Complex Networks from Classical to Quantum

Jacob Biamonte,1, ∗ Mauro Faccin,2, † and Manlio De Domenico3, ‡ 1Deep Quantum Labs Skolkovo Institute of Science and Technology, Skoltech Building 3, Moscow Russia 143026 2ICTEAM, Universit´eCatholique de Louvain, Euler Building 4, Avenue Lemaitre, B-1348 Louvain-la-Neuve, Belgium 3Fondazione Bruno Kessler, Via Sommarive 18, 38123 Povo (TN), Italy

Recent progress in applying complex network first consists of quantum systems whose connections are theory to problems in quantum information has represented by entangled states [6, 33, 34]. These quan- resulted in a beneficial crossover. Complex net- tum networks are used in secure quantum communication work methods have successfully been applied to systems. The second area of focus consists of networks of transport and entanglement models while infor- quantum systems, such as atoms or superconducting quan- mation physics is setting the stage for a theory tum electronics, whose connections are physical [35–40]. of complex systems with quantum information- Such systems are used to develop quantum-enhanced algo- inspired methods. Novel quantum induced effects rithms or quantum information transport systems, both have been predicted in random graphs—where modeled by quantum walks on complex networks. At a edges represent entangled links—and quantum fundamental level, the two types of quantum networks are computer algorithms have been proposed to offer described by quantum information theory, allowing one to enhancement for several network problems. Here extend the spectrum of network descriptors—such as rank- we review the results at the cutting edge, pin- ing indicators, similarity and correlation measures—inside pointing the similarities and the differences found the quantum domain. at the intersection of these two fields. Interestingly, the same tools can then be appropriately modified to apply to traditional complex networks, sug- Quantum mechanics has long been predicted to help gesting the existence of a framework—network informa- solve computational problems in physics [1], chemistry [2], tion theory—suitable for application to both classical and and machine learning [3] and to offer quantum security quantum networked systems [32, 41, 42]. This bidirec- enhancement in communications [4], including a quantum tional cross-over is carving out a coherent path forward secure Internet [5]. Rapid experimental progress has built fundamentally on the intersection of these two fields pushed quantum computing and communication devices (see Fig.1). Several quantum effects are still outside of into truly data-intensive domains, where even the classical the predictive range of applicability of complex network network describing a quantum system can exhibit complex theory. Future work should build on recent breakthroughs features, giving rise to what appears as a paradigm shift and head towards a new theory of complex networks which needed to face a fundamental type of complexity [6–13]. augments the current statistical mechanics approach to Methods originating in complex networks—traditionally complex networks, with a theory built fundamentally on based on statistical mechanics—are now being generalized quantum mechanics. Such a unified path forward appears to the quantum domain in order to address these new to be through the language of information theory. quantum complexity challenges. Here, we make an effort to review some of the crucial Building on several fundamental discoveries [14, 15], steps towards the creation of a network theory based complex network theory has demonstrated that many fundamentally on quantum effects. Therefore, we do (non-quantum) systems exhibit similarities in their com- not cover several topics that, nevertheless, deserve to be plex features [14–18], in the organization of their struc- mentioned as part of the field. These include, in no par- ture and dynamics [19–24], the controllability of their ticular order, quantum gravity theories based on complex arXiv:1702.08459v4 [quant-ph] 16 Apr 2019 constituents [25] and their resilience to structural and dy- networks [43–46], synchronization in and on quantum namical perturbations [26–31]. Certain quantum systems networks [47], quantum random circuits [48, 49], classical have been shown to indeed exhibit complex features re- spin models and quantum statistics successfully used in lated to classical systems, as well as novel mechanisms and complex network theory [50–52] (see [53] for a thorough principles that interrelate complex features in quantum review). systems [6–12, 32]. Two types of quantum networks have been of primary focus in the series of pioneering results we review. The NETWORKS IN QUANTUM PHYSICS VS COMPLEXITY

∗ [email protected]; DeepQuantum.AI Network and graph theory fundamentally arises in † [email protected] nearly all aspects of quantum information and computa- ‡ [email protected] tion. As is the case with traditional network science, not 2

Box 1 Cross-pollination between the ﬁelds of complex networks and quantum information science. In recent years, seminal work has been carried out at the intersection of quantum information and computation and complex network theory. We attempt to catalog the scope of this work in Fig.1.

