Cambridge University Press, 2000.

This course is a prerequisite for the course Quantum information and cryptography which covers advanced algorithms, quantum cryptography and post-quantum cryptography.

The following courses are strongly recommended.

• 2.11.1 Advanced algorithms
• 2.11.2 Randomness in Complexity

If you are interested in Algorithms and Complexity, we recommend taking courses from the following list.

1st quarter courses

• 2-11-1 Advanced Algorithms
• 2-11-2 Randomness in Complexity
• 2-12-1 Techniques in Cryptography and Cryptanalysis
• 2-18-2 Algorithmique distribuée avec mémoire partagée
• 2-38-1 Algorithms for embedded graphs

1st and 2nd quarter courses

• 2-13-2 Error correcting codes and applications to cryptography
• 2-18-1 Distributed algorithms for the networks
• 2-24-1 Optimisation
• 2-29-1 Graph algorithms
• 2-33-1 Theory of Computations

If you are particularly interested in quantum computing you can also take courses from the Physics masters program Dispositifs quantiques