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- 1. Google in a Quantum Network G.D. Paparo and M.A. Martin-Delgado Departamento de F´ ısica Te´rica I, Universidad Complutense, 28040. Madrid, Spain. o We introduce the characterization of a class of quantum PageRank algorithms in a scenario in which some kind of quantum network is realizable out of the current classical internet web, but no quantum computer is yet available. This class represents a quantization of the PageRank protocol currently employed to list web pages according to their importance. We have found an instance of this class of quantum protocols that outperforms its classical counterpart and may break the classical hierarchy of web pages depending on the topology of the web. PACS numbers: 03.67.Ac, 03.67.Hk, 89.20.Hh, 05.40.FbarXiv:1112.2079v1 [quant-ph] 9 Dec 2011 I. INTRODUCTION Property P1 reﬂects the fact originally we would have an internet that is a classical network represented by a The possibility of establishing a quantum network directed graph and then shall apply some kind of quan- of practical use is currently under active investigation. tization procedure in order to turn it into a quantum Some early versions of them, modest as they may be, network. The latter must be compatible with the clas- have been designed and realized in real world in the re- sical one, particularly preserving the directed structure cent years [1–6], or in some instances they are under way. which is crucial to measure a page’s authority. This is In fact, building a quantum network has been targeted non-trivial and some quantization methods may fail to as a fundamental goal in quantum information [7, 8] an produce a unitary quantum PageRank importance for the even a more feasible goal to accomplish than the ﬁrst quantum case, as shown in Sect.III. scalable quantum computer. There are other more ad- With property P2 we guarantee that we have a globally vanced proposals for quantum networks [9, 10] based on well-deﬁned notion of the importance of a web page at the entanglement connections that need quantum repeaters quantum level. This allows us to have the probabilistic [11–13] in order to function properly and being stable interpretation of the surfer’s position (see Sect.III). [14–16]. Related to this, some physical quantum mod- Property P3 is the key to a wide class of natural quan- els exhibit very remarkable long-distance entanglement tization methods for the classical PageRank based in the properties [17–20]. Another alternative to build diﬀerent equivalence of this one with a classical Markov chain pro- types of quantum networks make use of quantum perco- cess (see Sect.III). Thus, it is natural that the equivalent lation protocols [21–24]. property holds true in the quantum version of the PageR- Thus, it is interesting to study how diﬀerent possibil- ank, and consequently, its description in terms of a quan- ities of quantum networks would behave regarding what tized surfer’s motion. we know about the world wide web. In particular, an es- The reason for property P4 relies on the assumption sential ingredient in the classical network that we enjoy that we envisage a near-future scenario when a certain today is the ability to search web pages in the immense class of quantum network will be operative but not yet changing world that the web has come to be known. The a scalable quantum computer. Therefore, we demand key tool for performing those searches is the notion of that the computation of the quantum PageRank Iq be the PageRank algorithm [25–33]. eﬃciently carried out on a classical computer. Since the notion of a quantum PabeRank is by no In this paper we have constructed a valid quantum means unique, it is convenient to introduce a class or PageRank that fulﬁlls all these requirements. We remark category of possible quantum PageRanks. They must that there may be other solutions to the quantum ver- satisfy a set of properties that deﬁne an admissible class: sion of the PageRank within the class deﬁned above, but Quantum PageRank Class: nevertheless we shall show that ﬁnding one instance of P1 The classical PageRank must be embedded into the this quantum PageRank class is a non-trivial task. quantum class in such a way that the directed graph The explicit step-by-step description of our quantum structure is preserved at the quantum level. PageRank algorithm is presented in Sect.IV. A key dis- tinctive feature of this quantum algorithm is that the P2 The sum of all quantum PageRanks must add to 1 importance of the quantum pages exhibit quantum ﬂuc- i.e. i Iq (Pi ) = 1. tuations unlike its classical counterpart. These quantum P3 The Q-PageRank admits a quantized Markov Chain ﬂuctuations, as shown in the simulations in Sect.V, show (MC) description. up in the form of time dependent importances Iq (Pi , t), which causes in turn that sometimes one certain pair of P4 The classical algorithm to compute the quantum pages satisfy Iq (Pi , t1 ) > Iq (Pj , t1 ), and some other times PageRank belongs to the computational complex- the relative importance is reversed Iq (Pi , t2 ) < Iq (Pj , t2 ), ity class P. for time-steps such that t1 < t2 . We may use an anal-