tools for understanding physical resources network information theory quantum inspired network tools quantum inspired network measures

algorithms and descriptors quantum centrality measures polynomial and superpolynomial reductions for certain network problems such as flow and effective resistance detecting community structure in quantum systems polynomial and superpolynomial reductions for certain machine learning and data analysis problems

network analysis quantum algorithms adapted for network analysis to quantum complex networks and quantum physics (current status ) graphity and quantum transport emergent models on complex networks quantum walks solving graph recognition of space-time random network models and search engine ranking problems

walk models of exciton transport in photosynthetic complexes

related types quantum communication of quantum networks networks

random quantum circuits quantum generalizations of random graph models

random tensor networks and geometry entanglement percolation on complex networks

FIG. 1. Each of the shaded regions represent published findings that map out the field from theoretical, experimental and computational perspectives. The top area classifies the analytical tools inspired by quantum information for classical network analysis and vice versa—both covered in this review—as well as the quantum algorithms developed to address specific problems in network science. The classification in the bottom area includes quantum networks based on entangled states and on physical connections—also covered in this review—as well as quantum network models of space-time, random quantum circuits, random tensor networks and geometry.

all networks exhibit what is considered as ‘complexity’. (ii) quantum random walks on graphs; (iii) Quantum cir- Here we will recall briefly the basic definition of a network cuits/networks; (iv) Superconducting quantum (electrical) and mention several areas where network theory arises in circuits; (v) Tensor network states; (vi) Quantum graph quantum computation and contrast this with the concept states, etc. Although the idea of a complex network is not of a complex network. defined in a strict sense, the definition is typically that A network is an abstract representation of relationships of a network which exhibits an emergent property, such (encoded by edges) between units (encoded by nodes) of as a non-trivial distribution in node degree. This is in a complex system. Edges can be directed, i.e. they can contrast to graph theory, which applies graph theory or represent information incoming to or outgoing from a tensor network reasoning to deduce and determine prop- node and, in general, they can be weighted by real num- erties of quantum systems. Here we will focus on topics bers. The number of incoming, outgoing and total edges in quantum systems which are known to be connected is known as incoming, outgoing and total degree of a with the same sort of complexity considered in complex node, respectively. The sum of the corresponding weights networks. defines the incoming, outgoing and total strength of that node, respectively. Networks are often characterized by how node degree and strength are distributed and corre- QUANTUM NETWORKS BASED ON lated. Systems modeled by uncorrelated networks with ENTANGLED STATES homogeneous degree distribution are known as Erdos- Renyi networks, whereas systems with power-law degree To define quantum networks based on entangled states, distribution are known as scale-free networks. We refer let us start from the state of each ith qubit, written to [16, 54] for reviews of network concepts and models. without loss of generality, as

