**MASTER OF LOGIC**

If you are a Master of Logic student interested in quantum software, it is recommended to follow the Logic and Computation track.

## Quantum Cryptography, June

Quantum computing has important implications for the field of cryptography. On the one hand, techniques in quantum information open up new possibilities for secure communication (e.g. quantum key distribution), while on the other hand, achieving security might be harder in a world where attackers potentially possess quantum computers.

The aim of this project is to familiarize students with a variety of topics within quantum cryptography, so that they are able to read and understand research papers in the field. We will study quantum key distribution in more detail, consider hardware limitations, as well as cryptographic primitives such as oblivious transfer, bit commitment and secret sharing in a quantum setting. In particular, we will highlight the differences with the classical setting for these tasks. There will also be time to study recent developments based on the interest of the students. Read more

## Basic Probability: Theory & Programming, February-May

This course is designed to provide students with the background in discrete probability theory and programming that is necessary to follow other more advanced master-level courses in areas such quantum computing and information theory, linguistics, natural language processing, machine learning, complexity theory, cryptography, etc. The goal is to make students that have had not much prior exposure to probability theory and/or programming feel comfortable in these areas. To achieve this goal we will try to illustrate the theoretical concepts with real-life examples that relate to topics in, e.g., computer science, gambling, and the like. Moreover, we will make sure that there is a close tie between the theoretical and practical part of the course, thus enabling students to apply their newly acquired theoretical knowledge to real problems. Read more about the theory, and programming part.

## Information Theory, November-December

Information theory was developed by Claude E. Shannon in the 1950s to investigate the fundamental limits on signal-processing operations such as compressing data and on reliably storing and communicating data. These tasks have turned out to be fundamental for all of computer science. This course gives a thorough background in the basics of classical information theory which will be very useful to understand the quantum aspects treated in the course Quantum Information Theory. Read more

## Computational Complexity, February-March

Complexity theory deals with the fundamental question of how many resources, such as time, memory, communication, randomness, etc., are needed to perform a computational task. A fundamental open problem in the area is the well-known P versus NP problem, one of the Clay Millennium problems. In this course, the basics of classical complexity theory are treated: NP-completeness, diagonalization, Boolean circuits, randomized computation, interactive proofs, cryptography, quantum computing, and circuit lower bounds. The course does not treat any quantum aspects, but is considered an ideal stepping stone towards quantum complexity theory. Read more

**MASTER OF MATHEMATICS / MASTERMATH**

As a mathematics student interested in quantum software, it is recommended to follow the Mathematical Physics track.

## Quantum Computing, February-May

Today’s computers—both in theory (Turing machines) and practice (PCs and smart phones)—are based on classical physics. However, modern quantum physics tells us that the world behaves quite differently. Quantum computation is the field that investigates the computational power and other properties of computers based on quantum-mechanical principles. Its main building block is the qubit which, unlike classical bits, can take both values 0 and 1 at the same time, and hence affords a certain kind of parallelism. The laws of quantum mechanics constrain how we can perform computational operations on these qubits, and thus determine how efficiently we can solve a certain computational problem. Quantum computers generalize classical ones and hence are at least as efficient. However, the real aim is to find computational problems where a quantum computer is much more efficient than classical computers. For example, Peter Shor in 1994 found a quantum algorithm that can efficiently factor large integers into their prime factors. This problem is generally believed to take exponential time on even the best classical computers, and its assumed hardness forms the basis of much of modern cryptography (particularly the widespread RSA system). Shor’s algorithm breaks all such cryptography. A second important quantum algorithm is Grover’s search algorithm, which searches through an unordered search space quadratically faster than is possible classically. In addition to such algorithms, there is a plethora of other applications: quantum cryptography, quantum communication, simulation of physical systems, and many others. The course is taught from a mathematical and theoretical computer science perspective, but should be accessible for physicists as well. Check out Ronald de Wolf’s lecture notes or read more.

## Quantum Information Theory, February-May

With the birth of Quantum Mechanics a century ago, our understanding of the physical world has profoundly expanded, and so has our understanding of information. While a classical bit assumes only discrete values, represented by the binary zero and one, a quantum-mechanical bit or qubit can assume a continuum of intermediate states. Quantum Information Theory studies the remarkable properties of this new type of information, ways of processing it, as well as its advantages and limitations. This course provides a mathematical introduction to Quantum Information Theory, starting with the fundamentals (such as quantum states and measurements) while also introducing some more advanced topics (entanglement theory and quantum communication).

