arXiv subjects
Publications and source records attributed to Norio Konno.
In this paper we obtain some properties of orthogonal polynomials given by a weight function which is a limit density of a rescaled discrete-time quantum walk on the line.
We introduce a continuous-time quantum walk on an ultrametric space corresponding to the set of p-adic integers and compute its time-averaged probability distribution. It is shown that localization occurs for any location of the ultrametric space for the walk. This result presents a striking contrast to the classical random walk case. Moreover we clarify a difference between the ultrametric space and other graphs, such as cycle graph, line, hypercube and complete graph, for the localization of the quantum case. Our quantum walk may be useful for a quantum search algorithm on a tree-like hierarchical structure.
We obtain some rigorous results on limit theorems for quantum walks driven by many coins introduced by Brun et al. in the long time limit. The results imply that whether the behavior of a particle is quantum or classical depends on the three factors: the initial qubit, the number of coins M, d= [t/M], where t is time step. Our main theorem shows that we can see a transition from classical behavior to quantum one for a class of three factors.
One-parameter family of discrete-time quantum-walk models on the square lattice, which includes the Grover-walk model as a special case, is analytically studied. Convergence in the long-time limit $t \to \infty$ of all joint moments of two components of walker's pseudovelocity, $X_t/t$ and $Y_t/t$, is proved and the probability density of limit distribution is derived. Dependence of the two-dimensional limit density function on the parameter of quantum coin and initial four-component qudit of quantum walker is determined. Symmetry of limit distribution on a plane and localization around the origin are completely controlled. Comparison with numerical results of direct computer-simulations is also shown.
We study a series of one-dimensional discrete-time quantum-walk models labeled by half integers $j=1/2, 1, 3/2, ...$, introduced by Miyazaki {\it et al.}, each of which the walker's wave function has $2j+1$ components and hopping range at each time step is $2j$. In long-time limit the density functions of pseudovelocity-distributions are generally given by superposition of appropriately scaled Konno's density function. Since Konno's density function has a finite open support and it diverges at the boundaries of support, limit distribution of pseudovelocities in the $(2j+1)$-component model can have $2j+1$ pikes, when $2j+1$ is even. When $j$ becomes very large, however, we found that these pikes vanish and a universal and monotone convex structure appears around the origin in limit distributions. We discuss a possible route from quantum walks to classical diffusion associated with the $j \to \infty$ limit.
The threshold network model is a type of finite random graphs. In this paper, we introduce a generalized threshold network model. A pair of vertices with random weights is connected by an edge when real-valued functions of the pair of weights belong to given Borel sets. We extend several known limit theorems for the number of prescribed subgraphs to show that the strong law of large numbers can be uniform convergence. We also prove two limit theorems for the local and global clustering coefficients.
In this note we give a new method for getting a series of approximations for the extinction probability of the one-dimensional contact process by using the Gröbner basis.
Quantization of a random-walk model is performed by giving a qudit (a multi-component wave function) to a walker at site and by introducing a quantum coin, which is a matrix representation of a unitary transformation. In quantum walks, the qudit of walker is mixed according to the quantum coin at each time step, when the walker hops to other sites. As special cases of the quantum walks driven by high-dimensional quantum coins generally studied by Brun, Carteret, and Ambainis, we study the models obtained by choosing rotation as the unitary transformation, whose matrix representations determine quantum coins. We show that Wigner's $(2j+1)$-dimensional unitary representations of rotations with half-integers $j$'s are useful to analyze the probability laws of quantum walks. For any value of half-integer $j$, convergence of all moments of walker's pseudovelocity in the long-time limit is proved. It is generally shown for the present models that, if $(2j+1)$ is even, the probability measure of limit distribution is given by a superposition of $(2j+1)/2$ terms of scaled Konno's density functions, and if $(2j+1)$ is odd, it is a superposition of $j$ terms of scaled Konno's density functions and a Dirac's delta function at the origin. For the two-, three-, and four-component models, the probability densities of limit distributions are explicitly calculated and their dependence on the parameters of quantum coins and on the initial qudit of walker is completely determined. Comparison with computer simulation results is also shown.
Coexistence of individuals with different species or phenotypes is often found in nature in spite of competition between them. Stable coexistence of multiple types of individuals have implications for maintenance of ecological biodiversity and emergence of altruism in society, to name a few. Various mechanisms of coexistence including spatial structure of populations, heterogeneous individuals, and heterogeneous environments, have been proposed. In reality, individuals disperse and interact on complex networks. We examine how heterogeneous degree distributions of networks influence coexistence, focusing on models of cyclically competing species. We show analytically and numerically that heterogeneity in degree distributions promotes stable coexistence.