The use of various aspects of graph and network the- −iθi ψi = cos(αi) 0 + e sin(αI ) 1 , (1) ory can be found in all aspects of quantum theory, yet | i | i | i not all networks are complex. The commonly considered with 0 and 1 the preferred or ‘computational’ basis. networks include (i) Quantum spins arranged on graphs; The qubit| i is in| i a pure, coherent superposition of the two 3 basis states and any measurement in this same basis will system size, it is possible to control the number and type cause the state to collapse onto 0 or 1 , with probability of subgraphs present in a quantum network of N nodes. 2 2 | i | i cos (αi) and sin (αI ), respectively. Let us consider a This is useful to create special multipartite states, such as quantum system with two qubits, i.e. i = 1 and 2. The the Greenberger-Horne-Zeilinger state which exhibits non- basis of this system is given by the so called, tensor classical correlations [55]. The striking result is that it is product, of the two basis states: 00 , 01 , 10 and 11 . possible to obtain, with probability approaching unity, a If the two qubits are not entangled,| i i.e.| theiri | i states| arei quantum state with the topology of any finite subgraph independent from each other, then the state of the overall for N approaching infinity and z = 2. − system can be written as e.g. ψ12 = ψ1 ψ2 , whereas this is not possible if the two| qubitsi | arei ⊗ entangled. | i A This bridge between complex network theory and quan- generalization of this description to the case of mixed tum theory provides a powerful tool to investigate the states is obtained in terms of the non-negative density critical properties of a quantum system. For instance, in matrix ρ; a unit trace Hermitian operator representing the case of regular lattices, it has been shown that the the state of the system as an ensemble of (unknown) pure probability popt to establish a perfect quantum channel states. between the nodes can be mapped to the probability of dis- Instead of distributing entanglement on regular graphs, tributing links among each pair of nodes in the lattice [6], such as uniform lattices typically studied in condensed a scenario that can be studied using the well-established matter physics, it has been shown that it is possible to bond-percolation theory from statistical physics. This tune the amount of entanglement between two nodes result allows one to calculate the critical probability above in such a way that it equals the probability to have a which the system will exhibit an infinite connected clus- link in (classical) Erdos-Renyi graphs [6]. Such random ter and, in the case of qubits, it has been shown that graphs can be defined by the family of networks G(N, p), the probability of having an entangled path with infi- where N is the number of nodes and p the probability nite length—i.e., an infinite sequence of entangled states to find a link between any two nodes. The probability connecting an infinite number of qubits—is unity. How- scales with the size of the network following a power ever, for product states this probability is zero, denoting law p N −z, with z 0. In classical network theory, the existence of a sharp transition between these two ∝ ≥ there exists a critical value for the probability pc(N) for scenarios. However, local measurements based on this ap- which, if p > pc(N) a given subgraph of n nodes and l proach, called classical entanglement percolation (CEP), links has higher probability to be observed. The classical are not optimal, in general, to generate maximally en- result is that this critical probability scales with N as tangled states: CEP is not even asymptotically optimal −n/l pc(N) N . for two-dimensional lattices and new quantum protocols Acin,∝ Cirac, and Lewenstein [6] formulated an elegant based on quantum entanglement percolation have to be extension of this picture to the quantum realm by replac- used instead [6]. A novel critical phenomenon, defining ing each link with an entangled pair of particles, where an entanglement phase transition, emerges from this new the probability pi,j = p that the link exists between nodes strategy, where the critical parameter is the degree of i and j is substituted by a quantum state ρi,j := ρ of entanglement required to be distributed in order to es- two qubits, one at each node (see Fig.2). One can build tablish a quantum channel with probability that does a quantum network where each node consists of N 1 not decay exponentially with the size of the system, at qubits which are entangled, in pairs, with qubits of other− variance with CEP. This type of enhancement with re- nodes. However, in this case, although the connections spect to the classical case has been reported for different are identical and pure they encode non-maximally entan- network topologies, such as Erdos-Renyi, scale-free and gled pairs. For pure states of qubits the density matrix is small-world networks [33]. ρ = φ φ , with | i h | Unexpected quantum effects emerging from network 1 p effects have been reported. Cardillo et al. show that φ = 2 p 00 + √p 11 . (2) | i √2 − | i | i nodes which store the largest amount of information are the ones with intermediate connectivity and not the hubs, Here, 0 p 1 quantifies the entanglement of links and breaking down the usual hierarchical picture of classical ≤ ≤ the state of the overall quantum random graph can be networks [56]. More recently, Carvacho et al. measured denoted by G(N, p) . If each link, i.e. each entangled the emergence of special quantum correlations, named | i pair, attempts to convert its state to the maximally entan- non-bilocal, correlating distant qubits by means of several gled one (p = 1/2) through local operations and classical intermediate, typically independent, sources, and provid- communication (LOCC), the optimal probability of suc- ing evidence for violation of local causality in a quantum cessful conversion is exactly p. It follows that the fraction network [57]. of existing entangled states converted to maximally entangled ones by LOCC corresponds to the probability of The static entangled states providing the network con- having a link between nodes in the corresponding clas- nectivity described here, will be replaced in the next sical random network [6]. By varying the value of the section by dynamical processes on networked quantum parameter z, i.e. how the critical probability scales with systems. 4

