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 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 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 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 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 Efficient circuits for quantum walks

2010 ◽  
Vol 10 (5&6) ◽  
pp. 420-434
Author(s):  
C.-F. Chiang ◽  
D. Nagaj ◽  
P. Wocjan
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Probability Distributions ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Problem Size ◽  
Integrable Probability ◽  
The Difference ◽  
General Method

We present an efficient general method for realizing a quantum walk operator corresponding to an arbitrary sparse classical random walk. Our approach is based on Grover and Rudolph's method for preparing coherent versions of efficiently integrable probability distributions \cite{GroverRudolph}. This method is intended for use in quantum walk algorithms with polynomial speedups, whose complexity is usually measured in terms of how many times we have to apply a step of a quantum walk \cite{Szegedy}, compared to the number of necessary classical Markov chain steps. We consider a finer notion of complexity including the number of elementary gates it takes to implement each step of the quantum walk with some desired accuracy. The difference in complexity for various implementation approaches is that our method scales linearly in the sparsity parameter and poly-logarithmically with the inverse of the desired precision. The best previously known general methods either scale quadratically in the sparsity parameter, or polynomially in the inverse precision. Our approach is especially relevant for implementing quantum walks corresponding to classical random walks like those used in the classical algorithms for approximating permanents \cite{Vigoda, Vazirani} and sampling from binary contingency tables \cite{Stefankovi}. In those algorithms, the sparsity parameter grows with the problem size, while maintaining high precision is required.

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.

One-dimensional three-state quantum walks: Weak limits and localization

2016 ◽  
Vol 19 (04) ◽  
pp. 1650025 ◽  
Author(s):  
Chul Ki Ko ◽  
Etsuo Segawa ◽  
Hyun Jae Yoo
Keyword(s):  
Limit Distribution ◽  
Evolution Operator ◽  
Adjoint Operator ◽  
Quantum Walk ◽  
Vacuum Expectation ◽  
Weak Limit ◽  
Quantum Walks ◽  
Continuous Part ◽  
One Dimensional ◽  
Weak Limits

We investigate one-dimensional three-state quantum walks. We find a formula for the moments of the weak limit distribution via a vacuum expectation of powers of a self-adjoint operator. We use this formula to fully characterize the localization of three-state quantum walks in one dimension. The localization is also characterized by investing the eigenvectors of the evolution operator for the quantum walk. As a byproduct we clarify the concepts of localization differently used in the literature. We also study the continuous part of the limit distribution. For typical examples we show that the continuous part is the same kind as that of two-state quantum walks. We provide with explicit expressions for the density of the weak limits of some three-state quantum walks.

scholarly journals Multiple transitions between normal and hyperballistic diffusion in quantum walks with time-dependent jumps

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Marcelo A. Pires ◽  
Giuseppe Di Molfetta ◽  
Sílvio M. Duarte Queirós
Keyword(s):  
Probability Distribution ◽  
Uniform Distribution ◽  
Quantum Walk ◽  
Step Length ◽  
Quantum Walks ◽  
Time Dependent ◽  
Asymptotic Approach ◽  
Diffusive Regime ◽  
Dynamical Regimes ◽  
Functional Forms

AbstractWe extend to the gamut of functional forms of the probability distribution of the time-dependent step-length a previous model dubbed Elephant Quantum Walk, which considers a uniform distribution and yields hyperballistic dynamics where the variance grows cubicly with time, σ2 ∝ t3, and a Gaussian for the position of the walker. We investigate this proposal both locally and globally with the results showing that the time-dependent interplay between interference, memory and long-range hopping leads to multiple transitions between dynamical regimes, namely ballistic → diffusive → superdiffusive → ballistic → hyperballistic for non-hermitian coin whereas the first diffusive regime is quelled for implementations using the Hadamard coin. In addition, we observe a robust asymptotic approach to maximal coin-space entanglement.

scholarly journals Limit theorems for the interference terms of discrete-time quantum walks on the line

2013 ◽  
Vol 13 (7&8) ◽  
pp. 661-671
Author(s):  
Takuya Machida
Keyword(s):  
Density Matrix ◽  
Discrete Time ◽  
Limit Theorems ◽  
Probability Distributions ◽  
Quantum Walks ◽  
The Other ◽  
Density Matrices ◽  
Other Hand ◽  
Time Quantum

The probability distributions of discrete-time quantum walks have been often investigated, and many interesting properties of them have been discovered. The probability that the walker can be find at a position is defined by diagonal elements of the density matrix. On the other hand, although off-diagonal parts of the density matrices have an important role to quantify quantumness, they have not received attention in quantum walks. We focus on the off-diagonal parts of the density matrices for discrete-time quantum walks on the line and derive limit theorems for them.

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