Quantum Walks on Sierpinski Gaskets

Lara, Pedro; Portugal, Renato; Boettcher, Stefan

doi:10.1142/s021974991350069x

Cited by 12 publications

(9 citation statements)

References 13 publications

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“…The eigenvalues are α 1 = 1, α 2 = e iπ/3 and α 3 = e −iπ/3 . For α 1 = 1 we have an eigenvector of the form (24). Up to an overall phase, vector elements of each of the remaining eigenvectors can have only three possible values.…”

Section: Shift Operators With Cyclic Local Permutationsmentioning

confidence: 99%

“…Various types of the underlying graphs have been studied, e.g. quantum walks on general graphs [17], hypercubes [18,19], trees [20], honeycombs [21,22], spidernets [23] or fractal structures [24] (see also review [4]). One of the main driving forces of this research activities is interest in asymptotic properties of quantum walks, including limiting position distributions, speed of walker's propagation, and structure of trapped states.…”

Section: Introductionmentioning

confidence: 99%

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Percolated quantum walks with a general shift operator: From trapping to transport

2019

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We present a generalized definition of discrete-time quantum walks convenient for capturing a rather broad spectrum of walker's behavior on arbitrary graphs. It includes and covers both: the geometry of possible walker's positions with interconnecting links and the prescribed rule in which directions the walker will move at each vertex. While the former allows for the analysis of inhomogeneous quantum walks on graphs with vertices of varying degree, the latter offers us to choose, investigate, and compare quantum walks with different shift operators. The synthesis of both key ingredients constitutes a well-suited playground for analyzing percolated quantum walks on a quite general class of graphs. Analytical treatment of the asymptotic behavior of percolated quantum walks is presented and worked out in details for the Grover walk on graphs with maximal degree 3. We find, that for these walks with cyclic shift operators the existence of an edge-3-coloring of the graph allows for non-stationary asymptotic behavior of the walk. For different shift operators the general structure of localized attractors is investigated, which determines the overall efficiency of a source-to-sink quantum transport across a dynamically changing medium. As a simple nontrivial example of the theory we treat a single excitation transport on a percolated cube.

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Section: Shift Operators With Cyclic Local Permutationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Percolated quantum walks with a general shift operator: From trapping to transport

2019

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“…The internal coin degrees of freedom provide a labeling problem that severely complicates its consideration, already in evidence in Ref. 37. Interestingly, none of these problems exist for the classical random walk, and the Sierpinski gasket (or its dual) serves as a popular example of a simple demonstration of the RG, as its hierarchical structure and high degree of symmetry affects an RG-flow in a single real hopping parameter.…”

Section: Renormalization Of the Quantum Walk On The Dual Sierpinsmentioning

confidence: 99%

Renormalization and scaling in quantum walks

2014

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We show how to extract the scaling behavior of quantum walks using the renormalization group (RG). We introduce the method by efficiently reproducing well-known results on the one-dimensional lattice. As a nontrivial model, we apply this method to the dual Sierpinski gasket and obtain its exact, closed system of RG-recursions. Numerical iteration suggests that under rescaling the system length, L = 2L, characteristic times rescale as t = 2 dw t with the exact walk exponent dw = log 2 √ 5 = 1.1609 . . .. Despite the lack of translational invariance, this is very close to the ballistic spreading, dw = 1, found for regular lattices. However, we argue that an extended interpretation of the traditional RG formalism will be needed to obtain scaling exponents analytically. Direct simulations confirm our RG-prediction for dw and furthermore reveal an immensely rich phenomenology for the spreading of the quantum walk on the gasket. Invariably, quantum interference localizes the walk completely with a site-access probability that declines with a powerlaw from the initial site, in contrast with a classical random walk, which would pass all sites with certainty.

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“…Investigations, aiming first at the quantum walker on the line, have gradually broadened the scope of their interest to different graph geometries like e.g. cycles [6], hypercubes [9,10], trees [11], honeycombs [12,13], spidernets [14] or fractal structures [15] (for more see review [1]).…”

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confidence: 99%

Quantum walk transport on carbon nanotube structures

2020

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We study source-to-sink excitation transport on carbon nanotubes using the concept of quantum walks. In particular, we focus on transport properties of Grover coined quantum walks on ideal and percolation perturbed nanotubes with zig-zag and armchair chiralities. Using analytic and numerical methods we identify how geometric properties of nanotubes and different types of a sink altogether control the structure of trapped states and, as a result, the overall source-to-sink transport efficiency. It is shown that chirality of nanotubes splits behavior of the transport efficiency into a few typically well separated quantitative branches. Based on that we uncover interesting quantum transport phenomena, e.g. increasing the length of the tube can enhance the transport and the highest transport efficiency is achieved for the thinnest tube. We also demonstrate, that the transport efficiency of the quantum walk on ideal nanotubes may exhibit even oscillatory behavior dependent on length and chirality.Discrete coined quantum walks (CQWs) have become a standard tool in studying transport phenomena in the quantum domain [1]. Over the last two decades, quantum walker's behavior has been analyzed in the context of recurrence phenomena [2,3], state transfer [4, 5], speed of wave packet propagation [6], hitting times [7], or e.g. topological phenomena [8]. Investigations, aiming first at the quantum walker on the line, have gradually broadened the scope of their interest to different graph geometries like e.g. cycles [6], hypercubes [9, 10], trees [11], honeycombs [12, 13], spidernets [14] or fractal structures [15] (for more see review [1]).Analysis of complex graph structures has revealed that if vertices with degree at least three are present, the walker's dynamics may exhibit localisation originating from the presence of so-called trapped states [16][17][18][19][20][21][22]. They appear in various quantum systems and are known, in a different context, also as localized invariant or dark states [23][24][25][26]. These are eigenstates of the given dynamics, whose support does not spread over the whole position space. Due to that, the initial states having overlap with the trapped states can not fully propagate through the medium and the efficiency of quantum transport may be significantly reduced [27,28].On the other hand, trapped states were found to be fragile with respect to certain decoherence mechanisms arising in the presence of random external perturbations [29]. When the quantum walker moves on a changing graph whose edges are randomly and repeatedly closed and open again, we arrive at so-called dynamically percolated coined quantum walks (PCQWs) [30,31] capable to destroy some trapped states of the original nonpercolated CQW [32]. Recently, it was shown how the underlying geometry of Grover PCQW controls the structure of walker's trapped states [33]. The detailed analytical recipe for constructing the basis has essentially contributed to identify interesting counterintuitive quantum transport phenomena [34]. In partic...

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Quantum Walks on Sierpinski Gaskets

Cited by 12 publications

References 13 publications

Percolated quantum walks with a general shift operator: From trapping to transport

Percolated quantum walks with a general shift operator: From trapping to transport

Renormalization and scaling in quantum walks

Quantum walk transport on carbon nanotube structures

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