Degree Distribution in Quantum Walks on Complex Networks

Faccin, Mauro; Johnson, T. H.; Biamonte, Jacob; Kais, Sabre; Migdał, Piotr

doi:10.1103/physrevx.3.041007

Cited by 76 publications

(110 citation statements)

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“…Our results give a concrete quantitative measure of the size of the part of the Hilbert space accessible from |Ψ . This could be a useful tool in the analysis of complex quantum networks 35,36 . The expected return time T Ψ , or, for a more detailed picture, the partial expected return times defined in Eq.…”

Section: Discussionmentioning

confidence: 99%

Quantized recurrence time in unital iterated open quantum dynamics

et al. 2015

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The expected return time to the original state is a key concept characterizing systems obeying both classical or quantum dynamics. We consider iterated open quantum dynamical systems in finite dimensional Hilbert spaces, a broad class of systems that includes classical Markov chains and unitary discrete time quantum walks on networks. Starting from a pure state, the time evolution is induced by repeated applications of a general quantum channel, in each timestep followed by a measurement to detect whether the system has returned to the original state. We prove that if the superoperator is unital in the relevant Hilbert space (the part of the Hilbert space explored by the system), then the expectation value of the return time is an integer, equal to the dimension of this relevant Hilbert space. We illustrate our results on partially coherent quantum walks on finite graphs. Our work connects the previously known quantization of the expected return time for bistochastic Markov chains and for unitary quantum walks, and shows that these are special cases of a more general statement. The expected return time is thus a quantitative measure of the size of the part of the Hilbert space available to the system when the dynamics is started from a certain state.

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Section: Discussionmentioning

confidence: 99%

Quantized recurrence time in unital iterated open quantum dynamics

et al. 2015

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“…The classical PageRank algorithm has been the subject of exact studies yielding analytical results [5], and other studies have concentrated on properties of quantum walks on networks [80][81][82][83][84][85][86] or adiabatic quantum computations of the classical PageRank [87]. Similarly, it is desirable to obtain exact analytical results on the quantum algorithm, which would complement the understanding gained from numerical studies, such as the one of Ref.…”

Section: Resultsmentioning

confidence: 99%

Quantum Google algorithm

et al. 2014

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Abstract. We review the main findings on the ranking capabilities of the recently proposed Quantum PageRank algorithm [1,2] applied to large complex networks. The algorithm has been shown to identify unambiguously the underlying topology of the network and to be capable of clearly highlighting the structure of secondary hubs of networks. Furthermore it can resolve the degeneracy in importance of the low lying part of the list of rankings. Examples of applications include real world instances from the WWW, which typically display a scale-free network structure and models of hierarchical networks. The quantum algorithm has been shown to display an increased stability with respect to a variation of the damping parameter, present in the Google algorithm, and a more clearly pronounced power-law behaviour in the distribution of importance among the nodes, as compared to the classical algorithm.

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“…While this and other kinds of quantum complex networks have received increasing attention in recent years in the context of perfect state transfer [14,15], quantum random walks [16,17], efficient entanglement distribution [18 -20] and the unification of classical and quantum network theory [21,22], here the focus is on the interplay between the network structure and the reduced dynamics of an open quantum system attached to it. To this end, we investigate the impact of the structure on the network spectral density, excitation transport properties and non-Markovianity of the reduced dynamics.…”

Section: Introductionmentioning

confidence: 99%

Non-Markovianity over Ensemble Averages in Quantum Complex Networks

Nokkala

Maniscalco

Piilo

2017

Open Syst. Inf. Dyn.

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Abstract. We consider bosonic quantum complex networks as structured finite environments for a quantum harmonic oscillator and investigate the interplay between the network structure and its spectral density, excitation transport properties and non-Markovianity. After a review of the formalism used, we demonstrate how even small changes to the network structure can have a large impact on the transport of excitations. We then consider the non-Markovianity over ensemble averages of several different types of random networks of identical oscillators and uniform coupling strength. Our results show that increasing the number of interactions in the network tends to suppress the average non-Markovianity. This suggests that tree networks are the random networks optimizing this quantity.

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Degree Distribution in Quantum Walks on Complex Networks

Cited by 76 publications

References 50 publications

Quantized recurrence time in unital iterated open quantum dynamics

Quantized recurrence time in unital iterated open quantum dynamics

Quantum Google algorithm

Non-Markovianity over Ensemble Averages in Quantum Complex Networks

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