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

2015 ◽  
Vol 15 (1&2) ◽  
pp. 50-60
Author(s):  
F. Alberto Grunbaum ◽  
Takuya Machida
Keyword(s):  
Asymptotic Behavior ◽  
Limit Theorem ◽  
Time Evolution ◽  
Discrete Time ◽  
Limit Distribution ◽  
Shift Operator ◽  
Quantum Walk ◽  
Time Dependent ◽  
Long Time ◽  
Position Shift

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.

scholarly journals A limit theorem for a splitting distribution of a quantum walk

2018 ◽  
Vol 16 (03) ◽  
pp. 1850023
Author(s):  
Takuya Machida
Keyword(s):  
Limit Theorem ◽  
Probability Distribution ◽  
Random Walks ◽  
Discrete Time ◽  
Limit Distribution ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Unitary Evolution ◽  
Time Quantum ◽  
Long Time

Discrete-time quantum walks are considered a counterpart of random walks and their study 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.

scholarly journals A localized quantum walk with a gap in distribution

2016 ◽  
Vol 16 (5&6) ◽  
pp. 516-529
Author(s):  
Takuya Machida
Keyword(s):  
Limit Distribution ◽  
Limit Theorems ◽  
Shift Operator ◽  
Probability Distributions ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Time Dependent ◽  
View Point ◽  
Position Shift

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.

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 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 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.

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.

Central limit theorem for the solution to the heat equation with moving time

2016 ◽  
Vol 19 (01) ◽  
pp. 1650005 ◽  
Author(s):  
Junfeng Liu ◽  
Ciprian A. Tudor
Keyword(s):  
Asymptotic Behavior ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Heat Equation ◽  
White Noise ◽  
Limit Distribution ◽  
Malliavin Calculus ◽  
Time Variable ◽  
Time Space ◽  
Quadratic Variations

We consider the solution to the stochastic heat equation driven by the time-space white noise and study the asymptotic behavior of its spatial quadratic variations with “moving time”, meaning that the time variable is not fixed and its values are allowed to be very big or very small. We investigate the limit distribution of these variations via Malliavin calculus.

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