Quantum walks on directed graphs

2007 ◽  
Vol 7 (1&2) ◽  
pp. 93-102
Author(s):  
A. Montanaro
Keyword(s):  
Directed Graph ◽  
Discrete Time ◽  
Directed Graphs ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Time Quantum ◽  
Definition Of ◽  
Necessary And Sufficient

We consider the definition of quantum walks on directed graphs. Call a directed graph reversible if, for each pair of vertices $(v_i, v_j)$, if $v_i$ is connected to $v_j$ then there is a path from $v_j$ to $v_i$. We show that reversibility is a necessary and sufficient condition for a directed graph to allow the notion of a discrete-time quantum walk, and discuss some implications of this condition. We present a method for defining a "partially quantum'' walk on directed graphs that are not reversible.

scholarly journals Synchronization of a Generalized High-dimensional Kuramoto Model With Directed Graphs

2021 ◽  
Author(s):  
Jinxing Zhang ◽  
Jiandong Zhu ◽  
Xiaodi Li
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Kuramoto Model ◽  
High Dimensional ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Initial State ◽  
Necessary And Sufficient ◽  
Theoretical Results

Abstract In this paper, a generalized high-dimensional Kuramoto model with directed graphs is investigated. A necessary and sufficient condition for equilibria is given and the synchronization is proved under weaker directed graph conditions and more general initial state constrains. Finally, an example is given to validate the theoretical results.

On the Divisibility of Homogeneous Directed Graphs

1993 ◽  
Vol 45 (2) ◽  
pp. 284-294 ◽  
Author(s):  
M. El-Zahar ◽  
N. W. Sauer
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Isomorphic Copy ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finite Set ◽  
Necessary And Sufficient

AbstractLet be a finite set of finite tournaments. We will give a necessary and sufficient condition for the -free homogeneous directed graph to be divisible. That is, that there is a partition of into two classes such that neither of them contains an isomorphic copy of .

scholarly journals A generalized inversion formula for the continuous Jacobi transform

1987 ◽  
Vol 10 (4) ◽  
pp. 671-692 ◽  
Author(s):  
Ahmed I. Zayed
Keyword(s):  
Analytic Function ◽  
Inversion Formula ◽  
Generalized Functions ◽  
Generalized Function ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Generalized Inversion ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Jacobi Transform

In this paper we extend the definition of the continuous Jacobi transform to a class of generalized functions and obtain a generalized inversion formula for it. As a by-product of our technique we obtain a necessary and sufficient condition for an analytic functionF(λ)inReλ>0to be the continuous Jacobi transform of a generalized function.

On the Indispensability of Intentionality

1972 ◽  
Vol 2 (1) ◽  
pp. 127-133
Author(s):  
Harold Morick
Keyword(s):  
Recent Literature ◽  
Essential Property ◽  
Sufficient Condition ◽  
Conceptual Scheme ◽  
Necessary And Sufficient Condition ◽  
Definition Of ◽  
Necessary And Sufficient

In the last two decades, there has been a great deal of interest in providing an intentional criterion of the psychological. Of the various ones proferred, it seems to me that the best was the earliest, which was Chisholm’s initial criterion in his 1955 essay “Sentences about Believing.” In this present paper I first single out a basic misconception pervading the recent literature on intentionality and suggest that a consequence of this misconception has been the futile attempt to use the notion of intentionality to provide a kind of definition of “mind”; that is, to use intentionality to provide a necessary and sufficient condition for the psychological. Secondly, I point out how intentionality as captured by my own criterion is indispensable in that it is an essential property of certain particulars (persons) which are basic to our conceptual scheme and apparently basic to any conceptual scheme whatsoever.

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.