- 2. 2ogy to understand this situation: the classical PageR- other approaches were static and subjective w.r.t. con-ank gives us a snapshot or photo with a ﬁxed hierarchy tents of the pages.of web pages according to their calculated importance. The way most search engines, including Google, workOn the contrary, the quantum PageRank is more like a is to continually retrieve pages from the web, index themovie since the quantum importance of the pages vary words in each document, and store this information.with time. In order to produce a ﬁxed output made of Each time a user asks for a web search using a searcha list with the quantum pages sorted according to their phrase, such as ”search engine”, the search engine de-importance, a natural choice we make is to compute the termines outputs all pages on the web that contain thetemporal average of the quantum PageRanks and their words in the searched phrase or are semantically relatedstandard mean deviation. to it. There are additional features that a certain quantum Then, a problem arises naturally: Google now claimsPageRank may have as a consequence of the deﬁnition of to index 50 billion pages. Roughly 95 % of the text in webthe class above. We provide hereby several useful deﬁni- pages is composed from a mere 10,000 words. This meanstions: that, for most searches, there will be a huge number ofStrong Hierarchical Preserving PageRank: when pages containing the words in the search phrase. Whatthe classical hierarchical structure of a PageRank is pre- is needed is a mean of ranking the importance of theserved upon quantization. pages that ﬁt the search criteria so that the pages can beThis notion is too strong when the PageRank varies with sorted with the most important pages at the top of thetime as it is the case at the quantum level. list. Their success is largely due to PageRank’s ability toWeak Hierarchical Preserving PageRank: when rank the importance of pages in the WWW.the node with highest classical PageRank is preservedafter quantization, but not so for the rest of pages. A. Google PageRankOutperforming: when the highest classical PageRankof a page is overcome by the quantum PageRank of thatpage. The key idea of Google PageRank algorithm is that the importance of a page is given by how many pages link to Outperforming may occur at one given instant or on it. If we deﬁne I(Pi ) as the importance of a page Pi andaverage thereby leading to the natural extended concepts Bi as the set of pages linking to it, then we might thinkof instantaneously outperforming or average outperform- to put in equations the key idea put forward above asing, respectively. follows: In section VI we provide a list of main results that wehave obtained with our quantized version of the quantum I(Pj ) I(Pi ) := . (1)PageRank algorithm. Remarkably, quantum ﬂuctuations outdeg(Pj ) j∈Bimay change the classical hierarchy of web pages bothinstantaneously and in terms of mean values. Let us deﬁne a matrix, called the hyperlink matrix, in This paper is organized as follows: in Sect.II we give an which the entry in the ith row and jth column is:introduction to the classical notion of PageRank neededto present in Sect.III the quantum version of it. In 1/outdeg(Pj ) if Pj ∈ BiSect.IV we present our proposal for a quantum version of Hij := (2) 0 otherwiseGoogle PageRank algorithm. In Sect.V we perform nu-merical simulations of the quantum algorithm to repre- We will also form a vector I whose components aresentative directed graphs representing either an intranet the PageRanks I(Pi ). The condition above deﬁning theor a general web with no special symmetry. Sect.VI is PageRank can be expressed in matrix form as:devoted to conclusions. I = HI (3) Thus, we have recast the problem of ﬁnding the PageR- II. CLASSICAL PAGERANK anks as the problem of ﬁnding the eigenvalues of a ma- trix [28]. We are in for a special challenge since the ma- Brin and Page introduced Google in 1998 [25–27], a trix H is a square matrix with one column for each webtime when the pace at which the web was growing began page indexed by Google. This means that H has aboutto outstrip the ability of current search engines to yield n = 50 billion columns and rows. However H is a sparseuseable results. A major distinction between their algo- matrix, i.e. most of the entries in H are zero; in fact,rithm, called PageRank (PR), and previous approaches is studies show that web pages have an average of aboutthe fact that PR has an objective character, while other 10 links, meaning that, on average, all but 10 entries insearchers were based up on the subjective criterium of every column are zero.the contents of the pages , because they were built as a We will choose a method known as the power methodcollection of links that people in companies stored on a for ﬁnding the stationary vector I of the matrix H. Howregular basis. In order words, PR is dynamical while the does the power method work? We begin by choosing
- 3. 3a vector I0 as a candidate for I and then producing asequence of vectors Ik by: P1 P2 I k+1 = HI k (4) P3 P4However, as it is formulated the PageRank algorithm willnot output a meaningful vector. We will need to patchthe procedure in various ways. Figure 1: (Color online) A graph whose matrix E’s second eigenvalue, λ2 , is zero (see text in section II). B. Patching the Algorithm P1 P2 It can be seen that if there are dangling nodes, pagesthat are not linked to by any one, then the power methodwill output the null vector. If we consider the followingexample: P3 P4 P1 P2 Figure 2: (Color online) A graph that is not strongly con- whose hyperlink matrix is: nected, or equivalently, whose matrix E’s is reducible (see text in section II). 0 0 0 0 0 1 H= , 1 0 1 0 0 0 E= 0 1 0 0 and start from I0 = (1, 0)t one ends up with I = (0, 0)t . 