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Etsuo Segawa

Publications and source records attributed to Etsuo Segawa.

At least 19 recordsLinked to original sources
arXiv2026

Robustness of periodicity in Grover walks under a magnetic vector potential

We study the effect of magnetic vector potentials on periodic Grover walks on finite graphs. The magnetic vector potential is introduced through the framework of quantum graphs, which induces the Grover walk as a special case. We regard the vector potential as a perturbation of a periodic Grover walk and investigate the robustness of its periodicity. Our analysis reveals that the response to such perturbations depends on the spectral structure of the underlying graph. In particular, when the graph possesses at least one non-simple eigenvalue, we derive a Hermitian matrix that characterizes the robustness of its periodicity. As a consequence, we show that the perturbed dynamics is asymptotically described by a continuous-time quantum walk generated by this Hermitian matrix.

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arXiv2026

Entanglement entropy in two-particle Grover walks on graphs

We define a two-particle quantum walk of identical particles on a graph $G$ via the one-particle Grover walk on the Kronecker product $G \otimes G$, and call it the two-particle Grover walk. In systems of identical particles, quantum mechanics requires that quantum states have a certain invariance with respect to the exchange of particles. Focusing on the symmetry of the Kronecker product $G \otimes G$ as a graph, we show that the time evolution operator of this walk commutes with the swap operator, which ensures that this requirement is satisfied. Furthermore, we study the entanglement entropy of quantum states evolved by this walk. For the complete bipartite graph $K_{n,n}$, we completely determine the values of $n$ for which the quantum states evolved from specific initial states attain the upper bound of the entropy at some time, and prove that they are exactly $1$ and $2$.

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arXiv2026

Resonant scattering for tunable quantum walks on graphs with tails

We study the resonant scattering for discrete time quantum walks on graphs with some tails. In our arguments, we reduce the study of resonances to the perturbation of eigenvalues of a finite rank matrix associated with the internal graph. Then we can apply Kato's perturbation theory of matrices, and the reduction process of generalized eigenspaces allows us to derive an explicit asymptotic expansion of the scattering matrix. As a consequence, we obtain the resonant scattering at resonant energies.

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arXiv2025

Pulsation of quantum walk between two arbitrary graphs with weakly connected bridge

We consider the Grover walk on a finite graph composed of two arbitrary simple graphs connected by one edge, referred to as a bridge. The parameter $\epsilon>0$ assigned at the bridge represents the strength of connectivity: if $\epsilon=0$, then the graph is completely separated. We show that for sufficiently small values of $\epsilon$, a phenomenon called pulsation occurs. The pulsation is characterized by the periodic transfer of the quantum walker between the two graphs. An asymptotic expression with respect to small $\epsilon$ for the probability of finding the walker on either of the two graphs is derived. This expression reveals that the pulsation depends solely on the number of edges in each graph, regardless of their structure. In addition, we obtain that the quantum walker is transferred periodically between the two graphs, with a period of order $O(\epsilon^{-1/2})$. Furthermore, when the number of edges of two graphs is equal, the quantum walker is almost completely transferred.

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arXiv2025

Quantum spatial best-arm identification via quantum walks

Quantum reinforcement learning has emerged as a framework combining quantum computation with sequential decision-making, and applications to the multi-armed bandit (MAB) problem have been reported. The graph bandit problem extends the MAB setting by introducing spatial constraints, where the accessibility of arms is restricted by graph connectivity, yet quantum approaches to this setting remain limited. In this paper, we formulate best-arm identification in graph bandits and propose a quantum algorithmic framework, termed Quantum Spatial Best-Arm Identification (QSBAI), which is applicable to general graph structures. This framework uses quantum walks to encode superpositions over graph-constrained actions, thereby extending amplitude amplification and generalizing the quantum BAI algorithm via Szegedy's walk framework. We focus our theoretical analysis on complete and bipartite graphs, deriving the maximal success probability of identifying the best arm and the time step at which it is achieved. Our results clarify how quantum-walk-based search can be adapted to structurally constrained decision problems and provide a foundation for quantum best-arm identification in graph-structured environments.

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arXiv2025

Multi-player conflict avoidance through entangled quantum walks

Quantum computing has the potential to solve complex problems faster and more efficiently than classical computing. It can achieve speedups by leveraging quantum phenomena like superposition, entanglement, and tunneling. Quantum walks (QWs) form the foundation for many quantum algorithms. Unlike classical random walks, QWs exhibit quantum interference, leading to unique behaviors such as linear spreading and localization. These properties make QWs valuable for various applications, including universal computation, time series prediction, encryption, and quantum hash functions. One emerging application of QWs is decision making. Previous research has used QWs to model human decision processes and solve multi-armed bandit problems. This paper extends QWs to collective decision making, focusing on minimizing decision-conflict cases where multiple agents choose the same option, leading to inefficiencies like traffic congestion or overloaded servers. Prior research using quantum interference has addressed two-player conflict avoidance but struggled with three-player scenarios. This paper proposes a novel method using QWs to entirely eliminate decision conflicts in three-player cases, demonstrating its effectiveness in collective decision making.

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arXiv2025

Comfortability of quantum walks on embedded graphs on surfaces

The time evolutions of discrete-time quantum walks on graphs are determined by the local adjacency relations of the graphs. In this paper, first, we construct a discrete-time quantum walk model that reflects the embedding on the surface so that an underlying global geometric information is reflected. Second, we consider the scattering problem of this quantum walk model. We obtain the scattering matrix characterized by the faces on the surface and detect the orientablility of the embedding using scattering information. For the stationary state in the scattering problem, the comfortability is defined as the square norm of the stationary state restricted to the internal. This indicates how a quantum walker is stored in the internal under the embedding. Then we find that a quantum walker feels more comfortable on a surface with small genus in some natural setting. We illustrate our results with some interesting examples.

