Dynamics of continuous-time quantum walks in restricted geometries

Agliari, Elena; Blumen, A.; Muelken, Oliver

doi:10.1088/1751-8113/41/44/445301

Cited by 58 publications

(78 citation statements)

References 44 publications

(94 reference statements)

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“…Finally, in addition to the aforementioned dynamics, we expect that the obtained eigenvalues and eigenvectors can be adaptable to other dynamics in the smallworld networks, e.g., quantum walks. [57][58][59][60][61] …”

Section: Discussionmentioning

confidence: 99%

Laplacian spectra of recursive treelike small-world polymer networks: Analytical solutions and applications

Liu

Zhang²

2013

The Journal of Chemical Physics

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A central issue in the study of polymer physics is to understand the relation between the geometrical properties of macromolecules and various dynamics, most of which are encoded in the Laplacian spectra of a related graph describing the macrostructural structure. In this paper, we introduce a family of treelike polymer networks with a parameter, which has the same size as the Vicsek fractals modeling regular hyperbranched polymers. We study some relevant properties of the networks and show that they have an exponentially decaying degree distribution and exhibit the small-world behavior. We then study the Laplacian eigenvalues and their corresponding eigenvectors of the networks under consideration, with both quantities being determined through the recursive relations deduced from the network structure. Using the obtained recursive relations we can find all the eigenvalues and eigenvectors for the networks with any size. Finally, as some applications, we use the eigenvalues to study analytically or semi-analytically three dynamical processes occurring in the networks, including random walks, relaxation dynamics in the framework of generalized Gaussian structure, as well as the fluorescence depolarization under quasiresonant energy transfer. Moreover, we compare the results with those corresponding to Vicsek fractals, and show that the dynamics differ greatly for the two network families, which thus enables us to distinguish between them.

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Section: Discussionmentioning

confidence: 99%

Laplacian spectra of recursive treelike small-world polymer networks: Analytical solutions and applications

Liu

Zhang²

2013

The Journal of Chemical Physics

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“…Turning now to the quantum case and evaluating the lower bound α(t) 2 of π(t) of the quantum average return probabil-ity π(t), see Eq. (8), it has been found in [6] that its envelope does not show a strong dependence on the size of the DSG. Since the two eigenvalues 3 and 5 make up for about 1/3 of all eigenvalues, they control most of the behavior of π(t).…”

Section: Dual Sierpinski Gasketmentioning

confidence: 96%

“…Search via quantum walks on fractal graphs has been considered in Refs. [6,19,20]. A similar mathematical model arises for condensed matter systems, in which one considers a particle moving on an underlying fractal lattice (a Sierpinski gasket); here the solution of Schrödinger's equation has been studied within the tightbinding approximation [21,22].…”

Section: Introductionmentioning

confidence: 99%

Transport properties of continuous-time quantum walks on Sierpinski fractals

et al. 2014

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We model quantum transport, described by continuous-time quantum walks (CTQW), on deterministic Sierpinski fractals, differentiating between Sierpinski gaskets and Sierpinski carpets, along with their dual structures. The transport efficiencies are defined in terms of the exact and the average return probabilities, as well as by the mean survival probability when absorbing traps are present. In the case of gaskets, localization can be identified already for small networks (generations). For carpets, our numerical results indicate a trend towards localization, but only for relatively large structures. The comparison of gaskets and carpets further implies that, distinct from the corresponding classical continuous-time random walk, the spectral dimension does not fully determine the evolution of the CTQW.

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“…A recent study on the comparison of quantum energy transfer and classical energy transfer, by noisy cellular automata, can be found in [50]. Fractal geometries have also been exploited in studying the role of the topological structure in quantum transport [51] and for Grover-based searching problems [52]-for a review on continuous-time quantum walks on complex networks see also [15]. As regards the role of decoherence in the mixing time of both discrete-time and continuous-time quantum walks, especially for chains, cycles and hypercubes, a review appears in [21].…”

Section: Introductionmentioning

confidence: 99%

Universally optimal noisy quantum walks on complex networks

Caruso

2014

New J. Phys.

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Transport properties play a crucial role in several fields of science, for example biology, chemistry, sociology, information science and physics. The behavior of many dynamical processes running over complex networks is known to be closely related to the geometry of the underlying topology, but this connection becomes even harder to understand when quantum effects come into play. Here, we exploit the Kossakowski-Lindblad formalism of quantum stochastic walks to investigate the capability of quickly and robustly transmitting energy (or information) between two distant points in very large complex structures, remarkably assisted by external noise and quantum features such as coherence. An optimal mixing of classical and quantum transport is, very surprisingly, quite universal for a large class of complex networks. This widespread behavior turns out to be also extremely robust with respect to geometry changes. These results might pave the way for designing optimal bio-inspired geometries of efficient transport nanostructures that can be used for solar energy and also quantum information and communication technologies.

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Dynamics of continuous-time quantum walks in restricted geometries

Cited by 58 publications

References 44 publications

Laplacian spectra of recursive treelike small-world polymer networks: Analytical solutions and applications

Laplacian spectra of recursive treelike small-world polymer networks: Analytical solutions and applications

Transport properties of continuous-time quantum walks on Sierpinski fractals

Universally optimal noisy quantum walks on complex networks

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