networks. Recently, Faccin et al. have analytically solved a model which shed light on some key differences between stochastic and quantum walks on complex networks [7]. These differences push forward a general understanding which can lead to a theory explaining novel complex fea- A S B tures in quantum systems. Quantum walks on complex networks represent both a practical model of transport [70] as well as an interesting FIG. 2. Entanglement connecting distant network stage of comparison between the quantum and stochastic nodes. A pair of distant cavities in a quantum communi- cases. As a closed quantum system exhibits fluctuations cation network are driven by a shared squeezed light source S. in the probabilities in time, typically a long time average In the systems steady state the two atoms A and B entangle, is considered. Physically, this is the best approximation forming a network edge [34]. one can hope for, provided that there is no knowledge of when the walk started. In this case, the probability to find a quantum walker in the i-th node is given by QUANTUM NETWORKS BASED ON PHYSICAL Z T CONNECTIVITY 1 2 p(i) = lim dt i Ut 0 , (3) T →∞ T 0 | h | | i | Another wide area where network concepts have found where 0 is the initial state and U = e−iQt is the unitary applicability consists of quantum systems physically in- t evolution| i operator defined by the quantum generator Q. terconnected, such as atoms or superconducting quantum Interference between subspaces of different energy van- electronics [35–40]. These types of systems provide fer- ish in the long time average so we obtain an expression for tile ground where quantum algorithms are tested [58–62] the probability (P ) in terms of the energy eigen-space and quantum information transport systems are studied Q i projectors Π of the Hamiltonian H , [63–65]. Typical modeling approaches are based on so j Q called ‘quantum walks’ on complex networks, with recent X (PQ)i = i Πjρ(0)Πj i . (4) studies showing that quantum information tasks, typi- h | | i cally designed for simple topologies, retain performance in j very disordered structures [66]. Stochastic (non-quantum) P k k walks are also a central model in complex network theory— Here Πj = k φj φj projects onto the subspace | i h | k spanned by the eigenvalues φ of HQ corresponding see the review [67]. | j i Any quantum process can be viewed as a single particle to the same eigenvalue λj. walk on a graph. Single-particle quantum walks represent Quantum-enhanced page-ranking. The non- a universal model of quantum computation—-meaning symmetric adjacency matrix representing the directed that any algorithm for a quantum computer can be trans- connectivity of the World Wide Web, a.k.a. the Google lated into a quantum walk on a graph—and, additionally, matrix G, satisfies the Perron-Frobenius theorem [73] and quantum walks have been widely studied in the realm of hence there is a maximal eigenvalue corresponding to an quantum search on graphs, in both continuous and dis- eigenvector of positive entries Gp = p. The eigenvector crete time via coined walks (see e.g. [58–61]—in particular p corresponds to the limiting distribution of occupation the graph optimality results [66]). The computational probabilities of a random web surfer—it represents a advantages of quantum versus stochastic random walk unique attractor for the dynamics independently of the based algorithms has attracted wide interest with typical initial state. The vector p is known as the Page-Rank.[74] focus being on general graphs which consequently do not Several recent studies embed G into a quantum system exhibit complex features. However, many works have and consider quantum versions of Google’s Page-Rank [8– compared properties of stochastic [68, 69] and quantum 10]. Garnerone et al. [75] relied on an adiabatic quantum random walks [63–65] on complex networks [70]. algorithm to compute the Page-Rank of a given directed Network topology has further been shown as a means network, whereas Burillo et al. [11] rely on a mixture of to direct transport by adding complex numbers—while unitary and dissipative evolution to define a ranking that maintaining Hermiticity—to the networks adjacency ma- converges faster than classical PageRank. trix in ‘chiral quantum walks’ [65, 71, 72] (note that chiral The page-ranking vector ~p is an eigenvector of I G walks were realized experimentally in [12]). Open system corresponding to the zero eigenvalue (the lowest). This− walks which mix stochastic and quantum effects in ‘open’ fact leads to a definition of a Hermitian operator which evolutions [64] have aided in the study of quantum ef- can play the role of a Hamiltonian, defined as: fects in biological exciton transport (again, modeled as a quantum walk) and developments in a quantum version of hp = (I G)†(I G), (5) Google’s PageRank [8–10] has been seen, providing a prac- − − tical solution to overcome the degeneracy issues affecting though highly non-local, its ground state represents the the classical