Infectious diseases are practically represented by models with multiple states and complex transition rules corresponding to, for example, birth, death, infection, recovery, disease progression, and quarantine. In addition, networks underlying infection events are often much more complex than described by meanfield equations or regular lattices. In models with simple transition rules such as the SIS and SIR models, heterogeneous contact rates are known to decrease epidemic thresholds. We analyze steady states of various multi-state disease propagation models with heterogeneous contact rates. In many models, heterogeneity simply decreases epidemic thresholds. However, in models with competing pathogens and mutation, coexistence of different pathogens for small infection rates requires network-independent conditions in addition to heterogeneity in contact rates. Furthermore, models without spontaneous neighbor-independent state transitions, such as cyclically competing species, do not show heterogeneity effects.
A quantum central limit theorem for a continuous-time quantum walk on a homogeneous tree is derived from quantum probability theory. As a consequence, a new type of limit theorems for another continuous-time walk introduced by the walk is presented. The limit density is similar to that given by a continuous-time quantum walk on the one-dimensional lattice.
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.
Concerning a discrete-time quantum walk X^{(d)}_t with a symmetric distribution on the line, whose evolution is described by the Hadamard transformation, it was proved by the author that the following weak limit theorem holds: X^{(d)}_t /t \to dx / π(1-x^2) \sqrt{1 - 2 x^2} as t \to \infty. The present paper shows that a similar type of weak limit theorems is satisfied for a {\it continuous-time} quantum walk X^{(c)}_t on the line as follows: X^{(c)}_t /t \to dx / π\sqrt{1 - x^2} as t \to \infty. These results for quantum walks form a striking contrast to the central limit theorem for symmetric discrete- and continuous-time classical random walks: Y_{t}/ \sqrt{t} \to e^{-x^2/2} dx / \sqrt{2 π} as t \to \infty. The work deals also with issue of the relationship between discrete and continuous-time quantum walks. This topic, subject of a long debate in the previous literature, is treated within the formalism of matrix representation and the limit distributions are exhaustively compared in the two cases.
In this paper we consider the one-dimensional quantum random walk X^{varphi} _n at time n starting from initial qubit state varphi determined by 2 times 2 unitary matrix U. We give a combinatorial expression for the characteristic function of X^{varphi}_n. The expression clarifies the dependence of it on components of unitary matrix U and initial qubit state varphi. As a consequence of the above results, we present a new type of limit theorems for the quantum random walk. In contrast with the de Moivre-Laplace limit theorem, our symmetric case implies that X^{varphi}_n /n converges in distribution to a limit Z^{varphi} as n to infty where Z^{varphi} has a density 1 / pi (1-x^2) sqrt{1-2x^2} for x in (- 1/sqrt{2}, 1/sqrt{2}). Moreover we discuss some known simulation results based on our limit theorems.
We study a generalized Hadamard walk in one dimension with three inner states. The particle governed by the three-state quantum walk moves, in superposition, both to the left and to the right according to the inner state. In addition to these two degrees of freedom, it is allowed to stay at the same position. We calculate rigorously the wavefunction of the particle starting from the origin for any initial qubit state, and show the spatial distribution of probability of finding the particle. In contrast with the Hadamard walk with two inner states on a line, the probability of finding the particle at the origin does not converge to zero even after infinite time steps except special initial states. This implies that the particle is trapped near the origin after long time with high probability.
We study a quantum cellular automaton (QCA) whose time-evolution is defined from global transition function of classical cellular automata (CA). In order to investigate natural transformations from CA to QCA, the present QCA includes CA with Wolfram's rule 150 and 105 as special cases. We firstly compute the time-evolution of the QCA and examine its statistical properties. As a basic statistical value, the probability of finding an active cell averaged over a spatial-temporal space is introduced, and the difference between CA and QCA is considered. In addition, it is shown that statistical properties in QCA are related to the classical trajectory in the configuration space.
We study the one-dimensional two-state totalistic probabilistic cellular automata (TPCA) having an absorbing state with long-range interactions, which can be considered as a natural extension of the Domany-Kinzel model. We establish the conditions for existence of a site-bond representation and self-dual property. Moreover we present an expression of a set-to-set connectedness between two sets, a matrix expression for a condition of the self-duality, and a convergence theorem for the TPCA.
Motivated by the recent work of Patel et al., this paper clarifies a connection between coined quantum walks and quantum cellular automata in a general setting. As a consequence, their result is naturally derived from the connection.