arXiv subjects

Etsuo Segawa

Publications and source records attributed to Etsuo Segawa.

At least 37 records · Page 2Linked to original sources

arXiv2023

Resonance expansion for quantum walks and its applications to the long-time behavior

In this paper, resonances are introduced to a class of quantum walks on $\mathbb{Z}$. Resonances are defined as poles of the meromorphically extended resolvent of the unitary time evolution operator. In particular, they appear inside the unit circle. Some analogous properties to those of quantum resonances for Schrödinger operators are shown. Especially, the resonance expansion, an analogue of the eigenfunction expansion, indicates the long-time behavior of quantum walks. The decaying rate, the asymptotic probability distribution, and the weak limit of the probability density are described by resonances and associated (generalized) resonant states. The generic simplicity of resonances is also investigated.

math-ph↗

arXiv2023

Bandit Algorithm Driven by a Classical Random Walk and a Quantum Walk

Quantum walks (QWs) have a property that classical random walks (RWs) do not possess -- the coexistence of linear spreading and localization -- and this property is utilized to implement various kinds of applications. This paper proposes RW- and QW-based algorithms for multi-armed-bandit (MAB) problems. We show that, under some settings, the QW-based model realizes higher performance than the corresponding RW-based one by associating the two operations that make MAB problems difficult -- exploration and exploitation -- with these two behaviors of QWs.

quant-ph↗

arXiv2023

Asymmetric quantum decision-making

Collective decision-making is crucial to information and communication systems. Decision conflicts among agents hinder the maximization of potential utilities of the entire system. Quantum processes can realize conflict-free joint decisions among two agents using the entanglement of photons or quantum interference of orbital angular momentum (OAM). However, previous studies have always presented symmetric resultant joint decisions. Although this property helps maintain and preserve equality, it cannot resolve disparities. Global challenges, such as ethics and equity, are recognized in the field of responsible artificial intelligence as responsible research and innovation paradigm. Thus, decision-making systems must not only preserve existing equality but also tackle disparities. This study theoretically and numerically investigates asymmetric collective decision-making using quantum interference of photons carrying OAM or entangled photons. Although asymmetry is successfully realized, a photon loss is inevitable in the proposed models. The available range of asymmetry and method for obtaining the desired degree of asymmetry are analytically formulated.

quant-ph↗

arXiv2023

On symmetric spectra of Hermitian adjacency matrices for non-bipartite mixed graphs

We study the equivalence between bipartiteness and symmetry of spectra of mixed graphs, for $θ$-Hermitian adjacency matrices defined by an angle $θ\in (0, π]$. We show that this equivalence holds when, for example, an angle $θ$ is an algebraic number, while it breaks down for any angle $θ\in \mathbb{Q}π$. Furthermore, we construct a family of non-bipartite mixed graphs having the symmetric spectra for given $θ\in \mathbb{Q}π$.

math.CO↗

arXiv2023

Skeleton structure inherent in discrete-time quantum walks

In this paper, we claim that a common underlying structure--a skeleton structure--is present behind discrete-time quantum walks (QWs) on a one-dimensional lattice with a homogeneous coin matrix. This skeleton structure is independent of the initial state, and partially, even of the coin matrix. This structure is best interpreted in the context of quantum-walk-replicating random walks (QWRWs), i.e., random walks that replicate the probability distribution of quantum walks, where this newly found structure acts as a simplified formula for the transition probability. Additionally, we construct a random walk whose transition probabilities are defined by the skeleton structure and demonstrate that the resultant properties of the walkers are similar to both the original QWs and QWRWs.

math-ph↗

arXiv2022

Circuit equation of Grover walk

We consider the Grover walk on the infinite graph in which an internal finite subgraph receives the inflow from the outside with some frequency and also radiates the outflow to the outside. To characterize the stationary state of this system, which is represented by a function on the arcs of the graph, we introduce a kind of discrete gradient operator twisted by the frequency. Then we obtain a circuit equation which shows that (i) the stationary state is described by the twisted gradient of a potential function which is a function on the vertices; (ii) the potential function satisfies the Poisson equation with respect to a generalized Laplacian matrix. Consequently, we characterize the scattering on the surface of the internal graph and the energy penetrating inside it. Moreover, for the complete graph as the internal graph, we illustrate the relationship of the scattering and the internal energy to the frequency and the number of tails.

