A Path Integral Approach for Disordered Quantum Walks in One Dimension

Konno, Norio

doi:10.1142/s0219477505002987

Cited by 41 publications

(31 citation statements)

References 14 publications

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“…The method permits the analysis only for the QWs with one defect at the origin, whose quantum coins are the same both in positive and negative parts. Recently, various kinds of methods have been constructed to investigate mathematically the asymptotic behavior of QWs, such as the Fourier analysis [24], the CGMV method [4], the stationary phase method [22], the path counting method [18], and the generating function method [9]. We can expect to analyze various kinds of inhomogeneous QWs by the generating function method, while the Fourier analysis and stationary phase method are useful to study homogeneous QWs.…”

Section: Introductionmentioning

confidence: 99%

Weak Limit Theorem of a Two-phase Quantum Walk with One Defect

Endo

Konno

et al. 2016

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We attempt to analyze a one-dimensional space-inhomogeneous quantum walk (QW) with one defect at the origin, which has two different quantum coins in positive and negative parts. We call the QW ''the two-phase QW with one defect'', which we treated concerning localization theorems [7]. The two-phase QW with one defect has been expected to be a mathematical model of topological insulator [15] which is an intense issue both theoretically and experimentally [3,5,11]. In this paper, we derive the weak limit theorem describing the ballistic spreading, and as a result, we obtain the mathematical expression of the whole picture of the asymptotic behavior. Our approach is based mainly on the generating function of the weight of the passages. We emphasize that the timeaveraged limit measure is symmetric for the origin [7], however, the weak limit measure is asymmetric, which implies that the weak limit theorem represents the asymmetry of the probability distribution.

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Section: Introductionmentioning

confidence: 99%

Weak Limit Theorem of a Two-phase Quantum Walk with One Defect

Endo

Konno

et al. 2016

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“…[31] gives the solution for the coined QW, with an arbitrary unitary acting on the coin space. The same author considered the path-integral formulation for disordered QWs [32] where the coin unitary is a varying function of time.…”

Section: Introductionmentioning

confidence: 99%

Path-integral solution of the one-dimensional Dirac quantum cellular automaton

et al. 2014

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Quantum cellular automata have been recently considered as a fundamental approach to quantum field theory, resorting to a precise automaton, linear in the field, for the Dirac equation in one dimension. In such linear case a quantum automaton is isomorphic to a quantum walk, and a convenient formulation can be given in terms of transition matrices, leading to a new kind of discrete path integral that we solve analytically in terms of Jacobi polynomials versus the arbitrary mass parameter.

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“…These impurities are static like the voids, and the small particles in the solid. It implies one of disordered models of quantum walk [K2,RMM,MBSS,Kem]. There we numerically show that such impurities suppress the coherent nature and the distribution of quantum walks behaves like diffusive one.…”

Section: Introductionmentioning

confidence: 73%

New theory of diffusive and coherent nature of optical wave via a quantum walk

et al. 2017

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We propose a new theory on a relation between diffusive and coherent nature in one dimensional wave mechanics based on a quantum walk. It is known that the quantum walk in homogeneous matrices provides the coherent property of wave mechanics. Using the recent result of a localization phenomenon in a one-dimensional quantum walk (Konno Quantum Inf. Proc. (2010) 9 405-418), we numerically show that the randomized localized matrices suppress the coherence and give diffusive nature.

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A Path Integral Approach for Disordered Quantum Walks in One Dimension

Abstract: Abstract. The present letter gives a rigorous way from quantum to classical random walks by introducing an independent random fluctuation and then taking expectations based on a path integral approach.

Cited by 41 publications

References 14 publications

Weak Limit Theorem of a Two-phase Quantum Walk with One Defect

Weak Limit Theorem of a Two-phase Quantum Walk with One Defect

Path-integral solution of the one-dimensional Dirac quantum cellular automaton

New theory of diffusive and coherent nature of optical wave via a quantum walk

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