arXiv subjects

Takuya Machida

Publications and source records attributed to Takuya Machida.

At least 19 recordsLinked to original sources

arXiv2025

A simple model of quantum walk with a gap in distribution

Quantum walks are quantum counterparts of random walks and their probability distributions are different from each other. A quantum walker distributes on a Hilbert space and it is observed at a location with a probability. The finding probabilities have been investigated and some interesting things have been analytically discovered. They are, for instance, ballistic behavior, localization, or a gap. We study a 1-dimensional quantum walk in this paper. Although the walker launches off a location under a localized initial state, some numerical experiments show that the quantum walker does not seem to distribute around the launching location, which suggests that the probability distribution holds a gap around the launching location. To prove the gap analytically, we derive a long-time limit distribution, from which one can tell more details about the finding probability.

quant-ph↗

arXiv2024

Phase transition of a continuous-time quantum walk on the half line

Quantum walks are referred to as quantum analogs to random walks in mathematics. They have been studied as quantum algorithms in quantum information for quantum computers. There are two types of quantum walks. One is the discrete-time quantum walk and the other is the continuous-time quantum walk. We study a continuous-time quantum walk on the half line and challenge to find a limit theorem for it in this paper. As a result, approximate behavior of the quantum walker is revealed after the system of quantum walk gets updated in long time.

quant-ph↗

arXiv2023

Limit distribution of a continuous-time quantum walk with a spatially 2-periodic Hamiltonian

Focusing on a continuous-time quantum walk on $\mathbb{Z}=\left\{0,\pm 1,\pm 2,\ldots\right\}$, we analyze a probability distribution with which the quantum walker is observed at a position. The walker launches off at a localized state and its system is operated by a spatially periodic Hamiltonian. As a result, we see an asymmetric probability distribution. To catch a long-time behavior, we also try to find a long-time limit theorem and realize that the limit distribution holds a symmetric density function.

quant-ph↗

arXiv2021

Phase transition of an open quantum walk

It has been discovered that open quantum walks diffusively distribute in space, since they were introduced in 2012. Indeed, some limit distributions have been demonstrated and most of them are described by Gaussian distributions. We operate an open quantum walk on $\mathbb{Z}=\left\{0, \pm 1, \pm 2,\ldots\right\}$ with parameterized operations in this paper, and study its 1st and 2nd moments so that we find its standard deviation. The standard deviation tells us whether the open quantum walker shows diffusive or ballistic behavior, which results in a phase transition of the walker.

quant-ph↗

arXiv2020

Limit distribution of a time-dependent quantum walk on the half line

We focus on a 2-period time-dependent quantum walk on the half line in this paper. The quantum walker launches at the edge of the half line in a localized superposition state and its time evolution is carried out with two unitary operations which are alternately cast to the quantum walk. As a result, long-time limit finding probabilities of the quantum walk turn to be determined by either one of the two operations, but not both. More interestingly, the limit finding probabilities are independent from the localized initial state. We will approach the appreciated features via a quantum walk on the line which is able to reproduce the time-dependent walk on the half line.

quant-ph↗

arXiv2019

A limit distribution for a quantum walk driven by a five-diagonal unitary matrix

In this paper, we work on a quantum walk whose system is manipulated by a five-diagonal unitary matrix, and present long-time limit distributions. The quantum walk launches off a location and delocalizes in distribution as its system is getting updated. The five-diagonal matrix contains a phase term and the quantum walk becomes a standard coined walk when the phase term is fixed at special values. Or, the phase term gives an effect on the quantum walk. As a result, we will see an explicit form of a long-time limit distribution for a quantum walk driven by the matrix, and thanks to the exact form, we understand how the quantum walker approximately distributes in space after the long-time evolution has been executed on the walk.

quant-ph↗

arXiv2018

Some limit laws for quantum walks with applications to a version of the Parrondo paradox