quant-ph↗

arXiv2022

A convergence time of Grover walk on regular graph to stationary state

We consider a quantum walk model on a finite graph which has an interaction with the outside. Here a quantum walker from the outside penetrates the graph and also a quantum walker in the graph goes out to the outside at every time step. This dynamics of the quantum walk converges to a stationary state. In this paper, we estimate the speed of the convergence to the stationary state on the $κ$-regular graph with the uniformly inserting of the inflow to the graph. We show that larger degree of the regular graph makes the convergence speed of this quantum walk model slower.

quant-ph↗

arXiv2022

Spectral mapping theorem of an abstract non-unitary quantum walk

This paper continues the previous work (Quantum Inf. Process (2019)) by two authors of the present paper about a spectral mapping property of chiral symmetric unitary operators. In physics, they treat non-unitary time-evolution operators to consider quantum walks in open systems. In this paper, we generalize the above result to include a chiral symmetric non-unitary operator whose coin operator only has two eigenvalues. As a result, the spectra of such non-unitary operators are included in the (possibly non-unit) circle and the real axis in the complex plane. We also give some examples of our abstract results, such as non-unitary quantum walks defined by Mochizuki et al. Moreover, we present an application to the Ihara zeta functions and correlated random walks on regular graphs, which are not quantum walks.

math-ph↗

arXiv2022

Directivity of quantum walk via its random walk replica

Quantum walks (QWs) exhibit different properties compared with classical random walks (RWs), most notably by linear spreading and localization. In the meantime, random walks that replicate quantum walks, which we refer to as quantum-walk-replicating random walks (QWRWs), have been studied in the literature where the eventual properties of QWRW coincide with those of QWs. However, we consider that the unique attributes of QWRWs have not been fully utilized in the former studies to obtain deeper or new insights into QWs. In this paper, we highlight the directivity of one-dimensional discrete quantum walks via QWRWs. By exploiting the fact that QWRW allows trajectories of individual walkers to be considered, we first discuss the determination of future directions of QWRWs, through which the effect of linear spreading and localization is manifested in another way. Furthermore, the transition probabilities of QWRWs can also be visualized and show a highly complex shape, representing QWs in a novel way. Moreover, we discuss the first return time to the origin between RWs and QWs, which is made possible via the notion of QWRWs. We observe that the first return time statistics of QWs are quite different from RWs, caused by both the linear spreading and localization properties of QWs.

math-ph↗

arXiv2022

Toward fixed point and pulsation quantum search on graphs driven by quantum walks with in- and out-flows: a trial to the complete graph

We treat a quantum walk model with in- and out- flows at every time step from the outside. We show that this quantum walk can find the marked vertex of the complete graph with a high probability in the stationary state. In exchange of the stability, the convergence time is estimated by $O(N\log N)$, where $N$ is the number of vertices. However until the time step $O(N)$, we show that there is a pulsation with the periodicity $O(\sqrt{N})$. We find the marked vertex with a high relative probability in this pulsation phase. This means that we have two chances to find the marked vertex with a high relative probability; the first chance visits in the pulsation phase at short time step $O(\sqrt{N})$ while the second chance visits in the stable phase after long time step $O(N\log N)$. The proofs are based on Kato's perturbation theory.

math-ph↗

arXiv2022

Comfortable place for quantum walkers on finite path

We consider the stationary state of a quantum walk on the finite path, where the sink and source are set at the left and right boundaries. The quantum coin is uniformly placed at every vertex of the path graph. At every time step, a new quantum walker penetrates into the internal from the left boundary and also some existing quantum walkers in the internal goes out to the sinks located in the left and right boundaries. The square modulus of the stationary state at each vertex is regarded as the comfortability for a quantum walker to this vertex in this paper. We show the weak convergence theorem for the scaled limit distribution of the comfortability in the limit of the length of the path.

quant-ph↗

arXiv2022

Design for implementation of discrete-time quantum walk with circulant matrix on graph by optical polarizing elements

In this paper, we introduce a quantum walk whose local scattering at each vertex is denoted by a unitary circulant matrix; namely the circulant quantum walk. We also introduce another quantum walk induced by the circulant quantum walk; namely the optical quantum walk, whose underlying graph is a $2$-regular directed graph and obtained by blowing up the original graph in some way. We propose a design of an optical circuit which implements the stationary state of the optical quantum walk. We show that if the induced optical quantum walk does not have $+1$ eigenvalue, then the stationary state of the optical quantum walk gives that of the original circulant quantum walk. From this result, we give a useful condition for the setting of the circulant quantum walks which can be implemented by this optical circuit.

