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In the introductory article of this theme issue, we provide an overview of quantum annealing and computation with a very brief summary of the individual contributions to this issue made by experts as well as a few young researchers. We hope the readers will get the touch of the excitement as well as the perspectives in this unusually active field and important developments there. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.
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By controlling quantum fluctuations via the Falk-Bruch inequality we give the first rigorous argument for the existence of a spin-glass phase in the quantum Sherrington-Kirkpatrick model with a "transverse" magnetic field if the temperature and the field are sufficiently low. The argument also applies to the generalization of the model with multispin interactions, sometimes dubbed as the transverse p-spin model.
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In a remarkable paper [Phys. Rev. Lett. 96, 100503 (2006)], Gioev and Klich conjectured an explicit formula for the leading asymptotic growth of the spatially bipartite von Neumann entanglement entropy of noninteracting fermions in multidimensional Euclidean space at zero temperature. Based on recent progress by one of us (A. V. S.) in semiclassical functional calculus for pseudodifferential operators with discontinuous symbols, we provide here a complete proof of that formula and of its generalization to Rényi entropies of all orders α>0. The special case α=1/2 is also known under the name logarithmic negativity and often considered to be a particularly useful quantification of entanglement. These formulas exhibiting a "logarithmically enhanced area law" have been used already in many publications.
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We consider Lifshitz's model of a quantum particle subject to a repulsive Poissonian random potential and address various issues related to the influence of a constant magnetic field on the leading low-energy tail of the integrated density of states. In particular, we propose the magnetic analog of a 40-year-old landmark result of Lifshitz for short-ranged single-impurity potentials U. The Lifshitz tail is shown to change its character from purely quantum, through quantum classical, to purely classical with an increasing range of U. This systematics is explained by the increasing importance of the classical fluctuations of the particle's potential energy in comparison to the quantum fluctuations associated with its kinetic energy.