Ballistic Quantum Walk in a Discrete One-Dimensional System

Endo, Takumi; Osano, Shin'ichi; Toyoshima, Kouichi; Hirayoshi, Yutaka

doi:10.1143/jpsj.78.064004

Cited by 5 publications

(8 citation statements)

References 12 publications

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“…( 19)). Indeed, according to Endo et al [46], the prefactor of t 2 in Eq. ( 21) is related to the square of the parameter characterizing the speed of the walker.…”

Section: (B)mentioning

confidence: 96%

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Continuous-time quantum walks in the presence of a quadratic perturbation

Candeloro,

Razzoli,

Cavazzoni

et al. 2020

Preprint

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Motivated by their potential use to describe gravity induced perturbations, or next-nearestneighbor tunnelling, we investigate in details the properties of continuous-time quantum walks (CTQW) with Hamiltonians of the form H = L + λL 2 , being L the Laplacian (Kirchoff) matrix of the underlying graph. In particular, we focus attention to CTQW on cycle, complete, and star graphs, as they describe paradigmatic models with low/high connectivity and/or symmetry. At first, we investigate the dynamics of an initially localized walker, looking at the resulting site distribution, mixing, inverse participation ratio, and coherence. We then devote attention to the characterization of perturbation, i.e. the estimation of the perturbation parameter λ using only a snapshot of the walker dynamics. Our analysis shows that a walker on a cycle graph is spreading ballistically independently of the perturbation, whereas on complete and star graphs one observes perturbation-dependent revivals and strong localization phenomena. Concerning characterization, we determine the walker preparations that maximize the Quantum Fisher Information, and assess the performance of position measurement, which turns out to be optimal, or nearly optimal, in several situations of interest. Our study is based on exact analytic derivations supplemented by numerical results, and besides fundamental interest, it may find applications in the design of enhanced algorithms on graphs.

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“…( 19)). Indeed, according to Endo et al [46], the prefactor of t 2 in Eq. ( 21) is related to the square of the parameter characterizing the speed of the walker.…”

Section: (B)mentioning

confidence: 96%

“…the set of integers Z. Based on Konno's work, Endo et al [46] proved that the spreading of a CTQW on a one-dimensional infinite 2: Probability distribution Pj(k, t|λ) of the walker as a function of time in the cycle graph. The walker is initially localized in the vertex |j = 2 .…”

Section: A Cycle Graphmentioning

confidence: 99%

Continuous-time quantum walks in the presence of a quadratic perturbation

Candeloro,

Razzoli,

Cavazzoni

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…3 The solution of the time-dependent Schröequation of the unperturbed system (λ = 0) can be expressed in terms of Bessel functions [33]. This allows us to analytically prove the ballistic spreading in a one-dimensional infinite lattice [45], i.e., that the variance of the position is σ 2…”

Section: A Cycle Graphmentioning

confidence: 99%

“…Nevertheless, increasing |λ − λ 0 | makes the walker spread faster by affecting the factor in front of t 2 . Indeed, such a factor is related to the square of the parameter characterizing the speed of the walker [45]. The lowest variance is for λ = λ 0 , which is the value for which the nearest-neighbor hopping −(1 + 4λ) equals the next-nearest-neighbor one λ [see Eq.…”

Section: A Cycle Graphmentioning

confidence: 99%

See 1 more Smart Citation

Continuous-time quantum walks in the presence of a quadratic perturbation

et al. 2020

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We address the properties of continuous-time quantum walks with Hamiltonians of the form H = L + λL 2 , with L the Laplacian matrix of the underlying graph and the perturbation λL 2 motivated by its potential use to introduce next-nearest-neighbor hopping. We consider cycle, complete, and star graphs as paradigmatic models with low and high connectivity and/or symmetry. First, we investigate the dynamics of an initially localized walker. Then we devote attention to estimating the perturbation parameter λ using only a snapshot of the walker dynamics. Our analysis shows that a walker on a cycle graph spreads ballistically independently of the perturbation, whereas on complete and star graphs one observes perturbation-dependent revivals and strong localization phenomena. Concerning the estimation of the perturbation, we determine the walker preparations and the simple graphs that maximize the quantum Fisher information. We also assess the performance of position measurement, which turns out to be optimal, or nearly optimal, in several situations of interest. Besides fundamental interest, our study may find applications in designing enhanced algorithms on graphs.

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