0 0 1 0The ﬁrst patch in the tinkering of the PageRank algo-rithm will be replacing the column corresponding to a One can see that the second eigenvalue of E, |λ2 | is equaldangling node with a column of all 1/n with n the num- to one in this case, and actually all the eigenvalues areber of nodes. This means that virtually every dangling on the circle of radius 1 in the complex plane. If wenode is linking to every single node in the web, including compute I with the power method starting from, say,itself. This prevents the power method from giving the I0 = (1, 0, 0, 0)t it will fail to converge.null vector. This way, the disconnected graph becomes We will need to patch the algorithm again to ensureeﬀectively connected at the price of giving a very low that |λ2 | < 1. In order to guarantee it, we will requireweight to the artiﬁcial bonds (added links). E to be primitive, i.e. that there is an integer m suchThe graph, with the addition of the extra links would that E m contains all positive entries. The meaning oflook like: this assumption is that the graph is such that any page is connected by a path of at least m links to any other. Anticipating the interpretation of a diﬀusion phe- P1 P2 nomenon associated to searching the web, we can inter- pret the requirement of E to be primitive as the require- ment of ﬁnding the walker with nonzero probability on any site after a minimum time m. Let us now considerwith a modiﬁed hyperlink matrix, E: the graph in ﬁg. 2. One can divide the graph in two subgraphs G1 and G2 . There are no links pointing from the subgraph G2 , made of the nodes 3 and 4 to the ﬁrst 0 1/2 subgraph G1 , made of nodes 1 and 2. If we write down E= 1 1/2 the matrix E: 0 1/2 0 0 The matrix E that we obtain is, in general, (column) 1/3 0 0 0 stochastic, i.e. its columns all sum up to one. From E= , 1/3 1/2 0 1 the theory of stochastic matrices one knows that 1 is 1/3 0 1 0always an eigenvalue. Furthermore, the convergence ofI k = EI k−1 to I depends on the second eigenvalue of E, we can see that there is a block that is zero, precisely theλ2 . If it is smaller than 1, then the power method will one that carries the information of the edges linking theconverge. In addition, it is more rapid if |λ2 | is as close nodes of G2 to the nodes of G1 .to 0 as possible. If we calculate I we ﬁnd: I = (0, 0, 3/5, 2/5)t. Yet, oneLet us consider the graph in ﬁg. 1, with E matrix: is not satisﬁed from giving an interpretation to the vector
- 4. 4I because the nodes from the ﬁrst subgraph G1 have zero one for every time step. For each step, the random vari-importance albeit being linked by other nodes. This is able can take on values in the set of nodes {Pi } of thecaused by the reducibility of E that causes a drain of web. We can recast Google PageRank in the language ofimportance from G1 to G2 . In order to have a meaningful a Markov Chain (MC). Thus, from eq. (6) written as:vector I, one that has all nonzero entries, one shoulddemand that the matrix be irreducible. A necessary and Pr(X = Pi ) = Gij Pr(X (n) = Pj ) (7)suﬃcient condition for it is that the graph is strongly jconnected, i.e. that given two pages there is always apath connecting one to the other (see [29]) chap. 8). and from the law of total probability: Pr(X (n+1) = Pi ) = (n+1) (8) j Pr(X = Pi |X (n) = Pj ) Pr(X (n) = Pj ), 1. The Patched Algorithm one can interpret the stochastic matrix G as the condi- In order to implement all these patches, let us imagine tional probability linking one time step to the other, i.e.:a walk of a surfer on the graph. With probability α Gij = Pr(X (n+1) = Pi |X (n) = Pj ). (9)the surfer will follow the web with stochastic matrix Eand with probability 1 − α, it will jump to any page at We will make use in the following of the latter interpreta-random. The matrix of this process would be: tion of Google PageRank to devise methods to quantize it. (1 − α) G := αE + 1, (5) n III. QUANTUM PAGERANKSwhere 1 is a matrix with entries all set to 1. Now, the ma-trix G is irreducible because the matrix 1 is irreducible.Furthermore, it is also primitive since it has all positive Quantum walks in their discrete time formulation wereentries. We have thus obtained a matrix that is both known already to Feynman [34] and since then, they wereprimitive and irreducible. This means it has a unique rediscovered many times [35] and in contexts as diﬀerentstationary vector that may be calculated with the power as quantum cellular automata [36, 37] and the haltingmethod. Furthermore, the result does not depend on the problem of the quantum Turing machine [38, 39]. Forinitial value I0 because the underlying graph is strongly simplicity, let us discuss possible ways for quantizing aconnected, which is equivalent to the irreducibility of G, quantum walk taking place on the line. Later, we shallsee [29]. generalize it to an arbitrary graph. From now on, weThe parameter α is free and needs to be tuned. It is shall make the notation lighter denoting each node (orknown [30] that the second eigenvalue of G λ2 is such that page) Pi simply by i as shown in the following ﬁgure:λ2 ≤ α , so one would choose α as close to zero as possiblebut in this way the structure of the web, described by Ewould not be taken into account at all. Brin and pagechose α = 0.85 to optimize the calculation. −2 −1 0 1 2 2. Formulation as a Random Walk The naive way of quantizing this random walk would be It is very appealing and useful to rethink the problem to go from the index set {i | i ∈ Z} to a Hilbert space ofof assigning the importance of a page as the task of cal- states H = span{| i | i ∈ Z} as shown in ﬁgure 3:culating the fraction of time a walker diﬀusing on the Following the key idea outlined above, one could deﬁnegraph according to the stochastic process given by the the quantum importance of a page as:Google matrix G. In fact we, can reformulate the GooglePageRank as the algorithm that computes the fraction I(Pi ) = Pr(X = Pi ) := i| ψ (10)of the time the walker spends on each node by deﬁningthe fraction on the j th page Tj as: Ti = Gij Tj . (6) |−2 |−1 |0 |1 |2 jEquivalently, one can say that the operational meaning ofthe PageRank algorithm is to give the probability to ﬁndthe walker on the node Pi . Let us make it clearer by deﬁn- Figure 3: (Color online) The naive way to quantize a randoming a set of random variables: X (0) , X (1) , . . . , X (n) , . . . , walk on a line.
