Asymptotic behavior of decoherent and interacting quantum walks

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Monodromy analysis of the computational power of the Ising topological quantum computer

We show that all quantum gates which could be implemented by braiding of Ising anyons in the Ising topological quantum computer preserve the n-qubit Pauli group. Analyzing the structure of the Pauli group's centralizer, also known as the Clifford group, for n\geq 3 qubits, we prove that the image of the braid group is a non-trivial subgroup of the Clifford group and therefore not all Clifford gates could be implemented by braiding. We show explicitly the Clifford gates which cannot be realized by braiding estimating in this way the ultimate computational power of the Ising topological quantum computer.Comment: 10 pages, 2 figures and 1 table; v2: one more reference added and some typos corrected; Talk given at the VIII International Workshop "Lie Theory and its Applications in Physics", 15-21 June 2009, Varna, Bulgari

Molecular binding in interacting quantum walks

We show that the presence of an interaction in the quantum walk of two atoms leads to the formation of a stable compound, a molecular state. The wave function of the molecule decays exponentially in the relative position of the two atoms; hence it constitutes a true bound state. Furthermore, for a certain class of interactions, we develop an effective theory and find that the dynamics of the molecule is described by a quantum walk in its own right. We propose a setup for the experimental realization as well as sketch the possibility to observe quasi-particle effects in quantum many-body systems.DFG/FOR/635European Commission/CORNEREuropean Commission/COQUITEuropean Commission/AQUTENRW Nachwuchsgruppe ‘Quantenkontrolle auf der Nanoskala’Alexander von Humboldt Foundatio

Disordered Quantum Walks in one lattice dimension

We study a spin-1/2-particle moving on a one dimensional lattice subject to disorder induced by a random, space-dependent quantum coin. The discrete time evolution is given by a family of random unitary quantum walk operators, where the shift operation is assumed to be deterministic. Each coin is an independent identically distributed random variable with values in the group of two dimensional unitary matrices. We derive sufficient conditions on the probability distribution of the coins such that the system exhibits dynamical localization. Put differently, the tunneling probability between two lattice sites decays rapidly for almost all choices of random coins and after arbitrary many time steps with increasing distance. Our findings imply that this effect takes place if the coin is chosen at random from the Haar measure, or some measure continuous with respect to it, but also for a class of discrete probability measures which support consists of two coins, one of them being the Hadamard coin.Comment: minor change

Asymptotic behavior of quantum walks with spatio-temporal coin fluctuations

Quantum walks subject to decoherence generically suffer the loss of their genuine quantum feature, a quadratically faster spreading compared to classical random walks. This intuitive statement has been verified analytically for certain models and is also supported by numerical studies of a variety of examples. In this paper we analyze the long-time behavior of a particular class of decoherent quantum walks, which, to the best of our knowledge, was only studied at the level of numerical simulations before. We consider a local coin operation which is randomly and independently chosen for each time step and each lattice site and prove that, under rather mild conditions, this leads to classical behavior: With the same scaling as needed for a classical diffusion the position distribution converges to a Gaussian, which is independent of the initial state. Our method is based on non-degenerate perturbation theory and yields an explicit expression for the covariance matrix of the asymptotic Gaussian in terms of the randomness parameters

Comparative in vitro treatment of mesenchymal stromal cells with GDF‐5 and R57A induces chondrogenic differentiation while limiting chondrogenic hypertrophy

Abstract Purpose Hypertrophic cartilage is an important characteristic of osteoarthritis and can often be found in patients suffering from osteoarthritis. Although the exact pathomechanism remains poorly understood, hypertrophic de‐differentiation of chondrocytes also poses a major challenge in the cell‐based repair of hyaline cartilage using mesenchymal stromal cells (MSCs). While different members of the transforming growth factor beta (TGF‐β) family have been shown to promote chondrogenesis in MSCs, the transition into a hypertrophic phenotype remains a problem. To further examine this topic we compared the effects of the transcription growth and differentiation factor 5 (GDF‐5) and the mutant R57A on in vitro chondrogenesis in MSCs. Methods Bone marrow‐derived MSCs (BMSCs) were placed in pellet culture and in‐cubated in chondrogenic differentiation medium containing R57A, GDF‐5 and TGF‐ß1 for 21 days. Chondrogenesis was examined histologically, immunohistochemically, through biochemical assays and by RT‐qPCR regarding the expression of chondrogenic marker genes. Results Treatment of BMSCs with R57A led to a dose dependent induction of chondrogenesis in BMSCs. Biochemical assays also showed an elevated glycosaminoglycan (GAG) content and expression of chondrogenic marker genes in corresponding pellets. While treatment with R57A led to superior chondrogenic differentiation compared to treatment with the GDF‐5 wild type and similar levels compared to incubation with TGF‐ß1, levels of chondrogenic hypertrophy were lower after induction with R57A and the GDF‐5 wild type. Conclusions R57A is a stronger inducer of chondrogenesis in BMSCs than the GDF‐5 wild type while leading to lower levels of chondrogenic hypertrophy in comparison with TGF‐ß1