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

2015 ◽  
pp. 406-418
Author(s):  
Takuya Machida
Keyword(s):  
Limit Theorems ◽  
Quantum Walk ◽  
Time Limit ◽  
The Other ◽  
Convergence In Distribution ◽  
Initial State ◽  
Uniform Limit ◽  
Limit Measure ◽  
Long Time ◽  
Discrete Uniform

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.

scholarly journals A Discontinuity of the Energy of Quantum Walk in Impurities

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1134
Author(s):  
Kenta Higuchi ◽  
Takashi Komatsu ◽  
Norio Konno ◽  
Hisashi Morioka ◽  
Etsuo Segawa
Keyword(s):  
Unit Circle ◽  
Stationary State ◽  
Discrete Time ◽  
Quantum Walk ◽  
Time Limit ◽  
The Other ◽  
Unitary Matrix ◽  
Initial State ◽  
Time Quantum ◽  
Long Time

We consider the discrete-time quantum walk whose local dynamics is denoted by a common unitary matrix C at the perturbed region {0,1,⋯,M−1} and free at the other positions. We obtain the stationary state with a bounded initial state. The initial state is set so that the perturbed region receives the inflow ωn at time n(|ω|=1). From this expression, we compute the scattering on the surface of −1 and M and also compute the quantity how quantum walker accumulates in the perturbed region; namely, the energy of the quantum walk, in the long time limit. The frequency of the initial state of the influence to the energy is symmetric on the unit circle in the complex plain. We find a discontinuity of the energy with respect to the frequency of the inflow.

Limit theorems of a two-phase quantum walk with one defect

2015 ◽  
Vol 15 (15&16) ◽  
pp. 1373-1396
Author(s):  
Shimpei Endo ◽  
Takako Endo ◽  
Norio Konno ◽  
Masato Takei ◽  
Etsuo Segawa
Keyword(s):  
Generating Function ◽  
Function Method ◽  
Limit Theorems ◽  
Quantum Walk ◽  
Stationary Measure ◽  
Evolution Operators ◽  
Initial State ◽  
Two Phase ◽  
Limit Measure ◽  
Starting Point

We treat a position dependent quantum walk (QW) on the line which we assign two different time-evolution operators to positive and negative parts respectively. We call the model “the two-phase QW” here, which has been expected to be a mathematical model of the topological insulator. We obtain the stationary and time-averaged limit measures related to localization for the two-phase QW with one defect. This is the first result on localization for the two-phase QW. The analytical methods are mainly based on the splitted generating function of the solution for the eigenvalue problem, and the generating function of the weight of the passages of the model. In this paper, we call the methods “the splitted generating function method” and “the generating function method”, respectively. The explicit expression of the stationary measure is asymmetric for the origin, and depends on the initial state and the choice of the parameters of the model. On the other hand, the time-averaged limit measure has a starting point symmetry and localization effect heavily depends on the initial state and the parameters of the model. Regardless of the strong effect of the initial state and the parameters, the time-averaged limit measure also suggests that localization can be always observed for our two-phase QW. Furthermore, our results imply that there is an interesting relation between the stationary and time-averaged limit measures when the parameters of the model have specific periodicities, which suggests that there is a possibility that we can analyze localization of the two-phase QW with one defect from the stationary measure.

scholarly journals Realization of the probability laws in the quantum central limit theorems by a quantum walk

2013 ◽  
Vol 13 (5&6) ◽  
pp. 430-438
Author(s):  
Takuya Machida
Keyword(s):  
Probability Theory ◽  
Limit Theorem ◽  
Central Limit ◽  
Discrete Time ◽  
Limit Theorems ◽  
Quantum Walk ◽  
Quantum Probability ◽  
Quantum Walks ◽  
Central Limit Theorems ◽  
Initial State

Since a limit distribution of a discrete-time quantum walk on the line was derived in 2002, a lot of limit theorems for quantum walks with a localized initial state have been reported. On the other hand, in quantum probability theory, there are four notions of independence (free, monotone, commuting, and boolean independence) and quantum central limit theorems associated to each independence have been investigated. The relation between quantum walks and quantum probability theory is still unknown. As random walks are fundamental models in the Kolmogorov probability theory, can the quantum walks play an important role in quantum probability theory? To discuss this problem, we focus on a discrete-time 2-state quantum walk with a non-localized initial state and present a limit theorem. By using our limit theorem, we generate probability laws in the quantum central limit theorems from the quantum walk.

scholarly journals Nonstandard limit theorems and large deviations for the Jacobi beta ensemble

2014 ◽  
Vol 03 (03) ◽  
pp. 1450012 ◽  
Author(s):  
Jan Nagel
Keyword(s):  
Weak Convergence ◽  
Spectral Measure ◽  
Limit Theorems ◽  
Large Deviation ◽  
The Other ◽  
Limit Measure ◽  
Jacobi Ensemble ◽  
Large Deviation Principles ◽  
Beta Ensemble ◽  
Rate Functions

In this paper, we show weak convergence of the empirical eigenvalue distribution and of the weighted spectral measure of the Jacobi ensemble, when one or both parameters grow faster than the dimension n. In these cases, the limit measure is given by the Marchenko–Pastur law and the semicircle law, respectively. For the weighted spectral measure, we also prove large deviation principles under this scaling, where the rate functions are those of the other classical ensembles.