- 5. 5where | ψ is the state of the system after it has diﬀused is then a projector onto the equal superposition of theon the graph. However, this quantization procedure is vectors |ψj for j = 1, 2, . . . , N. With this, the step of thenot viable. This is because the direct quantization of the quantum walk is then given bytime step evolution operator as U := S(2Π − 1) (14) √ U = p |i + 1 i| + 1 − p |i − 1 i| (11) where S is the swap operator i.e.is not unitary. Indeed, while 0|2 = 0 one has that N 0|U † U |2 = 0 in the general case of p ∈ [0, 1]. S= |j, k k, j|. (15)This diﬃculty can be overcome enlarging the Hilbert j,k=1space. There are various ways to do it: Before continuing let us point out that the time step • adding a coin space HC to each site on the quantum operator is unitary: network. A coin space encodes the possibility to go left or right i.e. HC = span{|L , |R }. U U † = S(2Π−1)(2Π−1)S † = S(4Π2 −2Π−2Π+1)S † = 1 (16) • deﬁning the Hilbert space as the space of (ori- from the unitarity of S and the fact that Π is a projector ented) edges and treating the vertices as scattering and squares to itself. Analogously: centers. The resulting walk is called a Scattering Quantum Walk. U † U = (2Π − 1)S † S(2Π − 1) = 4Π2 − 2Π − 2Π + 1 = 1 (17) • using Szegedy’s [40] procedure to quantize Markov The time step is thus the eﬀect of a coin ﬂip followed by chains. a swap operator. Let us look more closely at the coin ﬂipWe will discuss the latter way for it will give us a operation:valid quantization of Google’s PageRank that satisﬁes N 1the properties 1 − 4 introduced in Sect.I. 2Π − 1 = 2 |ψj ψj | − 1 . (18) j=1 N 3. Szegedy’s Quantization of Markov Chains The vectors |ψj contain the information of the directed links that connect the j th node to all its neighbors to We have seen that the Google matrix G can be seen which it is connected through the stochastic matrix G. 1as the time step evolution operator of a M C Markov The operator 2 |ψj ψj | − N 1 is nothing but a reﬂectionchain, or equivalently, of a discrete-time classical ran- around |ψj and has the eﬀect of enhancing the ampli-dom walk: the two terms will be used interchangeably tudes of the mentioned directed edges at the expenses offrom now on. Szegedy put forward a general scheme to the others. The swap operator ensures that the step bequantize a Markov chain. Let G be a N × N stochastic unitary.matrix representing a MC Markov chain on an N -vertexgraph. In order to introduce a discrete-time quantum 4. Solving the Eigenvalue Problem for the Walk Operatorwalk on the same graph we use as the Hilbert space thespan of all vectors representing the |V | × |V | (directed)edges of the graphs i.e. H = span{|i 1 |j 2 , with i, j ∈ The quantization procedure based on the unitary op-|V | × |V |} = C|V | ⊗ C|V | = CN ⊗ CN . The order of the erator (14) allows us to get remarkable insight on thespaces in the tensor product is important here because properties of the walk by a systematic analysis of thewe are dealing with a directed graph. We stress this fact spectral properties of U . The spectrum of the quantizedin the notation using subindices 1 and 2. Let us deﬁne walk is related to the spectrum of the original stochasticthe vectors: matrix that in turn we have seen has a key role on the properties of the related classical random walk. N Let us deﬁne the following N × N matrix D that will |ψj := |j 1⊗ Gkj |k 2 , (12) play an important role in relating the spectra of the clas- k=1 sical and quantum walks by specifying its entries as fol-that is a superposition of the vectors representing the lows:edges outgoing from the j th vertex. The weights are given Dij := Gij Gji , (19)by the (square root of the) stochastic matrix G. One can easily verify that due to the stochasticity of where there is no sum over repeated indexes. Let us alsoG the vectors |ψj for j = 1, 2, . . . , N are normalized. The deﬁne the following operator A from the space of vertices,operator CN , to the space of edges, CN ⊗ CN : N N Π := |ψj ψj |, (13) A := |ψj j| (20) j=1 j=1