version and enhancing node ranking in large target Page-Rank which could be found by adiabatic 5 quantum annealing into the ground state. Using a quan- order to quantize the Markov chain defined by the Google tum computer to accelerate the calculation of various matrix G of N nodes, one introduces a Hilbert space network properties has been considered widely [58–61]. = span i 1 j 2 , i, j [0,N] and the superposition of As Page-Rank relies on finding the vector corresponding outgoingH edges{| i | fromi node∈ i: } to the lowest eigenvalue of the Google matrix, the adia- X p batic algorithm opens the door up to accelerate network ψi = i Gki k (8) | i | i1 ⊗ | i2 calculations using quantum computers. k Directing transport by symmetry breaking in and chiral walks. Chiral quantum walks, introduced by X Π = ψk ψk . (9) Zimboráset al. in [65] and realized experimentally in [12], | i h | append complex numbers to the adjacency matrix (playing k the role of the system Hamiltonian) while still maintaining Each step of the quantum walk U is defined by a coin flip the Hermitian property [12, 65, 71, 72]. These complex 2Π 1 and a swap operation S which ensures unitarity [8] phases in many cases do not affect transfer probabilities: − the theory explaining this finding was developed in [12, 65], without relying on approximations or averaging. The case U = S(2Π 1) (10) of open systems has been investigated as well in [65]. In − the scenarios where the addition of complex phases affects the swap operator is S = P ij ji . ij | i h | transfer probabilities, the underlying system breaks time- In the case of quantum Page-Rank, this is set to the reversal symmetry and, consequently, the probability flow instantaneous probability P (i, t) of finding the walker into the quantum system is biased. This fact enables at node i at the time-step t. To obtain a fixed value directed state transfer without requiring a biased (or non- for the quantum Page-Rank a time average is calculated local) distribution in the initial states, or coupling to an as long as with its variance as a measure for quantum environment. fluctuations. When the underlying graph is bipartite (e.g. a graph Another approach [11] involves defining a Markov quan- whose vertices can be divided into two disjoint sets such as tum evolution similar to Eq. (7), with a tuning parameter a square lattice), time-reversal symmetry in the transport α: probabilities can not be broken. Transport suppression " # is indeed possible however [65]. Bipartite graphs include X † 1 † ρ˙ = i(1 α)[H, ρ]+α LkρL L Lk, ρ (11) − − k − 2{ k } trees, linear chains and generally, graphs with only even k cycles. These results point to a subtle interplay between the topology of the underlying graph, giving rise to a new where the Hamiltonian H is the symmeterized adjacency p challenge for dynamical control of probability transfer matrix and Lk = Lij = Gij i i represent the jump | i h | when considering walks on complex networks [12, 65, 71, operators which consider the directness of the network. 72]. With this definition, for values of α (0, 1], a stationary state is guaranteed. In this case, for∈α = 0 we revert to Open quantum walks. The area of open quan- the unitary evolution while for α = 1 we revert to the tum systems [76] studies noise and its effects in quantum stochastic case. systems. The adiabatic version of Page-Rank [9] uses a The authors [8, 10, 11] show how this definition of quantum stochastic quantum walk as proposed by [64] quantum Page-Rank resolves problems of degeneracy in (see also [77, 78] for studies on open walks). the classical Page-Rank definition, enhances the impor- Quantum stochastic walks are defined by a quantum tance of secondary hubs and, for certain values of α, the walk undergoing dissipative dynamics. The latter follows algorithm exhibits faster convergence. the quantum master equation in the Lindbladian form: L ρ˙ = [ρ] (6) TOWARDS UNIFIED ANALYSIS OF NETWORK X † 1 n † o COMPLEXITY = i[H, ρ] + LkρL L Lk, ρ (7) − k − 2 k k The interaction between network science and quantum where Lk represents a jump operator while [ , ] and , information science has led to the development of theoret- are commutator and anti-commutator respectively.· · {· ·} ical and computational tools that benefitted from both The network topology is embedded by choosing H equal fields. On the one hand, quantum-inspired tools, such to the adjacency matrix of the symmetrized network and as information entropies and quantum distance measures, p Lk = Lij = Gij i j . In this picture, node ranking is have been successfully applied to practical problems con- defined by an activity| i h | vector α computed at the steady cerning classical complex networks [32]. On the other state ρss. hand, classical network descriptors have been ported to Paparo et al. [8, 10] introduced a Szegedy type of the quantum realm to gain better insights about the struc- Markov chain quantization [79] of the random walk. In ture and the dynamics of networked quantum systems [13]. 6

