Localization of quantum walks induced by recurrence properties of random walks

Abstract. A spidernet is a graph obtained by adding large cycles to an almost regular tree and considered as an example having intermediate properties of lattices and trees in the study of discrete-time quantum walks on graphs. We introduce the Grover walk on a spidernet and its one-dimensional reduction. We derive an integral representation of the n-step transition amplitude in terms of the free Meixner law which appears as the spectral distribution. As an application we determine the class of spidernets which exhibit localization. Our method is based on quantum probabilistic spectral analysis of graphs.

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“…Several quantum walks are known to exhibit the localization, see e.g., [8,10,17,24,25,35]. For relevant discussion see also [29].…”

Section: Introductionmentioning

confidence: 99%

Localization of the Grover Walks on Spidernets and Free Meixner Laws

Konno

Obata

Segawa

2013

Commun. Math. Phys.

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“…Quantum walks (QWs) [1] are the quantum mechanical analog of classical random walks (RWs), and hence can be used to develop quantum algorithms [2][3][4][5], emerge as an alternative to the standard circuit model for quantum computing [6][7][8], and represent one of the most promising resources for the simulation of physical system and important phenomena such as topological phases [9], energy transport in photosynthesis [10,11], Anderson localization [12][13][14][15][16][17][18] and quantum chaos [19][20][21][22].…”

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confidence: 99%

Trapping photons on the line: controllable dynamics of a quantum walk

Xue

Tang

2014

Sci Rep

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Optical interferometers comprising birefringent-crystal beam displacers, wave plates, and phase shifters serve as stable devices for simulating quantum information processes such as heralded coined quantum walks. Quantum walks are important for quantum algorithms, universal quantum computing circuits, quantum transport in complex systems, and demonstrating intriguing nonlinear dynamical quantum phenomena. We introduce fully controllable polarization-independent phase shifters in optical pathes in order to realize site-dependent phase defects. The effectiveness of our interferometer is demonstrated through realizing single-photon quantum-walk dynamics in one dimension. By applying site-dependent phase defects, the translational symmetry of an ideal standard quantum walk is broken resulting in localization effect in a quantum walk architecture. The walk is realized for different site-dependent phase defects and coin settings, indicating the strength of localization signature depends on the level of phase due to site-dependent phase defects and coin settings and opening the way for the implementation of a quantum-walk-based algorithm.

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“…Setting some suitable weight w and integer s, we can easily obtain from Theorem 1.1 all the previous results concerning spectra of quantum walks found in [4,7,14,16]. Details will be discussed in Section 2.…”

Section: Introductionmentioning

confidence: 82%

“…[4,7,14]); moreover the spectra of the positive support U + (G) of U(G) and the positive support (U 2 ) + (G) of its square of a regular graph G are also expressed in terms of that of A(G). On the other hand, a mapping property from the spectrum of the transition operator of a random walk on G to that of the Szegedy evolution operator of a quantum walk, which is introduced firstly in [18], is shown in [16]. One of our main purposes in this note is to give a generalized formula of the above.…”

Section: Introductionmentioning

confidence: 99%

A note on the discrete-time evolutions of quantum walk on a graph

Higuchi,

Konno,

Sato

et al. 2012

Preprint

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For a quantum walk on a graph, there exist many kinds of operators for the discretetime evolution. We give a general relation between the characteristic polynomial of the evolution matrix of a quantum walk on edges and that of a kind of transition matrix of a classical random walk on vertices. Furthermore we determine the structure of the positive support of the cube of some evolution matrix, which is said to be useful for isospectral problem in graphs, under a certain condition.

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Localization of quantum walks induced by recurrence properties of random walks

Cited by 5 publications

References 18 publications

Localization of the Grover Walks on Spidernets and Free Meixner Laws

Localization of the Grover Walks on Spidernets and Free Meixner Laws

Trapping photons on the line: controllable dynamics of a quantum walk

A note on the discrete-time evolutions of quantum walk on a graph

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