Quantum-walk approach to searching on fractal structures

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

doi:10.1103/physreva.82.012305

Cited by 91 publications

(95 citation statements)

References 34 publications

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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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“…Within this approach, the so-called √ N 0 speedup occurs on a complete graph and on cubic graphs whose dimension d is larger than 4 [11]. Unfortunately, in low-dimensional cubic graphs (d ≤ 3), the search algorithm does not provide substantial speedup, a negative feature that has been observed also on fractal networks by Mulken et al [20].…”

Section: Introductionmentioning

confidence: 95%

Exciton-mediated quantum search on a star graph

Pouthier

2015

Quantum Inf Process

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“…Examples include the element distinctness problem [13], the spatial search problem [14], and the hitting problem [15,16]. Consequently, CTQW and excitonmediated quantum walk have been studied in a large variety of networks such as binary and glued trees [16,17], Apollonian networks [18,19], fractal networks [20,21], sequentially growing networks [22] and star graphs [23][24][25][26][27][28].…”

Section: Introductionmentioning

confidence: 99%

The excitonic qubit coupled with a phonon bath on a star graph: anomalous decoherence and coherence revivals

Yalouz

Falvo

Pouthier

2017

Quantum Inf Process

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Based on the operatorial formulation of perturbation theory, the dynamical properties of a Frenkel exciton coupled with a thermal phonon bath on a star graph are studied. Within this method, the dynamics is governed by an effective Hamiltonian which accounts for exciton-phonon entanglement. The exciton is dressed by a virtual phonon cloud, whereas the phonons are dressed by virtual excitonic transitions. Special attention is paid to the description of the coherence of a qubit state initially located on the central node of the graph. Within the nonadiabatic weak coupling limit, it is shown that several timescales govern the coherence dynamics. In the short time limit, the coherence behaves as if the exciton was insensitive to the phonon bath. Then, quantum decoherence takes place, this decoherence being enhanced by the size of the graph and by temperature. However, the coherence does not vanish in the long time limit. Instead, it exhibits incomplete revivals that occur periodically at specific revival times and it shows almost exact recurrences that take place at particular super-revival times, a singular behavior that has been corroborated by performing exact quantum calculations.

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Quantum-walk approach to searching on fractal structures

Cited by 91 publications

References 34 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

Exciton-mediated quantum search on a star graph

The excitonic qubit coupled with a phonon bath on a star graph: anomalous decoherence and coherence revivals

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