516 research outputs found

    Discrete-time quantum walks on one-dimensional lattices

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    In this paper, we study discrete-time quantum walks on one-dimensional lattices. We find that the coherent dynamics depends on the initial states and coin parameters. For infinite size of lattice, we derive an explicit expression for the return probability, which shows scaling behavior P(0,t)t1P(0,t)\sim t^{-1} and does not depends on the initial states of the walk. In the long-time limit, the probability distribution shows various patterns, depending on the initial states, coin parameters and the lattice size. The average mixing time MϵM_{\epsilon} closes to the limiting probability in linear NN (size of the lattice) for large values of thresholds ϵ\epsilon. Finally, we introduce another kind of quantum walk on infinite or even-numbered size of lattices, and show that the walk is equivalent to the traditional quantum walk with symmetrical initial state and coin parameter.Comment: 17 pages research not

    On the relationship between continuous- and discrete-time quantum walk

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    Quantum walk is one of the main tools for quantum algorithms. Defined by analogy to classical random walk, a quantum walk is a time-homogeneous quantum process on a graph. Both random and quantum walks can be defined either in continuous or discrete time. But whereas a continuous-time random walk can be obtained as the limit of a sequence of discrete-time random walks, the two types of quantum walk appear fundamentally different, owing to the need for extra degrees of freedom in the discrete-time case. In this article, I describe a precise correspondence between continuous- and discrete-time quantum walks on arbitrary graphs. Using this correspondence, I show that continuous-time quantum walk can be obtained as an appropriate limit of discrete-time quantum walks. The correspondence also leads to a new technique for simulating Hamiltonian dynamics, giving efficient simulations even in cases where the Hamiltonian is not sparse. The complexity of the simulation is linear in the total evolution time, an improvement over simulations based on high-order approximations of the Lie product formula. As applications, I describe a continuous-time quantum walk algorithm for element distinctness and show how to optimally simulate continuous-time query algorithms of a certain form in the conventional quantum query model. Finally, I discuss limitations of the method for simulating Hamiltonians with negative matrix elements, and present two problems that motivate attempting to circumvent these limitations.Comment: 22 pages. v2: improved presentation, new section on Hamiltonian oracles; v3: published version, with improved analysis of phase estimatio

    Water-like anomalies for core-softened models of fluids: One dimension

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    We use a one-dimensional (1d) core-softened potential to develop a physical picture for some of the anomalies present in liquid water. The core-softened potential mimics the effect of hydrogen bonding. The interest in the 1d system stems from the facts that closed-form results are possible and that the qualitative behavior in 1d is reproduced in the liquid phase for higher dimensions. We discuss the relation between the shape of the potential and the density anomaly, and we study the entropy anomaly resulting from the density anomaly. We find that certain forms of the two-step square well potential lead to the existence at T=0 of a low-density phase favored at low pressures and of a high-density phase favored at high pressures, and to the appearance of a point CC' at a positive pressure, which is the analog of the T=0 ``critical point'' in the 1d1d Ising model. The existence of point CC' leads to anomalous behavior of the isothermal compressibility KTK_T and the isobaric specific heat CPC_P.Comment: 22 pages, 7 figure

    Gamma Ray Bursts as Probes of Quantum Gravity

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    Gamma ray bursts (GRBs) are short and intense pulses of γ\gamma-rays arriving from random directions in the sky. Several years ago Amelino-Camelia et al. pointed out that a comparison of time of arrival of photons at different energies from a GRB could be used to measure (or obtain a limit on) possible deviations from a constant speed of light at high photons energies. I review here our current understanding of GRBs and reconsider the possibility of performing these observations.Comment: Lectures given at the 40th winter school of theretical physics: Quantum Gravity and Phenomenology, Feb. 2004 Polan

    Low Q^2 Jet Production at HERA and Virtual Photon Structure

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    The transition between photoproduction and deep-inelastic scattering is investigated in jet production at the HERA ep collider, using data collected by the H1 experiment. Measurements of the differential inclusive jet cross-sections dsigep/dEt* and dsigmep/deta*, where Et* and eta* are the transverse energy and the pseudorapidity of the jets in the virtual photon-proton centre of mass frame, are presented for 0 < Q2 < 49 GeV2 and 0.3 < y < 0.6. The interpretation of the results in terms of the structure of the virtual photon is discussed. The data are best described by QCD calculations which include a partonic structure of the virtual photon that evolves with Q2.Comment: 20 pages, 5 Figure

    Hadron Production in Diffractive Deep-Inelastic Scattering

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    Characteristics of hadron production in diffractive deep-inelastic positron-proton scattering are studied using data collected in 1994 by the H1 experiment at HERA. The following distributions are measured in the centre-of-mass frame of the photon dissociation system: the hadronic energy flow, the Feynman-x (x_F) variable for charged particles, the squared transverse momentum of charged particles (p_T^{*2}), and the mean p_T^{*2} as a function of x_F. These distributions are compared with results in the gamma^* p centre-of-mass frame from inclusive deep-inelastic scattering in the fixed-target experiment EMC, and also with the predictions of several Monte Carlo calculations. The data are consistent with a picture in which the partonic structure of the diffractive exchange is dominated at low Q^2 by hard gluons.Comment: 16 pages, 6 figures, submitted to Phys. Lett.

    Measurement of D* Meson Cross Sections at HERA and Determination of the Gluon Density in the Proton using NLO QCD

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    With the H1 detector at the ep collider HERA, D* meson production cross sections have been measured in deep inelastic scattering with four-momentum transfers Q^2>2 GeV2 and in photoproduction at energies around W(gamma p)~ 88 GeV and 194 GeV. Next-to-Leading Order QCD calculations are found to describe the differential cross sections within theoretical and experimental uncertainties. Using these calculations, the NLO gluon momentum distribution in the proton, x_g g(x_g), has been extracted in the momentum fraction range 7.5x10^{-4}< x_g <4x10^{-2} at average scales mu^2 =25 to 50 GeV2. The gluon momentum fraction x_g has been obtained from the measured kinematics of the scattered electron and the D* meson in the final state. The results compare well with the gluon distribution obtained from the analysis of scaling violations of the proton structure function F_2.Comment: 27 pages, 9 figures, 2 tables, submitted to Nucl. Phys.
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