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Etsuo Segawa

Publications and source records attributed to Etsuo Segawa.

99 records · Page 6Linked to original sources
arXiv2012

Limit Theorems for the Discrete-Time Quantum Walk on a Graph with Joined Half Lines

We consider a discrete-time quantum walk $W_{t,κ}$ at time $t$ on a graph with joined half lines $\mathbb{J}_κ$, which is composed of $κ$ half lines with the same origin. Our analysis is based on a reduction of the walk on a half line. The idea plays an important role to analyze the walks on some class of graphs with \textit{symmetric} initial states. In this paper, we introduce a quantum walk with an enlarged basis and show that $W_{t,κ}$ can be reduced to the walk on a half line even if the initial state is \textit{asymmetric}. For $W_{t,κ}$, we obtain two types of limit theorems. The first one is an asymptotic behavior of $W_{t,κ}$ which corresponds to localization. For some conditions, we find that the asymptotic behavior oscillates. The second one is the weak convergence theorem for $W_{t,κ}$. On each half line, $W_{t,κ}$ converges to a density function like the case of the one-dimensional lattice with a scaling order of $t$. The results contain the cases of quantum walks starting from the general initial state on a half line with the general coin and homogeneous trees with the Grover coin.

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arXiv2011

Crossovers induced by discrete-time quantum walks

We consider crossovers with respect to the weak convergence theorems from a discrete-time quantum walk (DTQW). We show that a continuous-time quantum walk (CTQW) and discrete- and continuous-time random walks can be expressed as DTQWs in some limit. At first we generalize our previous study [Phys. Rev. A \textbf{81}, 062129 (2010)] on the DTQW with position measurements. We show that the position measurements per each step with probability $p \sim 1/n^β$ can be evaluated, where $n$ is the final time and $0<β<1$. We also give a corresponding continuous-time case. As a consequence, crossovers from the diffusive spreading (random walk) to the ballistic spreading (quantum walk) can be seen as the parameter $β$ shifts from 0 to 1 in both discrete- and continuous-time cases of the weak convergence theorems. Secondly, we introduce a new class of the DTQW, in which the absolute value of the diagonal parts of the quantum coin is proportional to a power of the inverse of the final time $n$. This is called a final-time-dependent DTQW (FTD-DTQW). The CTQW is obtained in a limit of the FTD-DTQW. We also obtain the weak convergence theorem for the FTD-DTQW which shows a variety of spreading properties. Finally, we consider the FTD-DTQW with periodic position measurements. This weak convergence theorem gives a phase diagram which maps sufficiently long-time behaviors of the discrete- and continuous-time quantum and random walks.

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arXiv2011

Localization of discrete-time quantum walks on a half line via the CGMV method

We study discrete-time quantum walks on a half line by means of spectral analysis. Cantero et al. [1] showed that the CMV matrix, which gives a recurrence relation for the orthogonal Laurent polynomials on the unit circle [2], expresses the dynamics of the quantum walk. Using the CGMV method introduced by them, the name is taken from their initials, we obtain the spectral measure for the quantum walk. As a corollary, we give another proof for localization of the quantum walk on homogeneous trees shown by Chisaki et al. [3].

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arXiv2010

Emergence of Randomness and Arrow of Time in Quantum Walks

Quantum walks are powerful tools not only to construct the quantum speedup algorithms but also to describe specific models in physical processes. Furthermore, the discrete time quantum walk has been experimentally realized in various setups. We apply the concept of the quantum walk to the problems in quantum foundations. We show that randomness and the arrow of time in the quantum walk gradually emerge by periodic projective measurements from the mathematically obtained limit distribution under the time scale transformation.

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arXiv2009

Limit theorems for discrete-time quantum walks on trees

We consider a discrete-time quantum walk W_t given by the Grover transformation on the Cayley tree. We reduce W_t to a quantum walk X_t on a half line with a wall at the origin. This paper presents two types of limit theorems for X_t. The first one is X_t as t\to\infty, which corresponds to a localization in the case of an initial qubit state. The second one is X_t/t as t\to\infty, whose limit density is given by the Konno density function [1-4]. The density appears in various situations of discrete-time cases. The corresponding similar limit theorem was proved in [5] for a continuous-time case on the Cayley tree.

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arXiv2009

Limit theorems for quantum walks driven by many coins

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.

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arXiv2005

One-Dimensional Three-State Quantum Walk

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.

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