Limit Theorems for Quantum Walks Driven by Many Coins

Segawa, Etsuo; Konno, Norio

doi:10.1142/s0219749908004456

Cited by 41 publications

(45 citation statements)

References 32 publications

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“…(23) and/or delta function turn out to be generic components of the density functions for many existing models of QW. Examples of these include, the three-state Grover walks by Inui et al [11], the QW with multiple coins by Segawa and Konno [15], and the four-state QW with a four-direction shift operator by Konno and Machida [12](we point out that this is a closer model to 2cQW considered in the present paper, which actually is a four-state QW with a three-direction shift operator. )…”

Section: Fourier Transform Formulation Of the Wave Function For 2cqwmentioning

confidence: 65%

“…As we will show in this work, the stationary probability of the QW is independent of the parity of time t, this is the major difference between the present model of QW and the one in [12]. In the case of the QW driven by many coins proposed and studied by Brun et al [14], the weak limit measures of the various scaled position operators were obtained by Segawa and Konno [15]. For a one-parameter family of the discretetime QW models in both one-dimensional and two-dimensional lattices, the convergence theorems for the moments of the walker's pseudovelocity were offered in Ampadu [16].…”

Section: Introductionmentioning

confidence: 67%

“…After taking derivatives of both sides of Eq. (29) with respect to y, we obtain the density as followsf (y) = dP(Y ≤ y) dy = tan β π(1 − y 2 ) 1 − y 2 sec 2 β | V 1 (k), Ψ ec 0 (k) | (k), Ψ ec 0 (k) | 2 + | V 3 (k), Ψ ec 0 (k) | 2 } y 2 sec 2 β | V 4 (k), Ψ ec 0 (k) | 2k=2 arccos(− tan β y √ 1−y 2 ) (30) Applying Eqs (11),(15) ,(16). and(18)to simplify Eq.…”

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confidence: 99%

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Asymptotic distributions of quantum walks on the line with two entangled coins

Liu

2012

Quantum Inf Process

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We advance the previous studies of quantum walks on the line with two coins. Such four-state quantum walks driven by a threedirection shift operator may have nonzero stationary distributions (localization), thus distinguishing themselves from the quantum walks on the line in the basic scenario (i.e., driven by a single coin). In this work, asymptotic position distributions of the quantum walks are examined. We derive a weak limit for the quantum walks and explicit formulas of stationary probability distribution, whose dependencies on the coin parameter and the initial state of quantum walks are presented. In particular, it is shown that the weak limit for the present quantum walks can be of the form in the basic scenario of quantum walks on the line, for certain initial states of the walk and certain values of the coin parameter. In the case where localization occurs, we show that the stationary probability exponentially decays with the absolute value of a walker's position, independent of the parity of time.keywords: quantum walks with two coins, three-direction shift operator, stationary distribution, localization, weak limit.

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Section: Fourier Transform Formulation Of the Wave Function For 2cqwmentioning

confidence: 65%

Section: Introductionmentioning

confidence: 67%

mentioning

confidence: 99%

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Asymptotic distributions of quantum walks on the line with two entangled coins

Liu

2012

Quantum Inf Process

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“…Other relevant works include [13,14], wherein weak limit results (Konnos density) are derived for QWs on the full line. Numerous variations of the same theme can be found in the literature, including [15,21,22,23,24,25,26,27,28], all of which exploit, to some degree, the notion of Konnos density.…”

Section: Weak Limit For Qw's On the Half Linementioning

confidence: 97%

Weak Limits for Quantum Walks on the Half-Line

Liu

Petulante

2013

Int. J. Quantum Inform.

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For a discrete two-state quantum walk on the half-line with a general condition at the boundary, we formulate and prove a weak limit theorem describing the terminal behavior of its transition probabilities. In this context, localization is possible even for a walk predicated on the assumption of homogeneity. For the Hadamard walk on the half line, the weak limit is shown to be independent of the initial coin state and to exhibit no localization.

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“…(1) generically holds with d w = 1, indicating a "ballistic" spreading of the quantum walk from its origin. This value has been obtained for various versions of one-and higher-dimensional quantum walks, for instance, with so-called weak-limit theorems [17,20,[25][26][27]. The renormalization-group (RG) method we have introduced recently [28] provides an alternative approach, expanding the analytic tools to understand quantum walks, since it works for networks that lack translational symmetries.…”

Section: Introductionmentioning

confidence: 99%

Relation between random walks and quantum walks

2015

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Based on studies of four specific networks, we conjecture a general relation between the walk dimensions d w of discrete-time random walks and quantum walks with the (self-inverse) Grover coin. In each case, we find that d w of the quantum walk takes on exactly half the value found for the classical random walk on the same geometry. Since walks on homogeneous lattices satisfy this relation trivially, our results for heterogeneous networks suggest that such a relation holds irrespective of whether translational invariance is maintained or not. To develop our results, we extend the renormalization-group analysis (RG) of the stochastic master equation to one with a unitary propagator. As in the classical case, the solution ρ(x,t) in space and time of this quantum-walk equation exhibits a scaling collapse for a variable x dw /t in the weak limit, which defines d w and illuminates fundamental aspects of the walk dynamics, e.g., its mean-square displacement. We confirm the collapse for ρ (x,t) in each case with extensive numerical simulation. The exact values for d w themselves demonstrate that RG is a powerful complementary approach to study the asymptotics of quantum walks that weak-limit theorems have not been able to access, such as for systems lacking translational symmetries beyond simple trees.

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Limit Theorems for Quantum Walks Driven by Many Coins

Cited by 41 publications

References 32 publications

Asymptotic distributions of quantum walks on the line with two entangled coins

Asymptotic distributions of quantum walks on the line with two entangled coins

Weak Limits for Quantum Walks on the Half-Line

Relation between random walks and quantum walks

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