scholarly journals A New Limit Theorem for Quantum Walk in Terms of Quantum Bernoulli Noises

Entropy ◽  
2020 ◽  
Vol 22 (4) ◽  
pp. 486
Author(s):  
Caishi Wang ◽  
Suling Ren ◽  
Yuling Tang
Keyword(s):  
Limit Theorem ◽  
Probability Distribution ◽  
Probability Distributions ◽  
Quantum Walk ◽  
Initial State ◽  
Wide Range ◽  
Gauss Type ◽  
Limit Probability

In this paper, we consider limit probability distributions of the quantum walk recently introduced by Wang and Ye (C.S. Wang and X.J. Ye, Quantum walk in terms of quantum Bernoulli noises, Quantum Inf. Process. 15 (2016), no. 5, 1897–1908). We first establish several technical theorems, which themselves are also interesting. Then, by using these theorems, we prove that, for a wide range of choices of the initial state, the above-mentioned quantum walk has a limit probability distribution of standard Gauss type, which actually gives a new limit theorem for the walk.

Higher-dimensional open quantum walk in environment of quantum Bernoulli noises

2021 ◽  
pp. 2250001
Author(s):  
Ce Wang
Keyword(s):  
Quantum Walk ◽  
Quantum Walks ◽  
Initial State ◽  
Quantum Random Walks ◽  
Open Quantum ◽  
Higher Dimensional ◽  
Gauss Type ◽  
Open Quantum Random Walks ◽  
Limit Probability ◽  
Creation Operators

Open quantum walks (OQWs) (also known as open quantum random walks) are quantum analogs of classical Markov chains in probability theory, and have potential application in quantum information and quantum computation. Quantum Bernoulli noises (QBNs) are annihilation and creation operators acting on Bernoulli functionals, and can be used as the environment of an open quantum system. In this paper, by using QBNs as the environment, we introduce an OQW on a general higher-dimensional integer lattice. We obtain a quantum channel representation of the walk, which shows that the walk is indeed an OQW. We prove that all the states of the walk are separable provided its initial state is separable. We also prove that, for some initial states, the walk has a limit probability distribution of higher-dimensional Gauss type. Finally, we show links between the walk and a unitary quantum walk recently introduced in terms of QBNs.

scholarly journals Higher-Dimensional Quantum Walk in Terms of Quantum Bernoulli Noises

Entropy ◽  
2020 ◽  
Vol 22 (5) ◽  
pp. 504
Author(s):  
Ce Wang ◽  
Caishi Wang
Keyword(s):  
Discrete Time ◽  
Quantum Information Processing ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Integer Lattice ◽  
Initial State ◽  
Higher Dimensional ◽  
Time Quantum ◽  
Gauss Type ◽  
Limit Probability

As a discrete-time quantum walk model on the one-dimensional integer lattice Z , the quantum walk recently constructed by Wang and Ye [Caishi Wang and Xiaojuan Ye, Quantum walk in terms of quantum Bernoulli noises, Quantum Information Processing 15 (2016), 1897–1908] exhibits quite different features. In this paper, we extend this walk to a higher dimensional case. More precisely, for a general positive integer d ≥ 2 , by using quantum Bernoulli noises we introduce a model of discrete-time quantum walk on the d-dimensional integer lattice Z d , which we call the d-dimensional QBN walk. The d-dimensional QBN walk shares the same coin space with the quantum walk constructed by Wang and Ye, although it is a higher dimensional extension of the latter. Moreover we prove that, for a range of choices of its initial state, the d-dimensional QBN walk has a limit probability distribution of d-dimensional standard Gauss type, which is in sharp contrast with the case of the usual higher dimensional quantum walks. Some other results are also obtained.

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 A Theory Of Errors in Quantum Measurement

2003 ◽  
Vol 18 (33n35) ◽  
pp. 2439-2450
Author(s):  
S. G. Rajeev
Keyword(s):  
Quantum Theory ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Probability Distribution ◽  
Quantum Measurement ◽  
Probability Distributions ◽  
Matrix Elements ◽  
Random Errors ◽  
The Central Limit Theorem ◽  
The Matrix

It is common to model the random errors in a classical measurement by the normal (Gaussian) distribution, because of the central limit theorem. In the quantum theory, the analogous hypothesis is that the matrix elements of the error in an observable are distributed normally. We obtain the probability distribution this implies for the outcome of a measurement, exactly for the case of traceless 2 × 2 matrices and in the steepest descent approximation in general. Due to the phenomenon of 'level repulsion', the probability distributions obtained are quite different from the Gaussian.

scholarly journals Quantum Walk in Terms of Quantum Bernoulli Noise and Quantum Central Limit Theorem for Quantum Bernoulli Noise

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Caishi Wang ◽  
Ce Wang ◽  
Yuling Tang ◽  
Suling Ren
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Degrees Of Freedom ◽  
Probability Distributions ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Quantum Central Limit Theorem ◽  
Creation Operators ◽  
Internal Degrees Of Freedom

As a unitary quantum walk with infinitely many internal degrees of freedom, the quantum walk in terms of quantum Bernoulli noise (recently introduced by Wang and Ye) shows a rather classical asymptotic behavior, which is quite different from the case of the usual quantum walks with a finite number of internal degrees of freedom. In this paper, we further examine the structure of the walk. By using the Fourier transform on the state space of the walk, we obtain a formula that links the moments of the walk’s probability distributions directly with annihilation and creation operators on Bernoulli functionals. We also prove some other results on the structure of the walk. Finally, as an application of these results, we establish a quantum central limit theorem for the annihilation and creation operators themselves.

