We study non-Markovianity and information flow for qubits experiencing local dephasing with an Ohmic class spectrum. We demonstrate the existence of a temperature-dependent critical value of the Ohmicity parameter s for the onset of non-Markovianity and give a physical interpretation of this phenomenon by linking it to the form of the reservoir spectrum. We demonstrate that this link holds also for more general spectra. We unveil a class of initial states for which discord is forever frozen at a positive value. We connect time invariant discord to non-Markovianity and propose a physical system in which it could be observed. PACS numbers: 03.65.Ta, 03.65.Yz, 03.67.Mn Introduction.-Qubits subjected to local purely dephasing noise are ubiquitous models of open quantum systems, and they have been studied extensively in the literature. Examples include dephasing in quantum registers [1-3], ultracold gases [4, 5], quantum metrology protocols [6], quantum biological systems [7], and dynamical decoupling theory [8]. From a theoretical point of view the dephasing model is exactly solvable [1-3], and hence it is an ideal testbed to investigate one of the most thrilling fields of the theory of open quantum systems, that of non-Markovian quantum processes [9].Recently, a great deal of attention has been devoted to the study of systems whose reduced dynamics are characterised by memory effects and recoherence phenomena, emerging from non-trivial correlations with an environment. Such dynamics are typically called non-Markovian. Memory effects and non-Markovianity have been shown to be a resource for quantum technologies [6, 10-13] and consequently measures of non-Markovianity have become important as quantifiers of this resource [14][15][16]. Moreover, it has been shown that non-Markovianity of a quantum probe can indicate a quantum phase transition occurring in a complex environment, with which the probe is interacting [17].Non-Markovian features play an important role in systems where the frequency spectrum of the environment is structured. However, a connection between the general form of the spectrum and the memory effects in the reduced system dynamics has not been elucidated until now. In this Letter we establish this connection by unveiling a necessary condition on the form of the spectrum to induce non-Markovian dynamics for a dephasing qubit. We then focus on the widely used Ohmic class of reservoir spectra and show that the condition is both necessary and sufficient for this type of spectra. Moreover, we demonstrate that only super-Ohmic environments can induce non-Markovian dynamics. This means that even if the reduced dynamics is exact, and hence no Markovian approximation has been performed, the time evolution of the qubit does not present any memory effects or reco-

In this theoretical study, we analyze quantum walks on complex networks, which model network-based processes ranging from quantum computing to biology and even sociology. Specifically, we analytically relate the average long time probability distribution for the location of a unitary quantum walker to that of a corresponding classical walker. The distribution of the classical walker is proportional to the distribution of degrees, which measures the connectivity of the network nodes and underlies many methods for analyzing classical networks including website ranking. The quantum distribution becomes exactly equal to the classical distribution when the walk has zero energy and at higher energies the difference, the so-called quantumness, is bounded by the energy of the initial state. We give an example for which the quantumness equals a Renyi entropy of the normalized weighted degrees, guiding us to regimes for which the classical degree-dependent result is recovered and others for which quantum effects dominate.Comment: 8 pages, 4 figures; improved description and new examples; accepted for publication in Phys. Rev.

Quantum simulators are devices that actively use quantum effects to answer questions about model systems and, through them, real systems. In this review we expand on this definition by answering several fundamental questions about the nature and use of quantum simulators. Our answers address two important areas. First, the difference between an operation termed simulation and another termed computation. This distinction is related to the purpose of an operation, as well as our confidence in and expectation of its accuracy. Second, the threshold between quantum and classical simulations. Throughout, we provide a perspective on the achievements and directions of the field of quantum simulation.