@article{GrossNesmeVogtsetal.2012, author = {Gross, David and Nesme, V. and Vogts, H. and Werner, Reinhard F.}, title = {Index theory of one dimensional quantum walks and cellular automata}, series = {Communications in mathematical physics}, volume = {310}, journal = {Communications in mathematical physics}, number = {2}, publisher = {Springer}, address = {New York}, issn = {0010-3616}, doi = {10.1007/s00220-012-1423-1}, pages = {419 -- 454}, year = {2012}, abstract = {If a one-dimensional quantum lattice system is subject to one step of a reversible discrete-time dynamics, it is intuitive that as much "quantum information" as moves into any given block of cells from the left, has to exit that block to the right. For two types of such systems - namely quantum walks and cellular automata - we make this intuition precise by defining an index, a quantity that measures the "net flow of quantum information" through the system. The index supplies a complete characterization of two properties of the discrete dynamics. First, two systems S-1, S-2 can be "pieced together", in the sense that there is a system S which acts like S-1 in one region and like S-2 in some other region, if and only if S-1 and S-2 have the same index. Second, the index labels connected components of such systems: equality of the index is necessary and sufficient for the existence of a continuous deformation of S-1 into S-2. In the case of quantum walks, the index is integer-valued, whereas for cellular automata, it takes values in the group of positive rationals. In both cases, the map S bar right arrow. ind S is a group homomorphism if composition of the discrete dynamics is taken as the group law of the quantum systems. Systems with trivial index are precisely those which can be realized by partitioned unitaries, and the prototypes of systems with non-trivial index are shifts.}, language = {en} } @article{GuetschowNesmeWerner2012, author = {Guetschow, Johannes and Nesme, Vincent and Werner, Reinhard F.}, title = {Self-similarity of cellular automata on abelian groups}, series = {Journal of cellular automata}, volume = {7}, journal = {Journal of cellular automata}, number = {2}, publisher = {Old City Publishing Science}, address = {Philadelphia}, issn = {1557-5969}, pages = {83 -- 113}, year = {2012}, abstract = {It is well known that the spacetime diagrams of some cellular automata have a self-similar fractal structure: for instance Wolfram's rule 90 generates a Sierpinski triangle. Explaining the self-similarity of the spacetime diagrams of cellular automata is a well-explored topic, but virtually all of the results revolve around a special class of automata, whose typical features include irreversibility, an alphabet with a ring structure, a global evolution that is a ring homomorphism, and a property known as (weakly) p-Fermat. The class of automata that we study in this article has none of these properties. Their cell structure is weaker, as it does not come with a multiplication, and they are far from being p-Fermat, even weakly. However, they do produce self-similar spacetime diagrams, and we explain why and how.}, language = {en} } @article{ArrighiNesmeWerner2011, author = {Arrighi, Pablo and Nesme, Vincent and Werner, Reinhard F.}, title = {One-Dimensional quantum cellular automata}, series = {International journal of unconventional computing : non-classical computation and cellular automata}, volume = {7}, journal = {International journal of unconventional computing : non-classical computation and cellular automata}, number = {4}, publisher = {Old City Publishing Science}, address = {Philadelphia}, issn = {1548-7199}, pages = {223 -- 244}, year = {2011}, abstract = {We define and study quantum cellular automata (QCA). We show that they are reversible and that the neighborhood of the inverse is the opposite of the neighborhood. We also show that QCA always admit, modulo shifts, a two-layered block representation. Note that the same two-layered block representation result applies also over infinite configurations, as was previously shown for one-dimensional systems in the more elaborate formalism of operators algebras [18]. Here the proof is simpler and self-contained, moreover we discuss a counterexample QCA in higher dimensions.}, language = {en} } @article{ZollerBethBinosietal.2005, author = {Zoller, Peter and Beth, Thomas and Binosi, D. and Blatt, Rainer and Briegel, Hans J. and Bruss, D. and Calarco, Tommaso and Cirac, Juan Ignacio and Deutsch, David and Eisert, Jens and Ekert, Artur and Fabre, Claude and Gisin, Nicolas and Grangiere, P. and Grassl, Markus and Haroche, Serge and Imamoglu, Atac and Karlson, A. and Kempe, Julia and Kouwenhoven, Leo P. and Kr{\"o}ll, S. and Leuchs, Gerd and Lewenstein, Maciej and Loss, Daniel and L{\"u}tkenhaus, Norbert and Massar, Serge and Mooij, J. E. and Plenio, Martin Bodo and Polzik, Eugene and Popescu, Sandu and Rempe, Gerhard and Sergienko, Alexander and Suter, David and Twamley, John and Wendin, G{\"o}ran and Werner, Reinhard F. and Winter, Andreas and Wrachtrup, J{\"o}rg and Zeilinger, Anton}, title = {Quantum information processing and communication : Strategic report on current status, visions and goals for research in Europe}, issn = {1434-6060}, year = {2005}, abstract = {We present an excerpt of the document "Quantum Information Processing and Communication: Strategic report on current status, visions and goals for research in Europe", which has been recently published in electronic form at the website of FET (the Future and Emerging Technologies Unit of the Directorate General Information Society of the European Commission, http://www.cordis.lu/ist/fet/qipc-sr.htm). This document has been elaborated, following a former suggestion by FET, by a committee of QIPC scientists to provide input towards the European Commission for the preparation of the Seventh Framework Program. Besides being a document addressed to policy makers and funding agencies (both at the European and national level), the document contains a detailed scientific assessment of the state-of-the-art, main research goals, challenges, strengths, weaknesses, visions and perspectives of all the most relevant QIPC sub-fields, that we report here}, language = {en} }