A quantum walker moves on the integers with four extra degrees of freedom, performing a coin-shift operation to alter its internal state and position at discrete units of time. The time evolution is described by a unitary process. We focus on finding the limit probability law for the position of the walker and study it by means of Fourier analysis. The quantum walker exhibits both localization and a ballistic behavior. Our two results are given as limit theorems for a 2-period time-dependent walk and they describe the location of the walker after it has repeated the unitary process a large number of times. The theorems give an analytical tool to study some of the Parrondo type behavior in a quantum game which was studied by J. Rajendran and C. Benjamin by means of very nice numerical simulations [1]. With our analytical tools at hand we can easily explore the "phase space" of parameters of one of the games, similar to the winning game in their papers. We include numerical evidence that our two games, similar to theirs, exhibit a Parrondo type paradox.

quant-ph↗

arXiv2017

A limit theorem for a splitting distribution of a quantum walk

Discrete-time quantum walks are considered a counterpart of random walks and the study for them has been getting attention since around 2000. In this paper, we focus on a quantum walk which generates a probability distribution splitting to two parts. The quantum walker with two coin states spreads at points, represented by integers, and we analyze the chance of finding the walker at each position after it carries out a unitary evolution a lot of times. The result is reported as a long-time limit distribution from which one can see an approximation to the finding probability.

quant-ph↗

arXiv2016

A quantum walk on the half line with a particular initial state

Quantum walks are considered to be quantum counterparts of random walks.They show us impressive probability distributions which are different from those of random walks.That fact has been precisely proved in terms of mathematics and some of the results were reported as limit theorems.When we analyze quantum walks, some conventional methods are used for the computations.Especially, the Fourier analysis has played a role to do that.It is, however, compatible with some types of quantum walks (e.g. quantum walks on the line with a spatially homogeneous dynamics) and can not well work on the derivation of limit theorems for all the quantum walks.In this paper we try to obtain a limit theorem for a quantum walk on the half line by the usage of the Fourier analysis.Substituting a quantum walk on the line for it, we will lead to a possibility that the Fourier analysis is useful to compute a limit distribution of the quantum walk on the half line.

quant-ph↗

arXiv2015

A localized quantum walk with a gap in distribution

Quantum walks behave differently from what we expect and their probability distributions have unique structures. They have localization, singularities, a gap, and so on. Those features have been discovered from the view point of mathematics and reported as limit theorems. In this paper we focus on a time-dependent three-state quantum walk on the line and demonstrate a limit distribution. Three coin states at each position are iteratively updated by a coin-flip operator and a position-shift operator. As the result of the evolution, we end up to observe both localization and a gap in the limit distribution.

quant-ph↗

arXiv2015

Localization and limit laws of a three-state alternate quantum walk on a two-dimensional lattice

A two-dimensional discrete-time quantum walk (DTQW) can be realized by alternating a two-state DTQW in one spatial dimension followed by an evolution in the other dimension. This was shown to reproduce a probability distribution for a certain configuration of a four-state DTQW on a two-dimensional lattice. In this work we present a three-state alternate DTQW with a parameterized coin-flip operator and show that it can produce localization that is also observed for a certain other configuration of the four-state DTQW and non-reproducible using the two-state alternate DTQW. We will present two limit theorems for the three-state alternate DTQW. One of the limit theorems describes a long-time limit of a return probability, and the other presents a convergence in distribution for the position of the walker on a rescaled space by time. We will also outline the relevance of these walks in physical systems.

quant-ph↗

arXiv2015

A limit law of the return probability for a quantum walk on a hexagonal lattice

A return probability of random walks is one of the interesting subjects. As it is well known, the return probability strongly depends on the structure of the space where the random waker moves. On the other hand, the return probability of quantum walks, which are quantum models corresponding to random walks, has also been investigated to some extend lately. In this paper, we present a limit of the return probability for a discrete-time 3-state quantum walk on a hexagonal lattice.

