Continuous-Time Classical and Quantum Random Walk on Direct Product of Cayley Graphs

Salimi, S.; Jafarizadeh, M. A.

doi:10.1088/0253-6102/51/6/08

Cited by 13 publications

(15 citation statements)

References 21 publications

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“…The quantum probabilistic aspect of the QW's has been discussed in a separate paper [10]. We remark that there already have been studies of continuous time QW's on the graphs [6,13,16,19,21], but we emphasize that the extension here is different from those. It is a natural extension of the discrete time QW on integer lattices in the sense that it agrees with the original discrete time QW for integer times.…”

Section: Introductionmentioning

confidence: 94%

The generator and quantum Markov semigroup for quantum walks

Yoo²

2013

Kodai Math. J.

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The quantum walks in the lattice spaces are represented as unitary evolutions. We find a generator for the evolution and apply it to further understand the walks.We first extend the discrete time quantum walks to continuous time walks. Then we construct the quantum Markov semigroup for quantum walks and characterize it in an invariant subalgebra. In the meanwhile, we obtain the limit distributions of the quantum walks in one-dimension with a proper scaling, which was obtained by Konno by a different method.

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Section: Introductionmentioning

confidence: 94%

The generator and quantum Markov semigroup for quantum walks

Yoo²

2013

Kodai Math. J.

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“…Then we multiply the above equation from the left side in e T κ and use the equations (4-21) and (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22) to prove that…”

Section: Calculating Bipartite Entanglement In Stratificatin Basis Ofmentioning

confidence: 99%

“…And the entanglement entropy can be obtained from equation (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18) and (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19).…”

Section: Examples:some Important Kinds Of Srgs Which Contain Nonisomomentioning

confidence: 99%

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Investigation graph isomorphism problem via entanglement entropy in strongly regular graphs

Jafarizadeh¹,

Eghbalifam²,

Nami³

2015

J. Stat. Mech.

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We investigate the quantum networks that their nodes are considered as quantum harmonic oscillators. The entanglement of the ground state can be used to quantify the amount of information one part of a network shares with the other part of the system.The networks which we studied in this paper, are called strongly regular graphs (SRG).These kinds of graphs have some special properties like they have three strata in the stratification basis. The Schur complement method is used to calculate the Schmidt number and entanglement entropy between two parts of graph. We could obtain analytically, all blocks of adjacency matrix in several important kinds of strongly regular graphs. Also * E-mail:jafarizadeh@tabrizu.ac.ir † E-mail:F.Egbali@tabrizu.ac.ir ‡ E-mail:S.Nami@tabrizu.ac.ir 1 Entanglement entropy 2 the entanglement entropy in the large coupling limit is considered in these graphs and the relationship between Entanglement entropy and the ratio of size of boundary to size of the system is found. Then, area-law is studied to show that there are no entanglement entropy for the highest size of system. Then, the graph isomorphism problem is considered in SRGs by using the elements of blocks of adjacency matrices. Two SRGs with the same parameters:(n, κ, λ, ν) are isomorphic if they can be made identical by relabeling their vertices. So the adjacency matrices of two isomorphic SRGs become identical by replacing of rows and columns.The nonisomirph SRGs could be distinguished by using the elements of blocks of adjacency matrices in the stratification basis, numerically.

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“…17À19. CTQWs have been studied on star graphs, 20,21 direct products of Cayley graphs, 22 quotient graphs, 23 odd graphs, 24 trees 25 and ultrametric spaces. 26 All of these articles have focused on coherent CTQWs.…”

Section: Introductionmentioning

confidence: 99%

The Effect of Decoherence on Mixing Time in Continuous-Time Quantum Walks on One-Dimensional Regular Networks

Salimi

Radgohar

2010

Int. J. Quantum Inform.

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In this paper, we study decoherence in continuous-time quantum walks (CTQWs) on onedimensional regular networks. For this purpose, we assume that every node is represented by a quantum dot continuously monitored by an individual point contact (Gurvitz's model). This measuring process induces decoherence. We focus on small rates of decoherence and then obtain the mixing time bound of the CTQWs on the one-dimensional regular network, whose distance parameter is l ! 2. Our results show that the mixing time is inversely proportional to the rate of decoherence, which is in agreement with the mentioned results for cycles in Refs. 29 and 37. Also, the same result is provided in Ref. 38 for long-range interacting cycles. Moreover, we¯nd that this quantity is independent of the distance parameter l ðl ! 2Þ and that the small values of decoherence make short the mixing time on these networks. 795 Int. J. Quantum Inform. 2010.08:795-806. Downloaded from www.worldscientific.com by UNIVERSITY OF AUCKLAND LIBRARY -SERIALS UNIT on 03/15/15. For personal use only. Ne ÀÀ NÀ1 N t : ð37Þ E®ect of Decoherence on Mixing Time in CTQWs 803 Int. J. Quantum Inform. 2010.08:795-806. Downloaded from www.worldscientific.com by UNIVERSITY OF AUCKLAND LIBRARY -SERIALS UNIT on 03/15/15. For personal use only.

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Continuous-Time Classical and Quantum Random Walk on Direct Product of Cayley Graphs

Cited by 13 publications

References 21 publications

The generator and quantum Markov semigroup for quantum walks

The generator and quantum Markov semigroup for quantum walks

Investigation graph isomorphism problem via entanglement entropy in strongly regular graphs

The Effect of Decoherence on Mixing Time in Continuous-Time Quantum Walks on One-Dimensional Regular Networks

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