(a) The internet can be thought of as a set of pages (nodes of a graph) connected by directed hyperlinks (edges of the graph). The classical (quantum) PageRank algorithm can be regarded as a single walker performing a directed classical (quantum) random walk on the graph. (b) The connectivity structure of the graph, captured by the connectivity matrix *C*, is of paramount importance to perform the ranking of the importance of pages both in the classical and the quantum case. In the classical (quantum) case the walker performs an incoherent (coherent) walk according to a combination of two hopping processes along the graph. The dynamics is governed by the “Google matrix” *G*, which describes the dynamics as a combination of contributions: (i) the first corresponds to a hopping according to a (patched) connectivity matrix *E* of the graph (parameter *α*, see main text and for more details). (ii) The second contribution represents a completely random hopping process (parameter 1 − *α*), where each node is connected to all other nodes of the graph. (c) In the quantum PageRank algorithm the Hilbert space is spanned by the set of directed links between all pairs *i* and *j* of nodes of the graph, as tensor product states |*i*〉_{1}|*j*〉_{2}. The initial state |*ψ*_{0}〉 as well as the coherent discrete time evolution operator *U*^{2} for the directed quantum walk (see main text) are determined by the Google matrix *G*. (d) Quantum fluctuations can lead to a reversal of the order of importances of pages at certain instances of (discrete) time (so-called instantaneous hierarchy non preserving property), as well as on the average over longer times (average hierarchy non preserving property). The latter effect is reflected by changes in the ordered list of pages (nodes) when the importance of pages according to the quantum PageRank algorithm is compared to its classical counterpart, the list of classical PageRank values. Red, blue and green curves in (d) show the instantaneous quantum PageRank of nodes # 4, 5 and 7 of the seven-node-graph shown in (c), which was explored in Ref. , and leads to the quantum PageRank and classical PageRank lists displayed in (e).

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