Quantitative Finance is a rapidly changing environment, and the financial industry is always on the lookout for new techniques and new technologies able to harness the rise of big data and the availability of computing power.

Quantum computing, though not a recent field, has gained huge popularity in the past few years with the development of small-scale quantum computers and quantum annealers. These have in turn set directions for new algorithms, hybrid between classical and quantum, and tailored for such computers. The financial industry is now looking at such developments and there is common agreement that this will be one of the leading advances in the coming decade.
The goal of this course is to provide an introduction to this new technology and these new algorithms and show them how they can be used to solve financial problems.

Plan : The course will walk along the following steps:

- Introduction to Quantum Mechanics and the basics of quantum computations
This part will introduce the key mathematical tools for quantum computing as well as the formalism in which it takes place. A large part of it is anchored in linear algebra, but expressed using Dirac's formulation.

- Quantum neural networks
Classical neural networks (for which there is a dedicated course in the programme) have become ubiquitous in Quantitative Finance, for synthetic data (Generative Adversarial Networks), Reinforcement Learning, or high-dimensional PDEs. We shall investigate how to build a quantum analogue to those and how (and if) quantum tools allow for stronger performance and better interpretability.

- Adiabatic Quantum annealing for (portfolio) optimisation
Quantum annealing using Hamiltonian formulations to express the objective function of an optimisation problem, and translates it into finding the minimal eigenvalue/eigenfunction of some matrix. We will try to understand the precise steps of this approach and how it allows to solve classically hard problems.

Each part will develop the theoretical aspects of the problem and show how to use them for practical problems in Quantitative Finance. To do so, we shall make heavy use of Python and Jupyter notebooks.