- 6. 6It has the following properties, which are straightforward To complete our analysis let us point out that on theto prove: orthogonal complement to the span of the vectors |ψj , the action of the walk operator U = S(2Π − 1) is just 1. A† A = 1 −S, which has eigenvalues ±1. This is because Π gives 2. AA† = Π 0 on this subspace. We conclude that the spectrum of U is the set 3. A† SA = D σ(U ) := {±1, exp(±i arccos λ)}, (29)The matrix D is symmetric by construction. The eigen-value problem D|λ = λ|λ can in principle be solved where λ are the eigenvalues of D. In the following weyielding N eigenvalues λ with the associated eigenvec- will need the quantum walk where two steps at a timetors |λ : are performed, with operator U 2 . We can advance that the interesting subspace, where we have dynamics, is the σ(D) := {λ, |λ }. (21) span of the vectors |ψj and S|ψj where the walk oper- Let us now consider the the following vectors out of ator acts nontrivially. ˜them: |λ := A|λ on the space of the quantized Markov Furthermore, considering the two-step evolution opera-chain i.e. CN ⊗ CN . In order to obtain the spectrum of tor, one can see that:U , we will ﬁrst isolate an invariant subspace for U and U 2 = (2SΠS − 1)(2Π − 1). (30)look for eigenvalues and eigenvectors in this space. Wewill then concentrate on the orthogonal complement of Therefore under the two-steps operator U 2 the Hilbertit. We will argue that the interesting part of the Hilbert space splits naturally into the subspaces Hdyn =space for the dynamics is the aforementioned invariant span{|ψj , S|ψj } where dynamics takes place and its or-subspace. In order to identify the invariant subspace of ⊥ ˜ thogonal complement Hnodyn = Hdyn as can be seenthe walk operator let us see the eﬀect of U on |λ : 2 from (30) U acts trivially in such a way that, obvi- ˜ ˜ U |λ = S(2AA† − 1)A|λ = S|λ , (22) ously, H = Hdyn ⊕ Hnodyn . The dimension of Hdyn is at most 2N and, remembering that the dimension ofwhere we used the property 1 and 2. Let us see also its H = C|V | ⊗C|V | is N 2 we can conclude that the spectrum ˜eﬀect on S|λ : of U 2 corresponding to Hdyn is composed by, at most, the 2N values ˜ ˜ U S|λ = S(2AA† − 1)SA|λ = (2λS − 1)|λ , (23) {exp(±2i arccos λ)}, (31)using properties 2 and 3 and the fact that the vectors |λare eigenvectors of D. From (22) and (23) we can deduce and the rest of the spectrum, corresponding to Hnodynthat the subspace where U 2 acts trivially, is composed of at least N 2 − 2N 1’s. ˜ ˜ IU := {|λ , S|λ }, (24)is invariant under the walk operator U . It is thus sensible IV. A QUANTIZATION OF GOOGLEto solve the eigenvalue problem: PAGERANK U |µ = µ|µ , (25) In this section we deﬁne a valid Quantum PageRankfor the walk operator restricted to the invariant subspace and take advantage of the analysis presented above to(24). Following what we have said, let us make an edu- provide an eﬃcient algorithm for its calculation as re-cated ansatz for the eigenstates of U: quested in Sect.I. A natural way to deﬁne a quantiza- tion of the importance of a node or page in the quantum ˜ ˜ |µ = |λ − µS|λ . (26) network associated to a directed graph is to exploit the connection with the Markov chain process in which theWe have that: ﬁctitious walker is now subjected to quantum superposi- ˜ ˜ U |µ = µ|λ + (1 − 2µλ)S|λ , (27) tions of paths throughout the quantum web. In this way, thethereby the condition for |µ to be eigenstate of U is instantaneous importance of a quantum web page, de- noted as Iq (Pi , m), is given by the probability of ﬁnding − µ2 = (1 − 2µλ), (28) the walker in that page Pi at the node i of the network √ after m time steps. As we have said before, the Hilbertwhich yields µ = λ ± λ2 − 1 = exp(±i arccos λ). space of this quantum walk is the set of directed links ˜ We note also that the span of the vectors |λ coin- of the graph, H = C|V | ⊗ C|V | where the numbering ofcides with the span of the vectors |ψj . Indeed we have the vector spaces is meaningful due to the directedness of ˜ ˜ † λ |λ λ| = A λ |λ λ|A = Π = j |ψj ψj | . the underlying Graph and the second space in the tensor
