scholarly journals Weak Limit Theorem of a Two-phase Quantum Walk with One Defect

2016 ◽  
Vol 22 (1) ◽  
pp. 17-29 ◽  
Author(s):  
Shimpei ENDO ◽  
Takako ENDO ◽  
Norio KONNO ◽  
Etsuo SEGAWA ◽  
Masato TAKEI
Keyword(s):  
Limit Theorem ◽  
Quantum Walk ◽  
Weak Limit ◽  
Two Phase ◽  
Weak Limit Theorem

scholarly journals Weak limit theorem for a nonlinear quantum walk

2018 ◽  
Vol 17 (9) ◽  
Author(s):  
Masaya Maeda ◽  
Hironobu Sasaki ◽  
Etsuo Segawa ◽  
Akito Suzuki ◽  
Kanako Suzuki
Keyword(s):  
Limit Theorem ◽  
Quantum Walk ◽  
Weak Limit ◽  
Nonlinear Quantum ◽  
Weak Limit Theorem

scholarly journals WEAK LIMITS FOR QUANTUM WALKS ON THE HALF-LINE

2013 ◽  
Vol 11 (06) ◽  
pp. 1350054 ◽  
Author(s):  
CHAOBIN LIU ◽  
NELSON PETULANTE
Keyword(s):  
Limit Theorem ◽  
General Condition ◽  
Transition Probabilities ◽  
Quantum Walk ◽  
Weak Limit ◽  
Quantum Walks ◽  
Half Line ◽  
Weak Limits ◽  
Terminal Behavior ◽  
Weak Limit Theorem

For a discrete two-state quantum walk (QW) on the half-line with a general condition at the boundary, we formulate and prove a weak limit theorem describing the terminal behavior of its transition probabilities. In this context, localization is possible even for a walk predicated on the assumption of homogeneity. For the Hadamard walk on the half-line, the weak limit is shown to be independent of the initial coin state and to exhibit no localization.

scholarly journals A Weak Limit Theorem for Generalized Jiřina Processes

2009 ◽  
Vol 46 (2) ◽  
pp. 453-462 ◽  
Author(s):  
Yuqiang Li
Keyword(s):  
Limit Theorem ◽  
Diffusion Process ◽  
Nonlinear Diffusion ◽  
Weak Limit ◽  
Weak Limit Theorem ◽  
Lévy Jumps

In this paper we prove that a sequence of scaled generalized Jiřina processes can converge weakly to a nonlinear diffusion process with Lévy jumps under certain conditions.

scholarly journals A Weak Limit Theorem for Generalized Jiřina Processes

2009 ◽  
Vol 46 (02) ◽  
pp. 453-462 ◽  
Author(s):  
Yuqiang Li
Keyword(s):  
Limit Theorem ◽  
Diffusion Process ◽  
Nonlinear Diffusion ◽  
Weak Limit ◽  
Weak Limit Theorem ◽  
Lévy Jumps

In this paper we prove that a sequence of scaled generalized Jiřina processes can converge weakly to a nonlinear diffusion process with Lévy jumps under certain conditions.

scholarly journals AN ALGEBRAIC STRUCTURE FOR ONE-DIMENSIONAL QUANTUM WALKS AND A NEW PROOF OF THE WEAK LIMIT THEOREM

2013 ◽  
Vol 16 (02) ◽  
pp. 1350018
Author(s):  
TATSUYA TATE
Keyword(s):  
Limit Theorem ◽  
Characteristic Function ◽  
Algebraic Structure ◽  
Chebyshev Polynomials ◽  
Transition Probability ◽  
Weak Limit ◽  
Quantum Walks ◽  
One Dimensional ◽  
Natural Computation ◽  
Weak Limit Theorem

An algebraic structure for one-dimensional quantum walks is introduced. This structure characterizes, in some sense, one-dimensional quantum walks. A natural computation using this algebraic structure leads us to obtain an effective formula for the characteristic function of the transition probability. Then, the weak limit theorem for the transition probability of quantum walks is deduced by using simple properties of the Chebyshev polynomials.

scholarly journals A new type of quantum walks based on decomposing quantum states

2021 ◽  
Vol 21 (7&8) ◽  
pp. 541-556
Author(s):  
Chusei Kiumi
Keyword(s):  
Quantum Walk ◽  
Weak Limit ◽  
Quantum Walks ◽  
Quantum States ◽  
The Other ◽  
Imaginary Component ◽  
Probability Measures ◽  
Real Component ◽  
New Type ◽  
Weak Limit Theorem

In this paper, the 2-state decomposed-type quantum walk (DQW) on a line is introduced as an extension of the 2-state quantum walk (QW). The time evolution of the DQW is defined with two different matrices, one is assigned to a real component, and the other is assigned to an imaginary component of the quantum state. Unlike the ordinary 2-state QWs, localization and the spreading phenomenon can coincide in DQWs. Additionally, a DQW can always be converted to the corresponding 4-state QW with identical probability measures. In other words, a class of 4-state QWs can be realized by DQWs with 2 states. In this work, we reveal that there is a 2-state DQW corresponding to the 4-state Grover walk. Then, we derive the weak limit theorem of the class of DQWs corresponding to 4-state QWs which can be regarded as the generalized Grover walks.

scholarly journals Limit theorems for quantum walks with memory

2010 ◽  
Vol 10 (11&12) ◽  
pp. 1004-1017
Author(s):  
Norio Konno ◽  
Takuya Machida
Keyword(s):  
Limit Theorems ◽  
Quantum Walk ◽  
Weak Limit ◽  
Quantum Walks ◽  
One Dimension ◽  
Counting Method ◽  
Return Probability ◽  
Weak Limit Theorem ◽  
Path Counting ◽  
With Memory

Recently Mc Gettrick introduced and studied a discrete-time 2-state quantum walk (QW) with a memory in one dimension. He gave an expression for the amplitude of the QW by path counting method. Moreover he showed that the return probability of the walk is more than 1/2 for any even time. In this paper, we compute the stationary distribution by considering the walk as a 4-state QW without memory. Our result is consistent with his claim. In addition, we obtain the weak limit theorem of the rescaled QW. This behavior is strikingly different from the corresponding classical random walk and the usual 2-state QW without memory as his numerical simulations suggested.

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