Final Report Summary - QUINTYL (Quantum Information Theory with Liouvillians) The main objectives of the Marie Curie Intra-European Fellowship project QUINTYL (Quantum Information Theory with Liouvillians) were, on the scientific side, to solidify and extend the mathematical framework underlying quantum dissipative time-evolutions, and on the academic training side, to establish the research fellow as an independent researcher in the field of Quantum Information Theory and to provide him with essential skills necessary to lead his own research group. Over the project time of two years, all objectives of the project have been achieved and from the project several new research directions have emerged on which work is continuing.More specifically, the scientific aspects of the project have addressed the idea of quantum computing by dissipative ("noisy") processes. The proposal of dissipative quantum computing has emerged about five years ago as a promosing and robust way to implement quantum algorithms. As one of several proposals for a universal quantum computer, it holds the hope of advancing technology in signifcant ways, e.g. by allowing for the simulation of large-scale quantum systems to assist in material design. The scientific aim of our project was to introduce more rigorous mathematical methods into this novel dissipative approach, and to establish dissipative versions of several procedures that have already been used in the unitary quantum computing framework or in the discrete-time framework. The most significant scientific results were, first, the introduction of methods from non-commutative harmonic analysis into the theory of quantum and classical Markov processes, in particular to control their convergence behaviour. This method has already found further use in the stability analysis of Markov chains. Secondly, we introduced a new information-theoretical coding and capacity framework that is suited to quantify the storage capacity of noisy quantum memories, where information is transmitted over time but not necessarily through space. We were able to quantify this storage capacity for several relevant classes of control operations that may be applied during the storage time; experimental quantum memory implementations may be benchmarked against our analytical results, mimicking the relation between practical classical coding schemes and Shannon's theorems. Thirdly, we established a so-called Landauer bound in the quantum domain and improved the usual bound to quantify both its gap and explicit dimension-dependence. These results give fundamental energy restrictions on any future implementations of quantum computers. Our finite-size bounds furthermore yield a tradeoff between the process time and energy comsumption. The fourth main result is our proposal of fault-ignorant algorithms, whose main feature is the reliable operation in a noisy environment where the noise is not precisely known. We have demonatrated the usefulness of this idea on the task of unstructured search, showing a speedup compared even to noiseless classical algorithms. We expect our proposal to be especially relevant for initial implementations of quantum computers which will feature a strong tradeoff between computer size and coherence time. The results of the scientific project part have been published in eight papers in scientific journals, with several papers on continuing work still in preparation. Furthermore, the results have been disseminated also by presentations given at scientific conferences in the field, among them the Quantum Information Processing conferences in 2013 and 2014, where three of the above main results were accepted and presented as contributed talks.The research fellow has, in close consultation with the scientist-in-change, successfully pursued various measures to develop skills beneficial for him to attain a position as an independent researcher leading his own research group. Firstly, within the host research group, the research fellow has (co-)supervised three PhD projects leading to scientific publications, apart from several master and bachelor projects.This has provided the research fellow with crucial experience in project planning and team management. Furthermore, the research fellow has taken great responsibility in organizing the weekly group research seminar, several group retreats with external guests, the managing and hosting of outside collaborators, and in the informal mentoring and advising. Secondly, the research fellow has gained substantial teaching experience, through the teaching of a specialized one-semester lecture class on the Mathematial Basis of Quantum Statistical Physics, which was designed and conducted solely by the research fellow, as well as through the co-teaching of bachelors and masters student seminars, including personal advising of students. Thirdly, over the course of the project, the research fellow has participated in many conferences and workshops and was invited to numerous research seminars. At these occasions, the project results have been presented in the scientfic public. Also, several ideas for the project were catalyzed through informal discussions with other researchers at these occasions, leading to an ongoing collaboration. These conference and research visits have furthermore provided the research fellow the opportunity to establish himself as an independent and visible figure in the field of Quantum Information Theory and Mathematical Physics. Fourthly, in the course of the project a habilitation committee, consisting of three internationally renowned researchers, has been assembled for the German habilitation process of the research fellow. The habilitation process is progressing well according to plan due to the fellow's scientific research outcomes, his student supervision and teaching, as well as grant and other organizatorial administration.In conclusion, both the scientific and academic training outcomes of the present project are significant. On the scientific side, the project results have strengthened and underscored the feasibility of dissipative approaches to the processing of quantum information. This may provide cucial steps towards the design of a practicable quantum computer, which has the ability to transform present-day computing technology and to influence society in manifold positive ways. On the academic training side, the project has greatly assisted the research fellow in his teaching and supervising abilities and in his scientific visibility, as well as the students under his supervision in their research abilities. Thus, both the scientific and academic training results of this project are expected to ultimately benefit the competitiveness of the European Research Area through technological advancement and through strengthening of the human potential in research in Europe.