quant-ph↗

arXiv2021

A variant of the discrete Burgers equation derived from the correlated random walk and its ultradiscretization

In this paper, we show that a variant of the discrete Burgers equation can be obtained through the Cole--Hopf transformation to a generalized discrete diffusion equation corresponding to the correlated random walk, which is also known as a generalization of the well known random walk. By applying the technique called ultradiscretization, we obtain the generalized ultradiscrete diffusion equation, the ultradiscrete Cole--Hopf transformation and a variant of the ultradiscrete Burgers equation. Moreover, we show that the resulting ultradiscrete Burgers equation yields cellular automata which can be interpreted as a traffic flow model.

math-ph↗

arXiv2021

On the average hitting times of the squares of cycles

The exact formula for the average hitting time (HT, as an abbreviation) of simple random walks from one vertex to any other vertex on the square $C^2_N$ of an $N$-vertex cycle graph $C_N$ was given by N. Chair [\textit{Journal of Statistical Physics}, \textbf{154} (2014) 1177-1190]. In that paper, the author gives the expression for the even $N$ case and the expression for the odd $N$ case separately. In this paper, by using an elementary method different from Chair (2014), we give a much simpler single formula for the HT's of simple random walks on $C^2_N$. Our proof is considerably short and fully combinatorial, in particular, has no-need of any spectral graph theoretical arguments. Not only the formula itself but also intermediate results through the process of our proof describe clear relations between the HT's of simple random walks on $C^2_N$ and the Fibonacci numbers.

math.CO↗

arXiv2021

Quantum walks driven by quantum coins with two multiple eigenvalues

We consider a spectral analysis on the quantum walks on graph $G=(V,E)$ with the local coin operators $\{C_u\}_{u\in V}$ and the flip flop shift. The quantum coin operators have commonly two distinct eigenvalues $κ,κ'$ and $p=\dim(\ker(κ-C_u))$ for any $u\in V$ with $1\leq p\leq δ(G)$, where $δ(G)$ is the minimum degrees of $G$. We show that this quantum walk can be decomposed into a cellular automaton on $\ell^2(V;\mathbb{C}^p)$ whose time evolution is described by a self adjoint operator $T$ and its remainder. We obtain how the eigenvalues and its eigenspace of $T$ are lifted up to as those of the original quantum walk. As an application, we express the eigenpolynomial of the Grover walk on $\mathbb{Z}^d$ with the moving shift in the Fourier space.

math-ph↗

arXiv2021

Perfect state transfer in Grover walks between states associated to vertices of a graph

We study perfect state transfer in Grover walks, which are typical discrete-time quantum walk models. In particular, we focus on states associated to vertices of a graph. We call such states vertex type states. Perfect state transfer between vertex type states can be studied via Chebyshev polynomials. We derive a necessary condition on eigenvalues of a graph for perfect state transfer between vertex type states to occur. In addition, we perfectly determine the complete multipartite graphs whose partite sets are the same size on which perfect state transfer occurs between vertex type states, together with the time.

math.CO↗

arXiv2021

Asymptotic properties of generalized eigenfunctions for multi-dimensional quantum walks

We construct a distorted Fourier transformation associated with the multi-dimensional quantum walk. In order to avoid the complication of notations, almost all of our arguments are restricted to two dimensional quantum walks (2DQWs) without loss of generality. The distorted Fourier transformation characterizes generalized eigenfunctions of the time evolution operator of the QW. The 2DQW which will be considered in this paper has an anisotropy due to the definition of the shift operator for the free QW. Then we define an anisotropic Banach space as a modified Agmon-Hörmander's $\mathcal{B}^*$ space and we derive the asymptotic behavior at infinity of generalized eigenfunctions in these spaces. The scattering matrix appears in the asymptotic expansion of generalized eigenfunctions.

math-ph↗

arXiv2021

Resonances in finitely perturbed quantum walks, resonance expansion and generic simplicity

We define resonances for finitely perturbed quantum walks as poles of the scattering matrix in the lower half plane. We show a resonance expansion which describes the time evolution in terms of resonances and corresponding Jordan chains. In particular, the decay rate of the survival probability is given by the imaginary part of resonances and the multiplicity. We prove generic simplicity of the resonances, although there are quantum walks with multiple resonances.

math.SP↗