quant-ph↗

arXiv2014

Trapping and spreading properties of quantum walk in homological structure

We attempt to extract a homological structure of two kinds of graphs by the Grover walk. The first one consists of a cycle and two semi-infinite lines and the second one is assembled by a periodic embedding of the cycles in $\mathbb{Z}$. We show that both of them have essentially the same eigenvalues induced by the existence of cycles in the infinite graphs. The eigenspace of the homological structure appears as so called {\it localization} in the Grover walks, in that the walk is partially trapped by the homological structure. On the other hand, the difference of the absolutely continuous part of spectrum between them provides different behaviors. We characterize the behaviors by the density functions in the weak convergence theorem: the first one is the delta measure at the bottom while the second one is expressed by two kinds of continuous functions which have different finite supports $(-1/\sqrt{10},1/\sqrt{10})$ and $(-2/7,2/7)$, respectively.

math-ph↗

arXiv2014

Turing instability in Reaction-Diffusion models on complex networks

In this paper, the Turing instability in reaction-diffusion models defined on complex networks is studied. Here, we focus on three types of models which generate complex networks, i.e. the Erd\H{o}s-R\'enyi, the Watts-Strogatz, and the threshold network models. From analysis of the Laplacian matrices of graphs generated by these models, we numerically reveal that stable and unstable regions of a homogeneous steady state on the parameter space of two diffusion coefficients completely differ, depending on the network architecture. In addition, we theoretically discuss the stable and unstable regions in the cases of regular enhanced ring lattices which include regular circles, and networks generated by the threshold network model when the number of vertices is large enough.

nlin.PS↗

arXiv2014

A limit theorem for a 3-period time-dependent quantum walk

We consider a discrete-time 2-state quantum walk on the line. The state of the quantum walker evolves according to a rule which is determined by a coin-flip operator and a position-shift operator. In this paper we take a 3-periodic time evolution as the rule. For such a quantum walk, we get a limit distribution which expresses the asymptotic behavior of the walker after a long time. The limit distribution is different from that of a time-independent quantum walk or a 2-period time-dependent quantum walk. We give some analytical results and then consider a number of variants of our model and indicate the result of simulations for these ones.

quant-ph↗

arXiv2014

Limit theorems of a 3-state quantum walk and its application for discrete uniform measures

We present two long-time limit theorems of a 3-state quantum walk on the line when the walker starts from the origin. One is a limit measure which is obtained from the probability distribution of the walk at a long-time limit, and the other is a convergence in distribution for the walker's position in a rescaled space by time. In addition, as an application of the walk, we obtain discrete uniform limit measures from the 3-state walk with a delocalized initial state.

quant-ph↗

arXiv2013

Self-avoiding quantum walks: realisations in subspaces and limit theorems

While completely self-avoiding quantum walks have the distinct property of leading to a trivial unidirectional transport of a quantum state, an interesting and non-trivial dynamics can be constructed by restricting the self-avoidance to a subspace of the complete Hilbert space. Here, we present a comprehensive study of three two-dimensional quantum walks, which are self-avoiding in coin space, in position space and in both, coin and position space. We discuss the properties of these walks and show that all result in delocalisation of the probability distribution for initial states which are strongly localised for a walk with a standard Grover coin operation. We also present analytical results for the evolution in the form of limit distributions for the self-avoiding walks in coin space and in both, coin and position space.

quant-ph↗

arXiv2013

Co-evolution of networks and quantum dynamics: a generalization of preferential attachment

We propose a model of network growth in which the network is co-evolving together with the dynamics of a quantum mechanical system, namely a quantum walk taking place over the network. The model naturally generalizes the Barab\'{a}si-Albert model of preferential attachment and has a rich set of tunable parameters, such as the initial conditions of the dynamics or the interaction of the system with its environment. We show that the model produces networks with two-modal power-law degree distributions, super-hubs, finite clustering coefficient, small-world behaviour and non-trivial degree-degree correlations.

physics.soc-ph↗