By exploiting the eigen-decomposition of the Laplacian matrix, it can be shown that this entropy corresponds to Q the Shannon entropy of the eigenvalue spectrum of ρ. This uan tu entropy has been generalized to the case of multilayer m systems [87], composite networks where units exhibit diﬀerent types of relationships that are generally modeled as diﬀerent layers (see [17, 18, 88] for a thorough review). It has been recently shown that the von Neumann entropy calculated from the rescaled Laplacian does ic st not satisfy the sub-additivity property in some circum- a

h r stances [32, 41]. This undesirable feature can be addressed c

o a l t by means of a more grounded deﬁnition [32], whose ratio-

i S m nale is to measure the entropy of a network by exploiting i S how information diffuses through its topology. Informa- al Spectr tion diffusion in this context is governed by the equation N X ψ˙i(t) = Ljiψj(t), (13) − j=1 FIG. 3. Known mappings between quantum and stochastic generators. Here G = (V,E) is a graph with with ψi(t) the amount of information in node i at time t. adjacency matrix A, D is a diagonal matrix of node degrees. The solution of this diffusion equation is given, in vector This yields the graph Laplacian L = D − A, and hence, the notation, by ψ(t) = exp( Lt)ψ(0), whose normalized −1 stochastic walk generator LS = LD , from this a similarity propagator is used to define− the density matrix as −1/2 −1/2 transform results in LQ = D LD , which generates a valid quantum walk and exhibits several interesting connec- e−τL > ρ = , (14) tions to the classical case. The mapping L −→ L L preserves Tr(e−τL) the lowest 0 energy ground state, opening the door for adiabatic quantum annealing which solves computational problems where time plays the role of a resolution parameter allow- by evolving a system into its ground state. ing one to probe entropy at different scales [32]. A similar approach, involving a modified Laplacian matrix, has been recently used for revealing the mesoscale structure The cross-pollination between the two fields—including, of complex directed networks [89, 90]. among others, quantum statistics for modeling the dynam- This quantum-inspired framework provides a power- ics of classical networks and their geometry [80, 81] (see ful basis to develop an information theory of complex also Ref. [82] and references therein)—is still ongoing with networks, with direct applications in classical network vibrant future research opportunities. Here we briefly re- science, such as system comparison. view the advances concerning quantum-inspired entropic Comparing classical networks. A known problem measures for networks and network-inspired measures for in network science is to compare two networks, without quantum systems. relying on a specific subset of indicators. Network infor- Information entropy of classical networks. His- mation entropy allows one to introduce relative entropies torically, the concept of entropy has been successfully such as the Kullback-Leibler divergence, to compare two used to quantify the complexity of many systems [83, 84]. networks with density matrices ρ and σ respectively: Recently, the possibility of using quantum entropy and (ρ σ) = Tr[ρ(log ρ log σ)]. (15) other quantum information theoretical measures has been D || 2 − 2 explored by the community of network scientists. By exploiting the well-known classical result that the For classical complex networks, von Neumann’s entropy minimization of Kullback-Leibler divergence between a has been applied over one decade ago [85]. The combina- reference distribution and its parametric model corre- torial Laplacian matrix L, obtained from the adjacency sponds to the maximization of the likelihood, it has been matrix representing the network, is rescaled by the num- shown that in a network context this allows one to define ber of edges in the network. The normalization of the the network log-likelihood by matrix L guarantees that the corresponding eigenvalues log (Θ) = Tr[ρ log σ(Θ)]. (16) are non-negative and sum up to 1—in order to be inter- 2 L 2 preted as probabilities [86]—and some other properties The introduction of network likelihood opens the door which makes the resulting object similar to a quantum to a variety of applications in statistical inference and density matrix ρ. Network entropy is defined according model selection, based on concepts such as the Fisher to von Neumann quantum entropy as information matrix, Akaike and Bayesian information criteria, and minimum description length, to cite some of S(ρ) = Tr(ρ log ρ). (12) them [32]. − 2 7

