Quantum Algorithms

Quantum computation has been a very active research area for the past decade. Feynman initiated the idea of building a computer that exploits laws of quantum mechanics in order to solve computing problems that could not be solved efficiently on classical computers. Today, two main results lend credence to this model of computation. P. Shor gave a quantum algorithm that, in polynomial time, decomposes any integer into its prime factors, whereas no such algorithm is known in the classical models of computation. Indeed, that best classical algorithms for factorizing integers run in sub-exponential time. A few years later, L. Grover designed an algorithm that finds any marked element in an unstructured list, and whose running time quadratically improves over the best possible randomized algorithm. The main motivation for a quantum computer, illustrated by the above results, is to solve tasks more efficiently than classically. This line of research targets such improvements of quantum computing over classical computing, among others, in the field of group theoretical algorithms and quantum walk based search procedures.

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Quantum Machine Learning (QML) and Learning Theory

Machine learning impacts society in areas such as autonomous vehicles, the internet of things, and e-commerce. Early breakthroughs such as Shor’s, Grover’s, and other algorithms promise quantum advantages for a substantial range of applications. The line of research on quantum machine learning explores the interface of quantum information, quantum computing, and machine learning. At the high level, efforts include designing quantum algorithms, developing fundamentally new algorithmic building blocks, and understanding more deeply the power of quantum computers. Specific focus areas include linear algebra problems such as matrix inversion, principal component analysis, and singular value transformations and optimization problems such as linear/semidefinite programming, support vector machines, generalised linear models, and neural networks. Another focus area is the design and implementation of quantum machine learning algorithms on presently available and near-term quantum computers. Such efforts have the potential to translate into quantum advantages for machine learning, AI, and optimisation.

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Quantum Finance

At the intersection of finance and quantum computing, it is interesting to ask about situations where the market and the assets traded on the market themselves have quantum properties. We investigate settings where instead of by classical probabilities the market is described by a pure quantum state or, more generally, a quantum density operator. This setting naturally leads to a new asset class, which we call quantum assets. Under the assumption that such assets have a price and can be traded, we develop an extended definition of arbitrage to quantify gains without the corresponding risk. Our main result is a quantum version of the first fundamental theorem of asset pricing. For future work, we consider the case of expected utility maximization and quantum assets. We would like to answer the question if and in what cases quantum assets can provide portfolios with larger utility than fully classical portfolios.

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