The fog on: Generalized teleportation by means of discrete-time quantum walks on N-lines and N-cycles

2019 ◽  
Vol 33 (23) ◽  
pp. 1950270 ◽  
Author(s):  
Duc Manh Nguyen ◽  
Sunghwan Kim
Keyword(s):  
Discrete Time ◽  
Quantum Teleportation ◽  
Quantum Walk ◽  
Quantum Walks ◽  
One Dimensional ◽  
Time Quantum ◽  
Infinite Line ◽  
The One

The recent paper entitled “Generalized teleportation by means of discrete-time quantum walks on [Formula: see text]-lines and [Formula: see text]-cycles” by Yang et al. [Mod. Phys. Lett. B 33(6) (2019) 1950069] proposed the quantum teleportation by means of discrete-time quantum walks on [Formula: see text]-lines and [Formula: see text]-cycles. However, further investigation shows that the quantum walk over the one-dimensional infinite line can be based over the [Formula: see text]-cycles and cannot be based on [Formula: see text]-lines. The proofs of our claims on quantum walks based on finite lines are also provided in detail.

A dynamical characterization of diagonal-preserving -isomorphisms of graph -algebras

2017 ◽  
Vol 38 (7) ◽  
pp. 2401-2421 ◽  
Author(s):  
SARA E. ARKLINT ◽  
SØREN EILERS ◽  
EFREN RUIZ
Keyword(s):  
Directed Graphs ◽  
Sufficient Condition ◽  
Orbit Equivalence ◽  
Necessary And Sufficient Condition ◽  
Graph Algebras ◽  
Boundary Path ◽  
Necessary And Sufficient

We characterize when there exists a diagonal-preserving $\ast$-isomorphism between two graph $C^{\ast }$-algebras in terms of the dynamics of the boundary path spaces. In particular, we refine the notion of ‘orbit equivalence’ between the boundary path spaces of the directed graphs $E$ and $F$ and show that this is a necessary and sufficient condition for the existence of a diagonal-preserving $\ast$-isomorphism between the graph $C^{\ast }$-algebras $C^{\ast }(E)$ and $C^{\ast }(F)$.

scholarly journals Convergence Theorems for Operators Sequences on Functionals of Discrete-Time Normal Martingales

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Jinshu Chen
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Discrete Time ◽  
Existence And Uniqueness ◽  
Continuous Dependence ◽  
Functional Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Convergence Of Operators

We aim to investigate the convergence of operators sequences acting on functionals of discrete-time normal martingales M. We first apply the 2D-Fock transform for operators from the testing functional space S(M) to the generalized functional space S⁎(M) and obtain a necessary and sufficient condition for such operators sequences to be strongly convergent. We then discuss the integration of these operator-valued functions. Finally, we apply the results obtained here and establish the existence and uniqueness of solution to quantum stochastic differential equations in terms of operators acting on functionals of discrete-time normal martingales M. And also we prove the continuity and continuous dependence on initial values of the solution.

scholarly journals Controllability of Linear Discrete-Time Systems with Both Delayed States and Delayed Inputs

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Hong Shi ◽  
Guangming Xie ◽  
Wenguang Luo
Keyword(s):  
Linear Systems ◽  
Discrete Time ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Control Delay ◽  
Discrete Time Systems ◽  
Necessary And Sufficient ◽  
And Control ◽  
Time Systems ◽  
Time Linear

The controllability issues for discrete-time linear systems with delay in state and control are addressed. By introducing a new concept, the controllability realization index (CRI), the characteristic of controllability is revealed. An easily testable necessary and sufficient condition for the controllability of discrete-time linear systems with state and control delay is established.

scholarly journals Sifat-Sifat Representasi Indekomposabel The Properties of Indecomposable Representations

2019 ◽  
Vol 8 (3) ◽  
Author(s):  
Vika Yugi Kurniawan
Keyword(s):  
Vector Space ◽  
Directed Graph ◽  
Linear Mapping ◽  
Sufficient Condition ◽  
Paper Study ◽  
Necessary And Sufficient Condition ◽  
Direct Sum ◽  
Indecomposable Representations ◽  
Finite Set ◽  
Necessary And Sufficient

A directed graph is also called as a quiver  where  is a finite set of vertices,  is a set of arrows, and  are two maps from  to . A representation  of a quiver  is an assignment of a vector space  to each vertex  of  and a linear mapping  to each arrow.  We denote by  the direct sum of representasions  and  of a quiver  . A representation  is called indecomposable if  is not ishomorphic to a direct sum of non-zero representations. This paper study about the properties of indecomposable representations. These properties will be used to investigate the necessary and sufficient condition of indecomposable representations.

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