This new framework has been used to compare networks The (dis)similarity between networks has also been for several purposes. For instance, in the case of pairs of quantified by using a combination of the classical Jensen- networks, the graphs are first merged by connecting each Shannon distance and the concept of network node dis- node from one network to any other node in the other persion, measuring the heterogeneity of a graph in terms network. Successively, continuous-time quantum walks of connectivity distance among its nodes [42]. are used to explore the composite system and the quantum Jensen-Shannon divergence between the evolution of two Degree distribution of quantum networks. walks is calculated. This divergence, that is a measure Nodes in a complex network have different roles and of (dis)similarity, is shown to be maximum when the two their influence on system dynamics can vary widely de- original networks are isomorphic [60]. pending on their topological characteristics. One of the The square root of quantum Jensen-Shannon divergence simpler (and widely applied) characteristics is the degree has the nice property of defining a metric, allowing one centrality, defined as the number of edges incident on to define a distance between networks. If ρ and σ are two that node. Many real world networks have been found density matrices corresponding to two networks with N to follow a widely heterogeneous distribution of degree nodes, their Jensen-Shannon divergence is defined by values [94]. Several models, based on mechanisms like preferential attachment [15], fitness [95] or constrained 1 1 JS(ρ σ) = KL(ρ µ) + KL(σ µ) (17) random wiring [54], to mention some of them, were de- D || 2D || 2D || veloped to reproduce degree distributions commonly ob- 1 = S(µ) [S(ρ) + S(σ)], (18) served in empirical systems. Despite the complexity of − 2 the linking pattern, the degree distribution of a network that is the difference between the entropy of the mix- affects in a simple way the ongoing dynamics. In fact, it 1 ture µ = 2 (ρ + σ) and the semi-sum of the entropies of can be shown that the probability of finding a memoryless the original systems. In the context of multilayer sys- random walker at a given node of a symmetric network tems, this measure has been used to quantify the distance at the stationary state, is just proportional to the degree between layers of a multiplex network, cluster and aggre- of such node [68]. gate them appropriately in order to reduce its structural In [7] the authors consider the relationship between complexity [41]. the stochastic and the quantum version of such processes, These ideas have quickly found direct applications in with the ultimate goal of shedding light on the meaning of biology. In genetic molecular systems, such as the ones degree centrality in the case of quantum networks. They described by gene-protein interactions, layers might en- consider a stochastic evolution governed by the Laplacian −1 code different relationships among molecules—functional, matrix LS = D , the stochastic generator that charac- e.g. additive, suppressive and other types of association, terizes classicalL random walk dynamics and leads to an or physical, e.g. co-localization or direct interaction. The occupation probability proportional to node degree. In information-theoretic framework described here allowed the quantum version, an hermitian generator is required to show that such systems exhibit a certain level of re- and the authors proposed the symmetric Laplacian ma- − 1 − 1 dundancy, larger than the one observed in man-made trix LQ = D 2 D 2 , generating a valid quantum walk systems [41], suggesting the existence of biological mech- that, however, doesL not lead to a stationary state, mak- anisms devoted to maximize diversity of interactions. ing difficult a direct comparison between classical and In computational neuroscience studies, the connectome quantum versions of the dynamics. A common and useful of the nematode C. elegans—one of the most studied in workaround to this issue is to average the occupation the field because of its small size, with approximately 300 probability over time, as in Eq. (4). neuronal cells—has been mapped to a multiplex network The generators of the two dynamics are spectrally where layers encode synaptic, gap junction, and neuro- similar (see Fig.3) and share the same eigenvalues, modulator interactions. Here, the analysis of reducibility while the eigenvectors are related by the transformation C − 1 Q revealed that the monoamine networks have a unique φi = D 2 φi . As a consequence, if the system is in the structure, with information complementary to that pro- ground state the average probability to find the walker vided by neuropeptide networks [91]. The same analysis, on a node will be the same as in the classic case, which applied to a multilayer functional representation of the will depend solely on the degree of each node. For the human brain revealed, quantitatively, the importance of cases in which the system is not in the ground state, it is not disregarding or aggregating connectivity information possible to define a quantumness measure for clinical classification of healthy and schizophrenic sub- Q Q ε = 1 φ ρ0 φ , jects [92]. − h 0 | | 0 i In another application the Jensen-Shannon distance describing how far from the classical case the probability between layers of a multiplex system has been used to distribution of the quantum walker will be. In the case identify community-based associations in the human mi- of uniformly distributed initial state ρ0, this provides a crobiome [32], where the microbial network corresponding measure for the heterogeneity of the degree distribution to each body site is represented by a layer in a multiplex of a quantum network. network, in perfect agreement with biological expectation [93]. Mesoscale organization of quantum systems. 8

