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Complex Networks from Classical to Quantum

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Complex Networks from Classical to Quantum Complex Networks from Classical to Quantum Jacob Biamonte,1, ∗ Mauro Faccin,2, y and Manlio De Domenico3, z 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 y [email protected] nearly all aspects of quantum information and computa- z [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 i = cos(αi) 0 + e sin(αI ) 1 ; (1) ory can be found in all aspects of quantum theory, yet j i j i j 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 qubitj i is inj 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 j i j 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,j i i.e.j theiri j i statesj 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.
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