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

2021 ◽  
Vol 21 (1&2) ◽  
pp. 0019-0036
Author(s):  
Takuya Machida
Keyword(s):  
Time Evolution ◽  
Limit Distribution ◽  
Quantum Walk ◽  
Time Limit ◽  
Unitary Matrix ◽  
Exact Form ◽  
Special Values ◽  
The Matrix ◽  
Long Time ◽  
Phase Term

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.

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

2015 ◽  
Vol 13 (07) ◽  
pp. 1550054 ◽  
Author(s):  
Takuya Machida
Keyword(s):  
Random Walks ◽  
Discrete Time ◽  
Hexagonal Lattice ◽  
Quantum Walk ◽  
Quantum Walks ◽  
The Other ◽  
View Point ◽  
Initial State ◽  
Random Walker ◽  
Return Probability

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 walker 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 take care of a discrete-time three-state quantum walk on a hexagonal lattice from the view point of mathematics. The mathematical result shows a limit of the return probability when the walker starts off at the origin. The result of the limit tells us about a possibility of localization at the position and a dependence of localization on the initial state.

scholarly journals Limit theorems for the discrete-time quantum walk on a graph with joined half lines

2012 ◽  
Vol 12 (3&4) ◽  
pp. 314-333
Author(s):  
Kota Chisaki ◽  
Norio Konno ◽  
Etsuo Segawa
Keyword(s):  
Asymptotic Behavior ◽  
Discrete Time ◽  
Limit Theorems ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Half Line ◽  
Initial State ◽  
Time Quantum ◽  
Weak Convergence Theorem ◽  
The One

We consider a discrete-time quantum walk W_{t,\kappa} at time t on a graph with joined half lines J_\kappa, which is composed of \kappa 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 symmetric initial states. In this paper, we introduce a quantum walk with an enlarged basis and show that W_{t,\kappa} can be reduced to the walk on a half line even if the initial state is asymmetric. For W_{t,\kappa}, we obtain two types of limit theorems. The first one is an asymptotic behavior of W_{t,\kappa} 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,\kappa}. On each half line, W_{t,\kappa} 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.

scholarly journals A QUANTUM WALK WITH A DELOCALIZED INITIAL STATE: CONTRIBUTION FROM A COIN-FLIP OPERATOR

2013 ◽  
Vol 11 (05) ◽  
pp. 1350053 ◽  
Author(s):  
TAKUYA MACHIDA
Keyword(s):  
Shift Operator ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Fourier Series Expansion ◽  
Initial Condition ◽  
Initial State ◽  
Limit Distributions ◽  
Time Quantum ◽  
Long Time ◽  
Position Shift

A unit evolution step of discrete-time quantum walks (QWs) is determined by both a coin-flip operator and a position-shift operator. The behavior of quantum walkers after many steps delicately depends on the coin-flip operator and an initial condition of the walk. To get the behavior, a lot of long-time limit distributions for the QWs starting with a localized initial state have been derived. In this paper, we compute limit distributions of a 2-state QW with a delocalized initial state, not a localized initial state, and discuss how the walker depends on the coin-flip operator. The initial state induced from the Fourier series expansion, which is called the (α, β) delocalized initial state in this paper, provides different limit density functions from the ones of the quantum walk with a localized initial state.

scholarly journals LIMIT THEOREMS FOR QUANTUM WALKS DRIVEN BY MANY COINS

2008 ◽  
Vol 06 (06) ◽  
pp. 1231-1243 ◽  
Author(s):  
ETSUO SEGAWA ◽  
NORIO KONNO
Keyword(s):  
Limit Theorems ◽  
Quantum Walks ◽  
Time Limit ◽  
Time Step ◽  
Rigorous Results ◽  
Long Time ◽  
Classical Behavior

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.

Leading with Pedagogical Tact- a Challenge in Children's Sports in Sweden

2010 ◽  
Vol 19 (1-2) ◽  
pp. 127-147 ◽  
Author(s):  
Krister Hertting
Keyword(s):  
Daily Activity ◽  
The Other ◽  
Competitive Sport ◽  
Sports Clubs ◽  
Youth Politics ◽  
Pedagogical Tact ◽  
Children And Parents ◽  
Swedish Society ◽  
Long Time ◽  
Voluntary Commitment

Leading with Pedagogical Tact- a Challenge in Children's Sports in Sweden The purpose of this article is to elucidate and problemize meetings between children and leaders in children's sport. The competitive sport is high valuated in the Swedish society and sport for children is central in the Swedish youth politics. The foundation in Swedish sport, as well as in the other Nordic countries, has for a long time relied on voluntary commitment. Approximately 650 000 people are voluntary engaged as leaders in sport in Sweden and 70% of children between 7 and 14 years compete in sports clubs. There is, however, a tension in the Swedish sport system. The sports for children has double missions - ‘association nurturing’ and ‘competition nurturing’, missions which are not always in harmony. In the daily activity it is the voluntary leaders who have to deal with these missions, which creates a field of tension. In this article I argue for a bridge between these missions by a leadership based on pedagogical tact. The empirical outlook is a narrative based on statements from leaders, children and parents in a study dealing with voluntary leadership within children's football. In the end I argue that focusing on this bridge is a win-win situation, both for children and sports.

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