DAVE3705 Mathematics 4000 Course description

The course shall prepare students for master’s degree programmes at universities and university colleges where different types of differential equations is used.

The elective course is initiated provided that a sufficient number of students choose the course.

None

No requirements over and above the admission requirements.

After completing the course, the student is expected to have achieved the following learning outcomes defined in terms of knowledge, skills and general competence:

Knowledge

The student is capable of:

• explaining the concepts of analytic function, ordinary, singular and regular singular points
• using series to solve differential equations
• defining the Laplace transform and derive it's basic properties
• explaining what characterize Fourier series and how they can be used to solve ordinary and partial differential equations
• giving examples of elliptical, parabolic and hyperbolic partial differential equations and how they are solved

Skills

The student is capable of:

• solving higher order linear differential equations with constant coefficients
• using power series and Frobenius series to solve second order linear differential equations with variable coefficients
• using the Laplace transform to solve non-homogeneous linear differential equations modelling oscillating systems
• determining the Fourier sine series and the Fourier cosine series of symmetrical expansions of non-periodic functions
• solving boundary value problems relating to partial differential equations in closed domains by separation of variables

General competence

The student:

• has acquired good skills in solving ordinary and partial differential equations

Lectures and exercises. Practical exercises are solved individually with the help of the pre-written compendium with solutions for all exercises and previous exams. At the end of the course, previous exams will be reviewed during the six weekly periods.

None

Individual written exam, 3 hours.

The exam result can be appealed.

Quantum information technology implements quantum phenomena to process information and communicate it beyond the limits of the classical world. According to the EU Quantum Technologies Flagship report, such technology is based on the following pillars:

• Quantum computation
• Quantum communication
• Quantum simulation
• Quantum metrology and sensing

This course will introduce students to the first three of these fields, by equipping them with knowledge of principles, ideas, and methods. Many of these methods are also applicable within several other fields.

Prior knowledge in quantum physics is not required. The first few weeks of the course is dedicated to an introduction to key concepts in quantum physics. These concepts are introduced in a practical manner - with emphasis on simulation and phenomenology rather than theory.

The students will be trained to create their own quantum algorithms, simulate quantum systems, and implement the corresponding programs on classical and quantum computers. By implementing calculations and simulations of quantum systems, the students will learn about the fundamental quantum phenomena and key concepts. Moreover, in order to lay the proper foundation, the fundamental concepts of classical information theory is introduced.

A selection of recently published quantum algorithms and methods, including communication protocols, computational methods of modern quantum physics, and optimization algorithms, will be presented and analysed. Particular focus will be given to applications in data science in order to address research challenges in sustainable systems. Finally, the most recent challenges and particular proof of concept problems, including so-called quantum supremacy, will be addressed.

Students taking the course should be familiar with elementary calculus, including the concepts of complex numbers and numerical methods, and with basic linear algebra. Moreover, the students should be in command of a programming language/computing environment such as, e.g., Python, MATLAB or C(++).

In this regard, it is worth mentioning that some relevant mathematical and numerical concepts will be revised during the the first lectures.

A student who has completed this course should have the following learning outcomes defined in terms of knowledge, skills and general competence:

Knowledge

On successful completion of the course the student

• is familiar with fundamental key concepts within information theory such as Shannon Entropy, noiseless and noisy-channel coding theorems, and optimal coding algorithms.
• knows what a qubit is and how the information content grows when qubits are connected.
• is familiar with the elementary operations, or gates, of quantum computing - including gates such as the Hadamard gate and CNOT.
• knows the present state of the art when it comes to existing quantum computers.
• can implement simple quantum algorithms and run them on actual quantum computers.
• knows basic quantum communication protocols such as key distributions and secret sharing and understands the ideas behind them
• is familiar with several methods, such as Shor’s algorithm and quantum annealing, which enables quantum computers to solve problems considerably faster than classical computers.
• is familiar with how quantum technology affects traditional encryption schemes, and provides novel ones.

Skills

On successful completion of the course the student

• is able to model and simulate numerically simple quantum systems and processes - both on classical and quantum computers.can independently devise, implement and run calculations and simulations of simple quantum systems.
• can design her/his own quantum algorithms.

General competence

On successful completion of the course the student

• is familiar with several phenomena specific to quantum physics - such as quantization, particle interference, collapse of the wave function, particle spin, entanglement and decoherence - and how they may manifest themselves within quantum computing.
• is familiar with how information may be described by quantitative means - both within a classical and a quantum context.
• knows how to revise and improve on implementations of quantum programs.
• can address some of the practical challenges related to building quantum computers.
• knows the importance of quantum computing within information technology and the open challenges yet to be solved in this scope.

The teaching is organized in sessions where the subject material is presented, and in sessions where the students solve problems on their laptops and prototype quantum computers. The latter is done using online cloud platforms currently provided by enterprises such as, e.g., IBM and D-Wave. Between these sessions, the students are expected to work independently, using their computers, access to quantum computers, and course notes.

In the last stage of the cource, the students are required to complete and present an individual project that involves (i) simulation of a quantum system/process, (ii) simulation of a quantum communications protocol, or (iii) creation of a quantum code and its implementation on a quantum processor using an online cloud platform. The project should be concluded by submitting a report which provides a description of the project, its motivation and implementation, and an analysis the obtained results.