Limit Theorems and Absorption Problems for One-Dimensional Correlated Random Walks

Konno, Norio

doi:10.1080/15326340802640941

Cited by 40 publications

(35 citation statements)

References 54 publications

(112 reference statements)

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“…This is defined as a quantum-mechanical analogue to the FTD stochastic coin of the correlated RW defined [42]. According to Ref.…”

Section: Final-time Dependent Discrete Time Quantum Walkmentioning

confidence: 99%

From Discrete Time Quantum Walk to Continuous Time Quantum Walk in Limit Distribution

Shikano¹

2013

Jnl of Comp & Theo Nano

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The discrete time quantum walk defined as a quantum-mechanical analogue of the discrete time random walk have recently been attracted from various and interdisciplinary fields. In this review, the weak limit theorem, that is, the asymptotic behavior, of the one-dimensional discrete time quantum walk is analytically shown. From the limit distribution of the discrete time quantum walk, the discrete time quantum walk can be taken as the quantum dynamical simulator of some physical systems.

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“…This is defined as a quantum-mechanical analogue to the FTD stochastic coin of the correlated RW defined [42]. According to Ref.…”

Section: Final-time Dependent Discrete Time Quantum Walkmentioning

confidence: 99%

From Discrete Time Quantum Walk to Continuous Time Quantum Walk in Limit Distribution

Shikano¹

2013

Jnl of Comp & Theo Nano

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“…According to Eqs. (12) and (14), we can get the walk matrix U . The size of both of shift matrix S and the walk matrix U are 4N × 4N .…”

Section: Numerical Simulationmentioning

confidence: 99%

Discrete-time quantum walk on the Cayley graph of the dihedral group

Dai

Yuan

2018

Quantum Inf Process

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The finite dihedral group generated by one rotation and one flip is the simplest case of the non-abelian group. Cayley graphs are diagrammatic counterparts of groups. In this paper, much attention is given to the Cayley graph of the dihedral group. Considering the characteristics of the elements in the dihedral group, we conduct the model of discrete-time quantum walk on the Cayley graph of the dihedral group by special coding mode. This construction makes Fourier transformation can be used to carry out spectral analysis of the dihedral quantum walk, i.e. the non-abelian case. Furthermore, the relation between quantum walk without memory on the Cayley graph of the dihedral group and quantum walk with memory on a cycle is discussed, so that we can explore the potential of quantum walks without and with memory. Here, the numerical simulation is carried out to verify the theoretical analysis results and other properties of the proposed model are further studied. † Electronic address:

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“…[1][2][3][4] It is often shown that a useful quantum search algorithm can be designed based on the quantum walk. [5][6][7] Moreover, the result on a quantum walk on Z with an absorbing wall 8 has been applied to solve the transport problems in solid-state physics of strongly correlated electron systems in Ref.…”

Section: Introductionmentioning

confidence: 99%

“…The M -CQW is a quantum generalization of the random walk depending on the previous M -step memory. 4,19 The quantum coin describing the one step dynamics can be obtained by replacing the nonzero entries of an adjacency matrix of the de-Bruijn digraph to some nonzero appropriate values. See Refs.…”

Section: Introductionmentioning

confidence: 99%

Limit Theorems for Quantum Walks Driven by Many Coins

Segawa

Konno

2008

Int. J. Quantum Inform.

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We obtain some rigorous results on limit theorems for quantum walks driven by many coins introduced by Brun et al. in the long time limit. The results imply that whether the behavior of a particle is quantum or classical depends on the three factors: the initial qubit, the number of coins M , d = [t/M ], where t is time step. Our main theorem shows that we can see a transition from classical behavior to quantum one for a class of three factors.

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Limit Theorems and Absorption Problems for One-Dimensional Correlated Random Walks

Cited by 40 publications

References 54 publications

From Discrete Time Quantum Walk to Continuous Time Quantum Walk in Limit Distribution

From Discrete Time Quantum Walk to Continuous Time Quantum Walk in Limit Distribution

Discrete-time quantum walk on the Cayley graph of the dihedral group

Limit Theorems for Quantum Walks Driven by Many Coins

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