- 7. 7product contains the information of where the directedlink points to. It seems natural then to project onto a 4vector of this second space |i 2 obtaining the quantum 2state |Iq (Pi , m) that contains the superposition of the 5nodes that were linked to it. To quantify the importanceone can then extract a positive number calculating the 1norm of |Iq (Pi , m) , and with this we obtain an instanta- 6neous list of page ranks including quantum ﬂuctuations 3of the network. Thus we expect its instantaneous value 7to oscillate in time as a result of the underlying coherentdynamics. The method for computing the instantaneous PageR-ank of the page Pi is to start from an initial vector |ψ(0) Figure 4: (Color online)The three level tree considered inand to let it evolve according to the two-step evolution the text to benchmark the Quantum PageRank (see text inoperator U 2 (in order to swap the directions of the edges section V). Each node represents a web page in an intranetan even number of times, thus preserving the graph’s with the root node being its home page.directedness). Then, we need to project onto |i 2 , andﬁnally to take the norm of the resulting quantum state: 2m as a measure of its ﬂuctuations. Iq (Pi , m) = ψ(0)| U † |i 2 i|U 2m |ψ(0) . (32) In order to obtain the Quantum PageRank values of theIn order to implement the full procedure, one starts from nodes of a digraph we apply the algorithm that comesthe stochastic matrix G representing the classical Google out of the analysis presented above. Namely the stepssearch that we want to quantize, forms the matrix D and one has to perform are:obtains its spectrum σ(D) = {λ}. One then forms the Quantum PageRank Protocol ˜states |λ = A|λ , in terms of which the eigenvectors ofthe walk operator, |µ , in the subspace where the dynam- Step 1/ Write the Google matrix for the digraph G.ics takes place are written. Using the spectral decomposition of U one can then Step 2/ Write down the matrix D (see eq. (19)) andarrive at a closed analytical expression for our quantum calculate its eigenvalues and eigenvectors.instantaneous PageRank: 2 Step 3/ Find the eigenvectors and eigenvalues of the 2m Iq (Pi , m) = µ 2 i|µ µ|ψ(0) , (33) two-step quantum diﬀusion operator U 2 in the dy- µ namical subspace Hdyn (see section III for the de- ˜ ˜ tails).where |µ = |λ − µS|λ and µ = exp(±i arccos λ). Notethat due to the fact that |i 2 are a basis of CN , in otherwords i |i 2 i| = 1 , from (32) one can see: Step 4/ Extract the Quantum PageRank value in time (33), its mean (35) and standard deviation i Iq (Pi , m) = (36) starting from the initial condition |ψ0 = † 2m 2m (34) 1 N = ψ(0)| U | i |i 2 i|U |ψ(0) = 1 ∀m, √ N i=1 |ψj .meaning that in the quantum version of Google PageR-ank we have that the normalization condition (34) ispreserved at all times allowing to interpret the quantity V. RESULTS: SIMULATIONS FOR QUANTUMIq (Pi , m) as the instantaneous relative importance of the PAGERANKSpage Pi , thereby reproducing a basic sum rule that alsoholds in the classical domain. In order to integrate out the ﬂuctuations arising from After developing a quantum version of the Googlethe coherent evolution we also introduce the average im- PageRank in the previous section, it is necessary to applyportance of the page i sitting on the ith node Iq (Pi ) it to speciﬁc networks and by means of simulations, seeas: how it behaves as compared with the classical algorithm of PageRank. M 1 We have put to test our new quantum version of the Iq (Pi ) := Iq (Pi , m). (35) M PageRank algorithm in the case of a binary directed tree m=0 with 3 levels (see ﬁg. 4) and a small albeit general, withWe also use its variance or standard mean deviation: no special property, directed graph (see ﬁg. 5). We will describe the results in subsections V A and V B respec- ∆Iq (Pi ) := Iq (Pi ) − Iq (Pi ) 2 , 2 (36) tively.