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10 6 9 7 8 (a)LHCIIcomplex (b)Monomericsubunits (c)Networkrepresentation

1 1 1 14 2 14 2 14 2 13 3 13 3 13 3

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11 5 11 5 11 5

10 6 10 6 10 6 9 7 9 7 9 7 8 8 8 (d) Transport; t 0 (e) Transport; t (f) Fidelity; t → → ∞ → ∞ FIG. 4. Community Detection in a light harvesting network (LHCII) [13]. (a) Monomeric subunit of the LHCII complex with pigments Chl-a (red) and Chl-b (green) packed in the protein matrix (gray). (b) Schematic representation of Chl-a and Chl-b in the monomeric subunit, here the labeling follows the usual nomenclature (b601, a602... ). (c) Network representation of the pigments in circular layout, colors represent the typical partitioning of the pigments into communities. The widths of the links represent the strength of the couplings |Hij | between nodes. Here the labels maintain only the ordering (b601→1, a602→2,... ). (d, e, f) Partition of LHCII applying the quantum community detection algorithm in [13]. Transport (for long times) and Fidelity approaches give similar results while short time transport is closer to classical community detection. Link width denotes the pairwise closeness of the nodes.

Community detection, and in general mesoscopic struc- was proposed in [101]. Here a node affinity measure based ture detection, has been widely studied in the literature on the response of node population density to link failure of classical complex networks [96, 97]. While the defini- was given. If the population on two nodes changes in a tion of community “a subset of nodes tightly connected similar manner after link removal, they are more likely to compared to what is expected” is in general ill defined belong to the same community. and part of an ongoing debate, the number of proposed A magnetic Laplacian, where a magnetic field is ex- algorithms is incredibly high and still growing. pected to traverse all cycles in the network, was used The cross-pollination of community detection with in [89]. With an approach similar to chiral walks, pre- quantum mechanics is in two levels. On the one hand, viously described, the symmetric Laplacian is amended chronologically, the first attempt was to borrow tools with the original link directionality by a phase term e±iθ, from quantum mechanics for applications to classical sys- with θ being a parameter for the method, and used for tems [89, 98–101]. On the other hand an algorithm to find community detection in directed networks, a longstanding communities in complex quantum systems was proposed problem in network science. in [13]. In [98, 99] the authors propose a method for data clus- In the case of quantum systems, partitioning in mod- tering similar to kernel density estimators, in a quantum ular units has been often carried out on the basis of ad framework. The given data points are mapped to a Gaus- hoc considerations. In an effort to extend community sian wave function and, supposing that the latter is an detection to the quantum mechanics realm, Faccin et eigenstate for some time-independent Schrödingherequa- al. [13] introduced several closeness matrices inspired by tion: different quantum quantities. Given the Hamiltonian P H = Hij i j of the quantum system of interest, ij | i h | Hψ = [T + V (x)]ψ = E0ψ , the authors consider a continuous-time random walk on the system topology. The first quantity is energy trans- the minimization of the potential V (x) leads to the desired port, porting to the quantum realm the concept applied clustering. An extension to dynamical quantum systems in several classical algorithms where communities are in- has been introduced in [100]. In this case the expectation terpreted as traps for the dynamical process. In this values of the position operator evolves in time toward the framework, two nodes are considered to be close if, on closer minimum of the potential. This formulation can average, their in-between transport is high. If this av- leverage the acceleration of graphics hardware. erage is computed over a short time period (compared A method based on continuous-time quantum walks to evolution time scales), then the closeness values are 9 proportional to the Hamiltonian terms Hij , providing a rather limited and quantum computing might overcome classical approach to community detection| (see| Fig.4). A such limitations [102, 103]. However, given that such second quantity, also proposed as a closeness measure, is systems are more sensitive to interactions with the envi- related to the average fidelity of the evolving process com- ronment, they are also more exposed to errors than their pared to the initial state. In this case the localization of classical counterparts. Quantum error-correcting codes eigenstates is the characteristic determining the closeness allow us to store and manipulate quantum information in of two nodes. These methods augment current ad hoc the presence of certain types of noise that, in this context, approaches to partitioning nodes in quantum transport might perturb the quantum system causing effects similar systems with enhanced methods based on community to random failures in classical complex networks. The detection algorithms. development of quantum error correction techniques that make quantum computing and quantum communication possible can not prescind from the study of ‘system re- OUTLOOK IN QUANTUM NETWORK SCIENCE silience’, a topic that found uncountable applications in classical network science [26, 28, 31, 104, 105]. Other Generalization of complex network methods to the types of perturbations that are natural for classical sys- quantum setting represents a foundational advancement tems, such as targeted attacks of network hubs [26] or required to understand complexity in physical systems. cascade-based attacks [106], still have no clear quantum These methods represent a change of paradigm which counterpart and their study, from both theoretical and bring several road blocks that must be faced. Centrally, experimental perspectives, will play a key role in the devel- the application domain of complex network methods to opment of a quantum Internet [5]. In fact, it is tantalizing quantum physics must be expanded, whereas studies in to think about how quantum hubs should be protected the other direction, i.e. where methods from the quantum by the quantum counterpart of typical denial of service domain have now been ported to network science. attacks. From a foundational perspective, as networks necessar- ily represent physical systems, such systems are inherently Continued advances in the theory of complexity in net- governed by the laws of information physics. In fact, a re- worked quantum systems will help address the challenges search line is emerging now that seeks to quantify, in terms faced as quantum technologies scale up to commercially of implicit information processing capacity, networked sys- feasible products. Work towards a quantum theory of tems, with several applications to social, technological complex networked systems is already opening up novel and biological systems [32, 41, 42, 91, 92]. Although this avenues when facing contemporary complexity challenges. interesting direction seems promising, yet it is compara- bly in its infancy, whereas it is still not known how to Author Contributions. JDB, MF and MDD designed and generalize classical concepts of complexity science to the wrote this review. quantum domain. Another relevant research direction, crucial for applica- Competing Financial Interests. The authors declare no tions in classical network science, concerns the interplay competing financial or non-financial interests. between structure and dynamics, which is almost entirely Data Availability. No dataset were generated or analysed unclear in the case of quantum networks. Although scale- during the current study. free networks have been considered in the quantum set- Acknowledgements. JDB acknowledges the Foundational ting [33], the result is an—albeit interesting—toy model Questions Institute (FQXi, under grant FQXi-RFP3-1322) for with theoretical predictions to be verified experimentally. financial support. MF acknowledges the MOVE-IN fellowship Therefore, further advancement along this track is of cen- program for financial support. MDD acknowledges financial tral interest, because it might play a fundamental role in support from the Spanish program Juan de la Cierva (IJCI- quantum enhanced technology and could lead to experi- 2014-20225). The authors thank Alex Arenas and Leonie ments devoted to test cross-disciplinary ideas in quantum Mueck for useful feedback, and the Institute for Quantum and complexity science [34]. Computing at the University of Waterloo and the Perimeter The quest for a theoretical foundation for quantum Institute for Theoretical Physics for funding and allowing us to complex networks might have a deep impact in informa- organize the first workshop on the intersection of these topics. tion and communication technology. While information Diagrams are courtesy of Lusa Zheglova (illustrator). processing in classical systems is well controlled, it is also

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