scholarly journals Sharpness of the Satisfiability Threshold for Non-Uniform Random k-SAT

2019 ◽  
Author(s):  
Tobias Friedrich ◽  
Ralf Rothenberger
Keyword(s):  
Probability Distribution ◽  
Power Law ◽  
General Model ◽  
Probability Distributions ◽  
Step Function ◽  
Propositional Satisfiability ◽  
Wide Range ◽  
Random Instances ◽  
Variable Probability

We study a more general model to generate random instances of Propositional Satisfiability (SAT) with n Boolean variables, m clauses, and exactly k variables per clause. Additionally, our model is given an arbitrary probability distribution (p_1, ..., p_n) on the variable occurrences. Therefore, we call it non-uniform random k-SAT. The number m of randomly drawn clauses at which random formulas go from asymptotically almost surely (a.a.s.) satisfiable to a.a.s. unsatisfiable is called the satisfiability threshold. Such a threshold is called sharp if it approaches a step function as n increases. We identify conditions on the variable probability distribution (p_1, ..., p_n) under which the satisfiability threshold is sharp if its position is already known asymptotically. This result generalizes Friedgut’s sharpness result from uniform to non-uniform random k -SAT and implies sharpness for thresholds of a wide range of random k -SAT models with heterogeneous probability distributions, for example such models where the variable probabilities follow a power-law.

scholarly journals Towards the Dependence on Parameters for the Solution of the Thermostatted Kinetic Framework

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 59
Author(s):  
Bruno Carbonaro ◽  
Marco Menale
Keyword(s):  
Complex System ◽  
Continuous Dependence ◽  
Social Role ◽  
Probability Distributions ◽  
Functional Dependence ◽  
Transition Probability ◽  
Classical Mechanics ◽  
Human Society ◽  
Wide Range ◽  
Probability Densities

A complex system is a system involving particles whose pairwise interactions cannot be composed in the same way as in classical Mechanics, i.e., the result of interaction of each particle with all the remaining ones cannot be expressed as a sum of its interactions with each of them (we cannot even know the functional dependence of the total interaction on the single interactions). Moreover, in view of the wide range of its applications to biologic, social, and economic problems, the variables describing the state of the system (i.e., the states of all of its particles) are not always (only) the usual mechanical variables (position and velocity), but (also) many additional variables describing e.g., health, wealth, social condition, social rôle ⋯, and so on. Thus, in order to achieve a mathematical description of the problems of everyday’s life of any human society, either at a microscopic or at a macroscpoic scale, a new mathematical theory (or, more precisely, a scheme of mathematical models), called KTAP, has been devised, which provides an equation which is a generalized version of the Boltzmann equation, to describe in terms of probability distributions the evolution of a non-mechanical complex system. In connection with applications, the classical problems about existence, uniqueness, continuous dependence, and stability of its solutions turn out to be particularly relevant. As far as we are aware, however, the problem of continuous dependence and stability of solutions with respect to perturbations of the parameters expressing the interaction rates of particles and the transition probability densities (see Section The Basic Equations has not been tackled yet). Accordingly, the present paper aims to give some initial results concerning these two basic problems. In particular, Theorem 2 reveals to be stable with respect to small perturbations of parameters, and, as far as instability of solutions with respect to perturbations of parameters is concerned, Theorem 3 shows that solutions are unstable with respect to “large” perturbations of interaction rates; these hints are illustrated by numerical simulations that point out how much solutions corresponding to different values of parameters stay away from each other as t→+∞.

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 Probability-Based Prediction of Activity in Multiple Arm Muscles: Implications for Functional Electrical Stimulation

2008 ◽  
Vol 100 (1) ◽  
pp. 482-494 ◽  
Author(s):  
Chad V. Anderson ◽  
Andrew J. Fuglevand
Keyword(s):  
Electrical Stimulation ◽  
Functional Electrical Stimulation ◽  
Muscle Activity ◽  
Probability Distributions ◽  
Probabilistic Methods ◽  
Muscle Stimulation ◽  
Implanted Electrodes ◽  
Motor Behaviors ◽  
Wide Range ◽  
Arm Muscles

Functional electrical stimulation (FES) involves artificial activation of muscles with implanted electrodes to restore motor function in paralyzed individuals. The range of motor behaviors that can be generated by FES, however, is limited to a small set of preprogrammed movements such as hand grasp and release. A broader range of movements has not been implemented because of the substantial difficulty associated with identifying the patterns of muscle stimulation needed to elicit specified movements. To overcome this limitation in controlling FES systems, we used probabilistic methods to estimate the levels of muscle activity in the human arm during a wide range of free movements based on kinematic information of the upper limb. Conditional probability distributions were generated based on hand kinematics and associated surface electromyographic (EMG) signals from 12 arm muscles recorded during a training task involving random movements of the arm in one subject. These distributions were then used to predict in four other subjects the patterns of muscle activity associated with eight different movement tasks. On average, about 40% of the variance in the actual EMG signals could be accounted for in the predicted EMG signals. These results suggest that probabilistic methods ultimately might be used to predict the patterns of muscle stimulation needed to produce a wide array of desired movements in paralyzed individuals with FES.

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