- 8. 8 root 6 0.6 Quantum PageRank 4 0.5 3 0.4 1 7 0.3 2 5 0.2 0.1Figure 5: (Color online) The general graph with 7 nodes con- 0 50 100 150 200 250 300 350 400 450 500 Timesidered in the text for benchmarking the Quantum PageRank(see text in section V) . Each node represents a web page inthis directed quantum network. Figure 6: (Color online) The evolution of the instantaneous Quantum PageRank Iq with time for the root page (home page) in the case of a directed binary tree in Fig.4. A. Case Study 1: A Tree Graph root In this subsection we will display the results obtained 0.6 level 2 Quantum PageRanksin the case of a tree graph (see ﬁg. 4). This type of level 3 0.5directed graph has a clear meaning in terms of a webnetwork: it represents an intranet with the root node 0.4being the home page of a certain website and its leavesrepresenting internal web pages. This case of study has 0.3been extensively studied classically [31]. The quantum 0.2algorithm presented above is implemented numerically.The quantum PageRank of the root page clearly oscil- 0.1lates in time and attains values that are higher than theclassical counterpart, as shown in Fig. 6. According to 0 0 50 100 150 200 250 300 350 400 450 500the properties studied in Sect.I, our quantum PageRank Timeis instantaneously outperforming. It is rather remarkable that a quantum version of the Figure 7: (Color online) The evolution with time of the instan-PageRank may have an enhancement of the importance taneous Quantum PageRank algorithm Iq deﬁned in Sect.IVin the home page with respect to its classical counter- for web pages (nodes) in the case of a directed binary treepart. This is achieved merely by quantum means, with- graph in Fig.4. Only one page per level of the tree is dis-out changing the connectivity of the original directed played because pages that are in the same level have equalgraph as has been proposed classically [31]. Quantum PageRank. As for the quantum ﬂuctuations present in the im-portance of the root page, it is important to emphasize clearly seen in ﬁgure 8 for this particular case of directedthat they remain bounded during the evolution as can be binary tree graph.checked from Fig. 6. In addition, the classical value is al- Furthermore we can calculate a coarse grained evolu-ways inside the range of the quantum ﬂuctuations. This tion in time of the instantaneous Quantum PageRank.feature is found to be true for all pages in the directed bi- We divide its total evolution time of M steps in L equalnary tree network conﬁguration (see Fig.7). A distinctive segments made of an integer number M/L steps. Wefeature of the Quantum PageRank is that the hierarchy calculate the mean in every segment:is not preserved at every time. Figure 7 clearly displaythe crossings of the instantaneous Quantum PageRanks. (n+1)M/L ¯ LIn order to extract a ﬁxed value for the relative impor- Iq (Pi , n) := Iq (Pi , m) , n = 0, . . . , L − 1.tances of the pages, we compute the mean value of the M m=nM/Linstantaneous quantum PageRank in time from eq. (35) (37)and its variance from eq. (36). The result of integrating out the oscillations in a coarse One can notice that the hierarchy of the pages is pre- grained time step is shown in ﬁgure 9. An interestingserved on average and that the errors i.e. the variances feature of this case is that the quantum PageRank oscil-are negligible compared to the means. We can thus infer lations shows a modulation with a clearly visible envelopethe pages’ hierarchy as predicted by the classical algo- for each node. It strongly enforces the idea that the pro-rithm from the Quantum PageRanks’ averages as can be posed algorithm has a solid grounding on the classical
- 9. 9 0.4 0.45 Quantum PageRank no de 7 0.35 Classical PageRank 0.4 no de 5 Quantum PageRank no de 4 0.3 0.35 PageRanks 0.3 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0 1 2 3 4 5 6 7 8 Nodes 0.05 0 0 50 100 150 200 250 300 350 400 450 500Figure 8: (Color online) Comparison of the hierarchies that Timeresult from the Classical and the Quantum PageRank in thecase of the directed graph tree shown in Fig.4. Error bars Figure 10: (Color online)The evolution in time of the Quan-in the quantum case are computed with the standard mean tum PageRank Iq of pages 7, 5 (that classically have the high-deviation (35),(36). est and nearly degenerate PageRank) and of page 4 (that clas- sically has the lowest PageRank) in the case of general graph shown in Fig.5. 0.4 0.35 Quantum PageRanks 0.3 ro ot 0.4 level 2 Quantum PageRank 0.35 0.25 level 3 PageRanks 0.3 Classical PageRank 0.2 0.25 0.15 0.2 0.15 0.1 0.1 0.05 0 10 20 30 40 50 60 0.05 Coarse grained time 0 0 1 2 3 4 5 6 7 8Figure 9: (Color online) The evolution in a coarse grained Nodestime (see text in section V) of the Quantum PageRanks inthe case of the directed graph tree shown in Fig.4. Again Figure 11: (Color online) Comparison of the hierarchies thatonly one page per level is being displayed because pages that result from the Classical and the Quantum PageRank in theare in the same level have equal Quantum PageRank. case of the general graph shown in Fig.5. Error bars in the quantum case are computed with the standard mean devia- tion (35),(36).PageRank algorithm and it is a valid quantization of it. B. Case Study 2: A General Graph the instantaneous Quantum PageRAnk even between the pages that classically have hishest and lowest PageRank. We have performed a calculation of the quantum Furthermore, the nodes that have a very close classicalPageRank also in the case of a general directed graph PageRank are shown to have a very similar behaviour inwith no particular symmetry (see ﬁg. 5). time of their instantaneous Quantum PageRank.The value of the instantaneous quantum PageRanks are Remarkably enough, we ﬁnd that the Quantum PageR-found to display oscillations that are bounded and in- anks’ averages do not give us the same hierarchy as in theclude the value found by the classical PageRank. The classical case (see Fig. 11). Nevertheless, it is possibleQPR of the node with the highest classical PageRank at- to clearly distinguish, within the error bar given by thetains, at given times, values of the QPR that are higher variance, which pages have highest and lowest classicalthan the classical counterpart (see ﬁg. 10). As seen in PageRank.the case of the binary tree treated above the QuantumPageRank is found to be instantaneously outperforming The analysis with a coarse graining in time of the in-according to the deﬁnition given in section I. stantaneous Quantum PageRank (see Fig. 12) reinforcesThe classical hierarchy is not preserved by the QPR at the conclusion that the classical PageRanks are still dis-any given time (as is clearly shown by Fig. 10, see cap- tillable in the case of pages with highest and lowest clas-tion). We can notice crossings in the importance given by sical importances.