## Symmetry and Quantum Information, February-March

(note that this course is not offered every year)

This course gives an introduction to quantum information theory. We use symmetries as a guiding principle and toolbox to study the fundamental features of quantum mechanics and solve quantum information processing tasks. Read more

## Semidefinite Optimization, February-May

The course aims at students with an interest in optimization, combinatorics, geometry and algebra. The purpose of the course is to give an introduction to the theoretical background, to the computational techniques, and to applications of semidefinite optimization. In particular, after successful participation in the course students will be able to: explain the theory and algorithmic approach to solve semidefinite optimization problems, give examples of problems in optimization, combinatorics, geometry and algebra to which semidefinite optimization is applicable, solve semidefinite optimization problems with the help of Matlab-based solvers, recognize problems which can be tackled using semidefinite optimization.

Even though this course focuses on classical techniques, semidefinite optimization has become of major importance in quantum information theory. Read more

## Machine Learning Theory, October-December

Machine learning is one of the fastest growing areas of science, with far-reaching applications. In this course we focus on the fundamental ideas, theoretical frameworks, and rich array of mathematical tools and techniques that power machine learning. The course covers the core paradigms and results in machine learning theory with a mix of probability and statistics, combinatorics, information theory, optimization and game theory. The course does not cover any aspects of quantum machine learning, but prepares you to dive into this area afterwards. Read more

**MASTER OF PHYSICS/ASTRONOMY**

As a mathematics student interested in quantum software, it is recommended to follow the Theoretical Physics track.

## Quantum in Business and Society, coming soon

The goal of this course is to gap the bridge between science and business/industry. It is designed for any (bachelor or master) students who have followed any of the other quantum courses. In every 2-hour lecture, the first 45 minutes will be used to explain the current cutting edge of science. After the break, the current real-word situation is sketched in about 15 minutes, and the rest of the time is used to explain how to possibly close this gap between theory and practice. Topics will include among others: quantum software, error correction, quantum simulations, algorithms, various quantum hardware platforms, quantum sensors, classical and quantum cryptography, quantum internet. In the last part of the course, participants learn about more business-oriented concepts such as international collaborations, societal impact of quantum technologies, various roles in quantum innovation. The course is currently being developed by QuSoft in collaboration with ATOS.

## Statistical Physics and Condensed Matter, May-June

Starting from basic notions of statistical mechanics and quantum theory, the students will be progressively introduced to the formalisms of second quantization, path integrals and functional field integration. The universal applicability of these methods in condensed matter and statistical physics will be emphasized through discussion of specific topics, to include (among others) low-dimensional interacting fermionic and spin systems, spin-charge separation and the Luttinger liquid, the Kondo effect, broken symmetry and Bose-Einstein condensation, superfluidity and superconductivity. Read more

## Mathematical Methods in Theoretical Physics, September-October

The course focuses on (algebraic) topological methods and group theory and their applications in theoretical physics. Subjects to be covered will be simplicial homology, homotopy, manifolds (real and complex), De Rham and Dolbeault cohomology. When time permits, the basics of fibre bundles and connections may also be covered. Read more

## Fermi Quantum Gases, February-March

This course will bring students close to current research topics in ultracold quantum gases and consists of an experimental and a theory part.

The experimental part, taught by Florian Schreck, introduces quantum simulation. Understanding quantum physics is often extremely hard or even impossible. This is especially true for emergent phenomena occurring in fermionic many-body systems, such as high-temperature superconductivity. Even more powerful classical supercomputers will not help. The problem is that quantum systems life in Hilbert space, which grows exponentially with system size and already for small systems exceeds the capacity of any classical computer. The solution is to use ultracold atoms or ions, over which we have perfect control, to simulate the difficult to manipulate quantum system we are interested in. Such quantum simulation even allows us to study systems that are of high theoretical interest but do not naturally occur. This course will introduce the basic building blocks of quantum simulations with ultracold atoms and explain their theoretical and experimental basis. We will use these building blocks to understand some of the most advanced quantum simulations performed to date, including quantum simulations based on Fermi gases.

The theory part treats the phenomenon of (low-temperature) superfluidity in Fermi systems. Starting from the properties of a normal Fermi gas, we give a detailed analysis of the BCS pairing instability and reveal its many-body nature. Various possible mechanisms of pairing together with the properties of superfluid fermionic systems will be discussed. We will then address the issues of superfluid pairing in two-dimensional Fermi systems and the phenomenon of spin-charge separation in one dimension. The lectures emphasize advances in theory and the description of the remarkable experimental progress with ultracold quantum gases over the last two decades.

At the end of the course, the student is able to derive and use the theory relevant for the implementation of important quantum simulations and the theory of degenerate Fermi gases. The student will also understand how a quantum simulator is constructed.

It is recommended, but not required, to take the *Bose-Einstein Condensates* course (5354BOEC6Y) offered October to December by the same lecturers before taking this course. Both courses together are an excellent starting point for a master thesis in condensed matter (theory or experiment) or in one of the labs exploring ultracold atoms, ions or molecules at VU, UvA, AMOLF or ARCNL.