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arXiv2024

A mathematical framework for maze solving using quantum walks

We provide a mathematical framework for identifying the shortest path in a maze using a Grover walk, which becomes non-unitary by introducing absorbing holes. In this study, we define the maze as a network with vertices connected by unweighted edges. Our analysis of the stationary state of the Grover walk on finite graphs, where we strategically place absorbing holes and self-loops on specific vertices, demonstrates that this approach can effectively solve mazes. By setting arbitrary start and goal vertices in the underlying graph, we obtain the following long-time results: (i) in tree structures, the probability amplitude is concentrated exclusively along the shortest path between start and goal; (ii) in ladder-like structures with additional paths, the probability amplitude is maximized near the shortest path.

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arXiv2024

Pulsation of quantum walk on Johnson graph

We propose a phenomenon of discrete-time quantum walks on graphs called the pulsation, which is a generalization of a phenomenon in the quantum searches. This phenomenon is discussed on a composite graph formed by two connected graphs $G_{1}$ and $G_{2}$. The pulsation means that the state periodically transfers between $G_{1}$ and $G_{2}$ with the initial state of the uniform superposition on $G_1$. In this paper, we focus on the case for the Grover walk where $G_{1}$ is the Johnson graph and $G_{2}$ is a star graph. Also, the composite graph is constructed by identifying an arbitrary vertex of the Johnson graph with the internal vertex of the star graph. In that case, we find the pulsation with $O(\sqrt{N^{1+1/k}})$ periodicity, where $N$ is the number of vertices of the Johnson graph. The proof is based on Kato's perturbation theory in finite-dimensional vector spaces.

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arXiv2024

QW-Search/Zeta Correspondence

We consider the connection between this zeta function and quantum search via quantum walk. First, we give an explicit expression of the zeta function on the one-dimensional torus in the general case of the number and position of marked vertices. Moreover, we deal with the two special cases of the position of the marked vertices on the $d$-dimensional torus $(d \ge 2)$. Additionally, we treat the property of the zeta function by using the Mahler measure. Our results show the relationship between the zeta function and quantum search algorithms for the first time.

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arXiv2024

Sensitivity of quantum walk to phase reversal and geometric perturbations: an exploration in complete graphs

In this paper, we analyze the dynamics of quantum walks on a graph structure resulting from the integration of a main connected graph $G$ and a secondary connected graph $G'$. This composite graph is formed by a disjoint union of $G$ and $G'$, followed by the contraction of a selected pair of vertices creating a cut vertex $v^*$ and leading to a unique form of geometric perturbation. Our study focuses on instances where $G$ is a complete graph $K_N$ and $G'$ is a star graph $S_m$. The core of our analysis lies in exploring the impact of this geometric perturbation on the success probability of quantum walk-based search algorithms, particularly in an oracle-free context. Despite initial findings suggesting a low probability of locating the perturbed vertex $v^*$, we demonstrate that introducing a phase reversal to the system significantly enhances the success rate. Our results reveal that with an optimal running time and specific parameter conditions, the success probability can be substantially increased. The paper is structured to first define the theoretical framework, followed by the presentation of our main results, detailed proofs, and concluding with a summary of our findings and potential future research directions.

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arXiv2024

Quantum walks on graphs embedded in orientable surfaces

A quantum walk model which reflects the $2$-cell embedding on the orientable closed surface of a graph in the dynamics is introduced. We show that the scattering matrix is obtained by finding the faces on the underlying surface which have the overlap to the boundary and the stationary state is obtained by counting two classes of the rooted spanning subgraphs of the dual graph on the underlying embedding.

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arXiv2024

Characterization of Green's function of discrete Schr\"odinger operator on a finite graph by its spanning subgraphs

The Green's function of the discrete Sch\"odinger operator on a finite graph is considered. This setting reproduces Laplacian and signless Laplacian by adjusting appropriate potentials. We show two ways of the expression for the Green's function by using graph structures. The first way is based on the factor of the graph by subtrees which have uni-self-loops; the second way is based on that by odd unicycle graphs.

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arXiv2023

A method of approximation of discrete Schr\"odinger equation with the normalized Laplacian by discrete-time quantum walk on graphs

We propose a class of continuous-time quantum walk models on graphs induced by a certain class of discrete-time quantum walk models with the parameter $\epsilon\in [0,1]$. Here the graph treated in this paper can be applied both finite and infinite cases. The induced continuous-time quantum walk is an extended version of the (free) discrete-Schr\"odinger equation driven by the normalized Laplacian: the element of the weighted Hermitian takes not only a scalar value but also a matrix value depending on the underlying discrete-time quantum walk. We show that each discrete-time quantum walk with an appropriate setting of the parameter $\epsilon$ in the long time limit identifies with its induced continuous-time quantum walk and give the running time for the discrete-time to approximate the induced continuous-time quantum walk with a small error $\delta$. We also investigate the detailed spectral information on the induced continuous-time quantum walk.

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arXiv2023

A comfortable graph structure for Grover walk

We consider a Grover walk model on a finite internal graph, which is connected with a finite number of semi-infinite length paths and receives the alternative inflows along these paths at each time step. After the long time scale, we know that the behavior of such a Grover walk should be stable, that is, this model has a stationary state. In this paper our objectives are to give some characterization upon the scattering of the stationary state on the surface of the internal graph and upon the energy of this state in the interior. For the scattering, we concretely give a scattering matrix, whose form is changed depending on whether the internal graph is bipartite or not. On the other hand, we introduce a comfortability function of a graph for the quantum walk, which shows how many quantum walkers can stay in the interior, and we succeed in showing the comfortability of the walker in terms of combinatorial properties of the internal graph.

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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.

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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.

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