- 10. 10 0.24 no de 1 root page without changing the topology of the original 0.22 no de 2 network. no de 3 Quantum PageRanks 0.2 no de 4 ii/ The instantaneous values of the quantum PageRanks 0.18 no de 5 for the nodes violate the hierarchy of the classical values. no de 6 0.16 iii/ The mean values of the quantum PageRanks includ- no de 7 0.14 ing its standard deviation preserve the hierarchy of the 0.12 classical values. 0.1 For the general directed graph: 0.08 i/ The quantum PageRank for the web page with the 0.06 highest classical PageRank is some times higher than the 0 10 20 30 40 coarse grained time 50 60 classical values obtained with the standard algorithm. This means that the quantum version of the PageRankFigure 12: (Color online) The evolution in a coarse grained is instantaneously outperforming with respect to the clas-time (see text in section V) of the Quantum PageRanks in sical value.the case of the general graph shown in Fig.5. ii/ The instantaneous values of the quantum PageRanks for the nodes violate the hierarchy of the classical values. iii/ Remarkably enough, there are pages with mean val- VI. CONCLUSIONS ues, including its standard deviation, of their quantum PageRank that violate the hierarchy of the classical val- In this paper we have proposed a notion of a class ues.of protocols that qualify to be considered a quantum These properties are a clear manifestation that our pro-version of the classical PageRank algorithm employed posal for a quantum version of the PageRank algorithmby the Google search engine (see Sect.I). In addition, behaves appropriately with respect to the classical algo-we have constructed a step-by-step protocol explicitly in rithm and exhibit non-trivial features.Sect.IV based on the quantization of Markov chains for As the main purpose of our work is to devise a quantumdirected graphs. This is a non-trivial problem since deal- PageRank algorithm by overcoming certain diﬃcultiesing with quantum versions of digraphs may produce uni- explained in the paper, thus far we have dealt only withtarity problems (see Sect.III). We have tested our quan- small networks representing diﬀerent types of web. Ittum PageRank algorithm with two web networks in order would be very interesting to perform computations withto gain insight into the speciﬁc behaviour of this protocol. the quantum PageRank applied to very large networksOne network is a binary tree graph representing an in- with the properties exhibited by the complex structuretranet with the root being the home page of it. The other of the real web [41–47].is a general directed graph with no speciﬁc structure. From our numerical simulations we have found thatour quantum PageRank has very interesting properties.For the directed binary tree: Acknowledgmentsi/ The quantum PageRank for the root page is instanta-neously outperforming with respect to the classical value. We thank the Spanish MICINN grant FIS2009-10061,This is a manifestation of the quantum ﬂuctuations in- CAM research consortium QUITEMAD S2009-ESP-herent to the quantum version of the algorithm and al- 1594, European Commission PICC: FP7 2007-2013,lows us to have an enhancement of the importance of the Grant No. 249958, UCM-BS grant GICC-910758. [1] C. Elliott; “The DARPA Quantum Network”; [6] T. L¨nger and G. Lenhart; “Standardization of quantum a arXiv:quant-ph/0412029. key distribution and the ETSI standardization initiative [2] A. Poppe, M. Peev and O. Maurhart; “Outline of the ISG-QKD”; New J. Phys. 11 055051 (2009). SECOQC Quantum-Key-Distribution Network in Vienna [7] M. A. Nielsen and I. L. Chuang, Quantum Computation ”; International Journal of Quantum Information Volume and Quantum Information (Cambridge University Press, 6, 209-218 ( 2008). Cambridge, 2000). [3] Tokyo Quantum Network 2010. www.uqcc2010.org [8] A. Galindo and M.A. Martin-Delgado; “Infor- [4] Swiss Quantum Network. mation and Computation: Classical and Quan- http://swissquantum.idquantique.com/ tum Aspects”; Rev.Mod.Phys. 74:347-423, (2002); [5] D. Lancho, J. Martinez, D. Elkouss, M. Soto and V. Mar- arXiv:quant-ph/0112105. tin; “QKD in Standard Optical Telecommunications Net- [9] H. J. Kimble; “The quantum internet”; Nature (London) works”; Lecture Notes of the Institute for Computer Sci- 453, 1023 (2008). ences, Social Informatics and Telecommunications Engi- [10] D. S. Wiersma; “Random Quantum Networks”; Science neering. Volume 36, pp 142-149 (2010). arXiv:1006.1858 327, 1333 (2010).
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