arXiv subjects

Etsuo Segawa

Publications and source records attributed to Etsuo Segawa.

At least 91 records · Page 5Linked to original sources

arXiv2015

Limit theorems of a two-phase quantum walk with one defect

We treat a position dependent quantum walk (QW) on the line which we assign two different time-evolution operators to positive and negative parts respectively. We call the model "the two-phase QW" here, which has been expected to be a mathematical model of the topological insulator. We obtain the stationary and time-averaged limit measures related to localization for the two-phase QW with one defect. This is the first result on localization for the two-phase QW. The analytical methods are mainly based on the splitted generating function of the solution for the eigenvalue problem, and the generating function of the weight of the passages of the model. In this paper, we call the methods "the splitted generating function method" and "the generating function method", respectively. The explicit expression of the stationary measure is asymmetric for the origin, and depends on the initial state and the choice of the parameters of the model. On the other hand, the time-averaged limit measure has a starting point symmetry and localization effect heavily depends on the initial state and the parameters of the model. Regardless of the strong effect of the initial state and the parameters, the time-averaged limit measure also suggests that localization can be always observed for our two-phase QW. Furthermore, our results imply that there is an interesting relation between the stationary and time-averaged limit measures when the parameters of the model have specific periodicities, which suggests that there is a possibility that we can analyze localization of the two-phase QW with one defect from the stationary measure.

math-ph↗

arXiv2015

Quantum walks on simplicial complexes

We construct a new type of quantum walks on simplicial complexes as a natural extension of the well-known Szegedy walk on graphs. One can numerically observe that our proposing quantum walks possess linear spreading and localization as in the case of the Grover walk on lattices. Moreover, our numerical simulation suggests that localization of our quantum walks reflect not only topological but also geometric structures. On the other hand, our proposing quantum walk contains an intrinsic problem concerning exhibition of nontrivial behavior, which is not seen in typical quantum walks such as Grover walks on graphs.

math-ph↗

arXiv2015

The spreading behavior of quantum walks induced by drifted random walks on some magnifier graph

In this paper, we consider the quantum walk on $\mathbb{Z}$ with attachment of one-length path periodically. This small modification to $\mathbb{Z}$ provides localization of the quantum walk. The eigenspace causing this localization is generated by finite length round trip paths. We find that the localization is due to the eigenvalues of an underlying random walk. Moreover we find that the transience of the underlying random walk provides a slow down of the pseudo velocity of the induced quantum walk and a different limit distribution from the Konno distribution.

quant-ph↗

arXiv2015

Weak limit theorem of a two-phase quantum walk with one defect

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", which we treated concerning localization theorems [10]. The two-phase QW has been expected to be a mathematical model of the topological insulator [16] 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 time-averaged limit measure is symmetric for the origin [10], however, the weak limit measure is asymmetric, which implies that the weak limit theorem represents the asymmetry of the probability distribution.

math-ph↗

arXiv2014

Spectral and asymptotic properties of Grover walks on crystal lattice

We propose a twisted Szegedy walk for estimating the limit behavior of a discrete-time quantum walk on a crystal lattice, an infinite abelian covering graph, whose notion was introduced by [14]. First, we show that the spectrum of the twisted Szegedy walk on the quotient graph can be expressed by mapping the spectrum of a twisted random walk onto the unit circle. Secondly, we show that the spatial Fourier transform of the twisted Szegedy walk on a finite graph with appropriate parameters becomes the Grover walk on its infinite abelian covering graph. Finally, as an application, we show that if the Betti number of the quotient graph is strictly greater than one, then localization is ensured with some appropriated initial state. We also compute the limit density function for the Grover walk on $\mathbb{Z}^d$ with flip flop shift, which implies the coexistence of linear spreading and localization. We partially obtain the abstractive shape of the limit density function: the support is within the $d$-dimensional sphere of radius $1/\sqrt{d}$, and $2^d$ singular points reside on the sphere's surface.

math.PR↗

arXiv2014

A one-dimensional Hadamard walk with one defect

We consider a one-dimensional space-inhomogeneous discrete time quantum walk. This model is the Hadamard walk with one defect at the origin which is different from the model introduced by Wojcik et al. [14]. We obtain a stationary measure of the model by solving the eigenvalue problem and an asymptotic behaviour of the return probability by the path counting approach. Moreover, we get the time-averaged limit measure using the space-time generating function method. The measure is symmetric for the origin and independent of the initial coin state at the starting point. So localization depends only on the parameter which determines the model.

math-ph↗

arXiv2014

Trapping and spreading properties of quantum walk in homological structure

We attempt to extract a homological structure of two kinds of graphs by the Grover walk. The first one consists of a cycle and two semi-infinite lines and the second one is assembled by a periodic embedding of the cycles in $\mathbb{Z}$. We show that both of them have essentially the same eigenvalues induced by the existence of cycles in the infinite graphs. The eigenspace of the homological structure appears as so called {\it localization} in the Grover walks, in that the walk is partially trapped by the homological structure. On the other hand, the difference of the absolutely continuous part of spectrum between them provides different behaviors. We characterize the behaviors by the density functions in the weak convergence theorem: the first one is the delta measure at the bottom while the second one is expressed by two kinds of continuous functions which have different finite supports $(-1/\sqrt{10},1/\sqrt{10})$ and $(-2/7,2/7)$, respectively.

