The present paper treats the period T_N of the Hadamard walk on a cycle C_N with N vertices. Dukes (2014) considered the periodicity of more general quantum walks on C_N and showed T₂=2, T₄=8, T₈=24 for the Hadamard walk case. We prove that the Hadamard walk does not have any period except for his case, i.e., N = 2,4,8. Our method is based on a path counting and cyclotomic polynomials which is different from his approach based on the property of eigenvalues for unitary matrix that determines the evolution of the walk.

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We define the quaternionic quantum walk on a finite graph and investigate its properties. This walk can be considered as a natural quaternionic extension of the Grover walk on a graph. We explain the way to obtain all the right eigenvalues of a quaternionic matrix and a notable property derived from the unitarity condition for the quaternionic quantum walk. Our main results determine all the right eigenvalues of the quaternionic quantum walk by using complex eigenvalues of the quaternionic weighted matrix which is easily derivable from the walk. Since our derivation is owing to a quaternionic generalization of the determinant expression of the second weighted zeta function, we explain the second weighted zeta function and the relationship between the walk and the second weighted zeta function.

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Properties of the probability distribution generated by a discrete-time quantum walk, such as the number of peaks it contains, depend strongly on the choice of the initial condition. In the present paper we discuss from this point of view the model of the two-dimensional quantum walk analyzed in K. Watabe et al., Phys. Rev. A 77, 062331, (2008). We show that the limit density can be altered in such a way that it vanishes on the boundary or some line. Using this result one can suppress certain peaks in the probability distribution. The analysis is simplified considerably by choosing a more suitable basis of the coin space, namely the one formed by the eigenvectors of the coin operator.

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In this expository work, we discuss spatially inhomogeneous quantum walks in one dimension and describe a genre of mathematical methods that enables one to translate information about the time-independent eigenvalue equation for the unitary generator into dynamical estimates for the corresponding quantum walk. To illustrate the general methods, we show how to apply them to a 1D coined quantum walk whose coins are distributed according to an element of the Thue–Morse subshift.

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One-dimensional discrete-time quantum walks (DTQWs) can simulate various quantum and classical dynamics and have already been implemented in several physical systems. This implementation needs a well-controlled quantum dynamical system, which is the same requirement for implementing quantum information processing tasks. Here, we consider how to realize DTQWs by Dirac particles toward a solid-state implementation of DTQWs.

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Dynamics of two-member Markov processes is formulated based on the binomial probability. Sets of initial states are then sought such that the final state reaches an equilibrium. On the two-parameter phase plane, such initial states are found to exhibit diverse geometric configurations depending on the source probability. Those initial-state boundaries undergo phase transitions ranging over pills, stripes, circles, ellipses, lemons, and even fuzzy shapes. These results are quite helpful in understanding several physical phenomena involving photons, electrons, and atoms. For convenience of discussion, deformations of vortices are taken as an example.

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We theoretically investigate the localization of population distribution in rotational excitation of diatomic molecules induced by a train of optical pulses in the terahertz region. In a simulation with parameters of real molecules, the localization is observed as a combined effect of several causes. For mathematical analysis, we classify the localization into four types based on the viewpoints of physical processes. We provide some extreme numerical examples of the four types of localizations.

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This study is motivated by the previous work [14]. We treat 3 types of the one-dimensional quantum walks (QWs), whose time evolutions are described by diagonal unitary matrices except at one defected point. In this paper, we call the QW defined by diagonal unitary matrices, ``the diagonal QW'', and we consider the stationary distributions of general 2-state diagonal QW with one defect, 3-state space-homogeneous diagonal QW, and 3-state diagonal QW with one defect. One of the purposes of our study is to characterize the QWs by the stationary measure, which may lead to answer the basic and natural question, ``What are stationary measures for one-dimensional QWs?''. In order to analyze the stationary distribution, we focus on the corresponding eigenvalue problems and the definition of the stationary measure.

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Two subjects are discussed in this work: localisation and recurrence in a model of quantum walk in a periodic potential, and a model of opinion dynamics with multiple choices of opinions.

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In this paper we discuss the periodicity of the evolution matrix of Szegedy walk, which is a special type of quantum walk induced by the classical simple random walk, on a finite graph. We completely characterize the periods of Szegedy walks for complete graphs, compete bipartite graphs and strongly regular graphs. In addition, we discuss the periods of Szegedy walk induced by a non-reversible random walk on a cycle.

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Based on the similarity between telegraph equation for transmission lines and Klein–Gordon equation, we have related a distributed element model in electrical engineering to a discrete-time quantum walk through Dirac equation. As a result, we have constructed a discrete transmission line model for a discrete-time quantum walk, and it enables us understanding the characteristics of quantum walks as those of the transmission line.

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PT symmetry, namely, a combined parity and time-reversal symmetry can make non-unitary quantum walks exhibit entirely real eigenenergy. However, it is known that the concept of PT symmetry can be generalized and an arbitrary anti-unitary symmetry has a possibility to substitute PT symmetry. The aim of the present work is to seek such non-unitary quantum walks with generalized PT symmetry by focusing on effects of spatially random disorder which breaks PT symmetry. We numerically find non-unitary quantum walks whose quasi-energy is entirely real despite PT symmetry is broken.

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We discuss the description of eigenspace of a quantum walk model U with an associating linear operator T in abstract settings of quantum walk including the Szegedy walk on graphs. In particular, we provide the spectral mapping theorem of U without the spectral decomposition of T. Arguments in this direction reveal the eigenspaces of U characterized by the generalized kernels of linear operators given by T.

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Quantum computers promise to be able to solve tasks beyond the reach of standard computational platforms. Among the others, photonic quantum walks prove to be great candidates for their implementation, since single photon sources, passive linear optics and photo-detectors are sufficient for universal quantum computation. To this aim, a device performing the quantum Fourier transform represents a fundamental building block for quantum algorithms, whose applications are not limited to the field of quantum computation. Recently, an algorithm has been developed to efficiently realize a quantum Fourier transform of an input photonic state by using a quantum walk on elementary linear-optical components. Here we provide a simple operative description of the algorithm, introducing a whole class of quantum transformations achievable through a generalization of this procedure. We finally discuss how femtosecond laser writing technology well represents an efficient and scalable platform for the implementation of this class of photonic quantum walks.

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We analyze the equivalence between discrete-time coined quantum walks and Szegedy's quantum walks. We characterize a class of flip-flop coined models with generalized Grover coin on a graph Γ that can be directly converted into Szegedy's model on the subdivision graph of Γ and we describe a method to convert one model into the other. This method improves previous results in literature that need to use the staggered model and the concept of line graph, which are avoided here.

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