Regular quantum graphs

Severini, Simone; Tanner, Gregor

doi:10.1088/0305-4470/37/26/005

Cited by 17 publications

(17 citation statements)

References 20 publications

(22 reference statements)

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“…Quantum walks are extremely useful either theoretically, as primitives of universal quantum computers [66][67][68], or operationally, as building blocks to quantum algorithms [65,[69][70][71]. Thus, since there is a close connection between quantum walks and quantum graphs [72][73][74][75], this might open the possibility of extending different techniques to treat quantum graphs to the study of quantum walks [76][77][78][79], therefore helping in the development of quantum algorithms.…”

Section: Introductionmentioning

confidence: 99%

Green’s function approach for quantum graphs: An overview

et al. 2016

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Here we review the many aspects and distinct phenomena associated to quantum dynamics on general graph structures. For so, we discuss such class of systems under the energy domain Green's function (G) framework. This approach is particularly interesting because G can be written as a sum over classical-like paths, where local quantum effects are taking into account through the scattering matrix amplitudes (basically, transmission and reflection amplitudes) defined on each one of the graph vertices. Hence, the exact G has the functional form of a generalized semiclassical formula, which through different calculation techniques (addressed in details here) always can be cast into a closed analytic expression. It allows to solve exactly arbitrary large (although finite) graphs in a recursive and fast way. Using the Green's function method, we survey many properties for open and closed quantum graphs as scattering solutions for the former and eigenspectrum and eigenstates for the latter, also considering quasi-bound states. Concrete examples, like cube, binary trees and Sierpiński-like topologies are presented. Along the work, possible distinct applications using the Green's function methods for quantum graphs are outlined.

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Section: Introductionmentioning

confidence: 99%

Green’s function approach for quantum graphs: An overview

et al. 2016

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“…We mention that related notions of symmetries on quantum graphs have been discussed by several authors, cf. [11]- [28]- [7]. Throughout this paper we will consider directed graphs.…”

Section: Introductionmentioning

confidence: 99%

Well-posedness and symmetries of strongly coupled network equations

Cardanobile¹,

Mugnolo²,

Nittka³

2008

J. Phys. A: Math. Theor.

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We consider a class of evolution equations taking place on the edges of a finite network and allow for feedback effects between different, possibly non-adjacent edges. This generalizes the setting that is common in the literature, where the only considered interactions take place at the boundary, i. e., in the nodes of the network. We discuss well-posedness of the associated initial value problem as well as contractivity and positivity properties of its solutions. Finally, we discuss qualitative properties that can be formulated in terms of invariance of linear subspaces of the state space, i. e., of symmetries of the associated physical system. Applications to a neurobiological model as well as to a system of linear Schrödinger equations on a quantum graph are discussed.2000 Mathematics Subject Classification. 34B45, 70S10, 47D06. Key words and phrases. parabolic diffusion equations on networks; symmetries of dynamical systems.

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“…Indeed, line digraphs of digraphs described by unitary matrices are used in quantum computation and in the study of quantum walks (see [28]), as their statistical dynamics models that of random matrix theory (see [24] and [28]). In particular, the use of regular quantum graphs was studied in [29] and in [30], as the line digraph of a regular digraph is the digraph of the transition matrix of a coined quantum walk. In addition, line digraphs have been used in information theory as solutions to the index coding with side information problem (see [13]), where the minimum rank of a digraph represents the length of an optimal scalar linear solution of the corresponding instance of the problem (see [4]).In this work, we extend to digraphs the relationship between zero forcing and power domination established for undirected graphs in [7].…”

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confidence: 99%

Zero forcing in iterated line digraphs

Ferrero

Kalinowski

Stephen

2019

Discrete Applied Mathematics

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Zero forcing is a propagation process on a graph, or digraph, defined in linear algebra to provide a bound for the minimum rank problem. Independently, zero forcing was introduced in physics, computer science and network science, areas where line digraphs are frequently used as models. Zero forcing is also related to power domination, a propagation process that models the monitoring of electrical power networks.In this paper we study zero forcing in iterated line digraphs and provide a relationship between zero forcing and power domination in line digraphs. In particular, for regular iterated line digraphs we determine the minimum rank/maximum nullity, zero forcing number and power domination number, and provide constructions to attain them. We conclude that regular iterated line digraphs present optimal minimum rank/maximum nullity, zero forcing number and power domination number, and apply our results to determine those parameters on some families of digraphs often used in applications.1 in network science, it models the spread of a disease over a population, or of an opinion in a social network (see [14]).In addition to its intrinsic relation to minimum rank, zero forcing is closely related to power domination, a graph theory concept introduced in [20] to optimize the monitoring process of electrical power networks. From the definitions of power domination and zero forcing, it follows that the closed out neighborhood of a power dominating set is a zero forcing set, and a stronger relationship between zero forcing and power domination was established in [7].Zero forcing, minimum rank and power domination are all NP-hard problems, as proven in [1], [10], and [20], respectively. Thus, it is important to obtain bounds for the minimum rank, the zero forcing and the power domination numbers, as well as closed formulas to calculate them for families of graphs. Although power domination and zero forcing where introduced on undirected graphs, they were extended to digraphs in [1] and [5], respectively. Further results on zero forcing on digraphs can be found in [5], [15], [21], [23] and [31]. Power domination in digraphs has not been so thoroughly explored, but recently zero forcing and power domination for de Bruijn and Kautz digraphs was studied in [19]. De Bruijn and Kautz digraphs are iterated line digraphs of the complete digraph, with and without loops, respectively. In this work, we extend the results in [19] to zero forcing and power domination of iterated line digraphs of any regular digraph.The line digraph has been used in a broad range of disciplines, but its large number of applications precludes us from including an exhaustive summary here. In the context of this work, since the line digraph of a digraph described by a unitary matrix can also be described by a unitary matrix, iterated line digraphs are used to obtain arbitrarily large digraphs described by unitary matrices (see [24] and [28]). Such digraphs are frequently used to model quantum systems in physics, chemistry, and engineering (see [22]), and...

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Regular quantum graphs

Cited by 17 publications

References 20 publications

Green’s function approach for quantum graphs: An overview

Green’s function approach for quantum graphs: An overview

Well-posedness and symmetries of strongly coupled network equations

Zero forcing in iterated line digraphs

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