math-ph↗

arXiv2014

Localization of discrete time quantum walks on the glued trees

In this paper, we consider the time averaged distribution of discrete time quantum walks on the glued trees. In order to analyse the walks on the glued trees, we consider a reduction to the walks on path graphs. Using a spectral analysis of the Jacobi matrices defined by the corresponding random walks on the path graphs, we have spectral decomposition of the time evolution operator of the quantum walks. We find significant contributions of the eigenvalues $\pm 1$ of the Jacobi matrices to the time averaged limit distribution of the quantum walks. As a consequence we obtain lower bounds of the time averaged distribution.

math-ph↗

arXiv2013

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

For a quantum walk on a graph, there exist many kinds of operators for the discrete-time 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.

math-ph↗

arXiv2013

Limit measures of inhomogeneous discrete-time quantum walks in one dimension

We treat three types of measures of the quantum walk (QW) with the spatial perturbation at the origin, which was introduced by [1]: time averaged limit measure, weak limit measure, and stationary measure. From the first two measures, we see a coexistence of the ballistic and localized behaviors in the walk as a sequential result following [1,2]. We propose a universality class of QWs with respect to weak limit measure. It is shown that typical spatial homogeneous QWs with ballistic spreading belong to the universality class. We find that the walk treated here with one defect also belongs to the class. We mainly consider the walk starting from the origin. However when we remove this restriction, we obtain a stationary measure of the walk. As a consequence, by choosing parameters in the stationary measure, we get the uniform measure as a stationary measure of the Hadamard walk and a time averaged limit measure of the walk with one defect respectively.

quant-ph↗

arXiv2013

One-dimensional quantum walks via generating function and the CGMV method

We treat a quantum walk (QW) on the line whose quantum coin at each vertex tends to be the identity as the distance goes to infinity. We obtain a limit theorem that this QW exhibits localization with not an exponential but a "power-law" decay around the origin and a "strongly" ballistic spreading called bottom localization in this paper. This limit theorem implies the weak convergence with linear scaling whose density has two delta measures at $x=0$ (the origin) and $x=1$ (the bottom) without continuous parts.

math-ph↗

arXiv2012

Quantum graph walks II: Quantum walks on graph coverings

We give a new determinant expression for the characteristic polynomial of the bond scattering matrix of a quantum graph G. Also, we give a decomposition formula for the characteristic polynomial of the bond scattering matrix of a regular covering of G. Furthermore, we define an L-function of G, and give a determinant expression of it. As a corollary, we express the characteristic polynomial of the bond scattering matrix of a regular covering of G by means of its L-functions. As an application, we introduce three types of quantum graph walks, and treat their relation.

math-ph↗

arXiv2012

Quantum graph walks I: mapping to quantum walks

We clarify that coined quantum walk is determined by only the choice of local quantum coins. To do so, we characterize coined quantum walks on graph by disjoint Euler circles with respect to symmetric arcs. In this paper, we introduce a new class of coined quantum walk by a special choice of quantum coins determined by corresponding quantum graph, called quantum graph walk. We show that a stationary state of quantum graph walk describes the eigenfunction of the quantum graph.

math-ph↗

arXiv2012

Weak convergence of complex-valued measure for bi-product path space induced by quantum walk

In this paper, a complex-valued measure of bi-product path space induced by quantum walk is presented. In particular, we consider three types of conditional return paths in a power set of the bi-product path space (1) $Λ\times Λ$, (2) $Λ\times Λ'$ and (3) $Λ'\times Λ'$, where $Λ$ is the set of all $2n$-length ($n\in \mathbb{N}$) return paths and $Λ'(\subseteq Λ)$ is the set of all $2n$-length return paths going through $nx$ ($x\in [-1,1]$) at time $n$. We obtain asymptotic behaviors of the complex-valued measures for the situations (1)-(3) which imply two kinds of weak convergence theorems (Theorems 1 and 2). One of them suggests a weak limit of weak values.

math-ph↗

arXiv2012

Localization of quantum walks induced by recurrence properties of random walks

We study a quantum walk (QW) whose time evolution is induced by a random walk (RW) first introduced by Szegedy (2004). We focus on a relation between recurrent properties of the RW and localization of the corresponding QW. We find the following two fundamental derivations of localization of the QW. The first one is the set of all the $\ell^2$ summable eigenvectors of the corresponding RW. The second one is the orthogonal complement, whose eigenvalues are $\pm 1$, of the subspace induced by the RW. In particular, as a consequence, for an infinite half line, we show that localization of the QW can be ensured by the positive recurrence of the corresponding RWs, and also that the existence of only one self loop affects localization properties.

quant-ph↗

arXiv2012

Localization of the Grover walks on spidernets and free Meixner laws

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.

quant-ph↗

arXiv2012

Time averaged distribution of a discrete-time quantum walk on the path

The discrete time quantum walk which is a quantum counterpart of random walk plays important roles in the theory of quantum information theory. In the present paper, we focus on discrete time quantum walks viewed as quantization of random walks on the path. We obtain a weak limit theorem for the time averaged distribution of our quantum walks.

quant-ph↗

arXiv2012

On the entropy of decoherence matrix for quantum walks

The decoherence matrix studied by Gudder and Sorkin (2011) can be considered as a map from the set of all the pairs of $n$-length paths to complex numbers, which is induced by the discrete-time quantum walk. The decoherence matrix is one of the decoherence functionals which present their historical quantum measure theory. In this paper, we compute the von Neumann entropy of the decoherence matrix. To do so, we use the result that the eigensystem of the decoherence matrix can be expressed by a corresponding correlated random walk.

quant-ph↗