Scattering quantum random-walk search with errors

Gábris, Aurél; Kiss, T.; Jex, Igor

doi:10.1103/physreva.76.062315

Cited by 22 publications

(25 citation statements)

References 27 publications

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“…Finally, we give the explicit form of the coefficients C 1 , C 2 and C 12 . We find that they have a particularly simple form in the basis formed by the tensor product of eigenvectors of the Hadamard coin |χ ± , which have been given in (16). With the decomposition of the initial coin state in the Hadamard basis as given in (22) we obtain the following expressions for the coefficients C 1,2 and C 12 :…”

Section: Discussionmentioning

confidence: 98%

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Directional correlations in quantum walks with two particles

Štefaňák

Barnett²,

Kollár³

et al. 2011

New J. Phys.

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Quantum walks on the line with a single particle possess a classical analog. Involving more walkers opens up the possibility to study collective quantum effects, such as many particle correlations. In this context, entangled initial states and indistinguishability of the particles play a role. We consider directional correlations between two particles performing a quantum walk on a line. For non-interacting particles we find analytic asymptotic expressions and give the limits of directional correlations. We show that introducing δ-interaction between the particles, one can exceed the limits for non-interacting particles.

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Section: Discussionmentioning

confidence: 98%

“…From the expression (16) we find the transformation between the coefficients in the standard and the Hadamard basis…”

Section: Separable Initial Statesmentioning

confidence: 99%

Directional correlations in quantum walks with two particles

Štefaňák

Barnett²,

Kollár³

et al. 2011

New J. Phys.

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“…Full-revival in the 2-D Grover walk. As the initial state we choose |ψ(0) given by (10) which is not an eigenvector of the evolution operator. Coin flip reverts the initial states of the coin, as can be seen from the upper center plot.…”

Section: Stationary States and Full-revivalsmentioning

confidence: 99%

Full revivals in 2D quantum walks

Štefaňák¹,

Kollár

Kiss

et al. 2010

Phys. Scr.

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Recurrence of a random walk is described by the Pólya number. For quantum walks, recurrence is understood as the return of the walker to the origin, rather than the full-revival of its quantum state. Localization for two dimensional quantum walks is known to exist in the sense of non-vanishing probability distribution in the asymptotic limit. We show on the example of the 2-D Grover walk that one can exploit the effect of localization to construct stationary solutions. Moreover, we find full-revivals of a quantum state with a period of two steps. We prove that there cannot be longer cycles for a four-state quantum walk. Stationary states and revivals result from interference which has no counterpart in classical random walks.

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“…Again, certain facts are clearly observed from Figs. [12][13][14][15][16]. First, as it should be, we can obtain one evolution from the other by correct projections.…”

Section: Examplesmentioning

confidence: 99%

Unveiling and exemplifying the unitary equivalence of discrete time quantum walk models

Venancio¹,

Andrade²,

Luz³

2013

J. Phys. A: Math. Theor.

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The two major discrete time formulations for quantum walks, coined and scattering, are unitarily equivalent for arbitrary position dependent transition amplitudes and any topology (PRA 80, 052301 (2009)). Although the proof explicit describes the mapping obtention, its high technicality may hinder relevant physical aspects involved in the equivalence. Discussing concrete examples -the most general constructions for the line, square and honeycomb lattices -here we unveil the similarities and differences of these two versions of quantum walks. We moreover show how to derive the dynamics of one from the other by means of proper projections. We perform calculations for different probability amplitudes like, Hadamard, Grover, Discrete Fourier Transform and the uncommon in the area (but interesting) Discrete Hartley Transform, comparing the evolutions. Our study illustrates the models interplay, an important issue for implementations and applications of such systems.

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Scattering quantum random-walk search with errors

Cited by 22 publications

References 27 publications

Directional correlations in quantum walks with two particles

Directional correlations in quantum walks with two particles

Full revivals in 2D quantum walks

Unveiling and exemplifying the unitary equivalence of discrete time quantum walk models

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