Quantum percolation in complex networks Author: Andr´esLaverde Mar´ın. Facultat de F´ısica, Universitat de Barcelona, Diagonal 645, 08028 Barcelona, Spain.∗ Advisor: Mari´anBogu~n´a Abstract: Complex quantum networks will be essential in the future either for the distribution of quantum information (telecommunications) or for studying complex quantical linked systems among others. Complex networks have a wide variety of properties that will help us understand how complex quantum networks behave. Here we will study how this complex networks perform using local quantum transformations at the nodes. We will focus specifically on the Internet network and we will study its behaviour from a current and quantum perspective. I. INTRODUCTION II. BOND PERCOLATION THEORY The objective of modeling complex networks is to un- A large part of the current systems, such as the Inter- derstand the systems they represent without having to net network, neural network or social networks amongst know its exact structure, i.e. only by knowing their many others, can be modeled with graph theory, where statistical properties we can understand their behavior. the system is described by a set of N nodes and L edges, Once measured and quantified these properties, we must each one of these links one node to another. Within com- transform these results into conclusions on how the sys- plex networks we can find both symmetric and antisym- tem can work. The phenomenon of percolation describes, metric networks, i.e. a square network is a totally ran- in a simple way, the criticality of some complex sys- dom network and a scientist connections lattice is totally tems, such as physical, biological and social phenomena. antisymmetric. Due to the great variety of systems that Canonical studies of this systems are within statistical can be described through a complex network, a line of mechanics and are fundamentally associated with phase research has been generated, which includes from math- transitions[3]. ematical to theoretical physical models in order to reveal the principles that govern these lattices. Knowing the Usually the percolation process is understood as the properties of these networks allows us to know how the movement of a fluid in a porous material. This defini- system will behave; for instance, if we study the internet tion can help us to understand the process of percolation. network we can see how it will behave in case of a global We can imagine this process as a fluid that can flow be- internet repeaters failure. tween two nodes with a probability p 2 [0; 1]. When the probability is p = 1, we will have all the edges con- In the future, we hope to develop communication sys- nected, therefore, the liquid can flow through the whole tems based on quantum physics, such as the quantum network. When the probability is p = 0 we will have all internet[1]. Quantum networks offer new possibilities and the edges disconnected and the liquid will not be able to phenomena, which we can compare with their classical flow between nodes as there will be no edges connecting equivalent. One of the new advantages is that complex it. Fig.(1). quantum network offers us the possibility of sending en- In our case, we study real networks, with a number of crypted messages securely, without anyone being able to fixed and connected nodes in a concrete way. To under- intercept the message and leaving a trace. stand how the network behaves we can imagine that each one of the edges has a probability p of being open and a The objective of this study is the analysis of a same probability 1 − p of being closed, this way of understand- network in a classic and quantum way. To compare ing the network is knowning as bond percolation [3]. The them, we can understand what advantages or disadvan- percolation threshold pc is a concentration p at which a tages these new complex quantum networks bring us macroscopic connected component (or giant component over the already known complex networks. We will focus (GC)) emerges [3], see Fig.(1). mainly on the Internet network, although, it does not imply that this comparative study can not be extended Below pc, the network is composed of a large number of to more than one network. small disconnected components with a size distribution approaching a power law as p tends to pc. Above pc, the giant component grows and becomes equal to the entire system when p = 1. For finite systems the giant component is defined as the largest connected component ∗Electronic address: [email protected] of the system. Quantum percolation in complex networks Andr´esLaverde Mar´ın applying this transformation to each edge of the classical network, whether optimally or not, is known as classical entanglement percolation [4]. So far we have only made a quantum transformation allowing the network to be maximally entangled. But we can obtain new characteristics if we apply other local quantum transformations that transform the geometry of the network, it must be taken into account that this change in geometry involves a change in the percolation FIG. 1: Square lattice. Each edge is represented with white threshold. [4]. To understand this new quantum prop- squares and black squares. The white one is disconected and erty first we need to comprehend the entaglement swap- black one is connected.We observe the emergence of GC with ping. Let us now consider a node (c) of our network, in pc = 0:59, with p < pc we only see disconnected islands. which we make a Bell measurement in two qubits that . belong to a jΨi state and each edge is shared with a dif- ferent node (a and b). Once this operation is done, the III. MODELLING THE NETWORK AS A nodes (a and b) of the central node (c) are untangled QUANTUM NETWORK and a tangle is created between nodes a and b [7] , see Fig.(3). Thanks to this type of quantum transformation we can converte a honeycomb lattice into a triangular We can consider this modelling, as in Fig.(2), where lattice, ensuring that no information is lost [4]. In this each edge correspond to pairs of pure entanglement state, type of transformation we can see that the percolation mathematically we can express it as jΨi⊗2, where the p p threshold is lower for the triangular lattice than for the wave function is defined as jΨi = λ j00i + λ j11i, p0 p 1 honeycomb lattice [3], which represents an improvement representing two qubit states, where λ ≥ λ ≥ 0 0 1 over the previous lattice. are the Schmidt coefficients. Each of these partially en- tangled states can be converted into a singlet (maximum entangled state) with the singlet conversion probability p = min[1; 2(1 − λ0)] [5] . With this transformation we guarantee that as we are in a maximally linked state, no information is lost between states [5].However, if the two nodes share two copies of jΨi, the probability that at least one of the two states is converted into a singlet 2 state is p2 = 2p − p , where p is the singlet conversion probability, previously defined. FIG. 3: Entangelment swapping . IV. INTERNET NETWORK Many times we believe that the WWW and the Internet network are the same, which is not true. They are very different networks from each other. In a general way, we can define the Internet network as a worldwide FIG. 2: Equivalence between a classic and a quantum node. network of computers interconnected by cables, this On the left we see a node with three classic connections, where allows an exchange of information between them. The there are three neighbours connected with a simple edge. On vertices of this network are: the right we can see these same neighbours connected quan- tum, i.e. with two links per neighbor. Figure from [4] - The computers of the users. - The servers (computers or programs providing a To apply the classic percolation in a quantum complex network service). network, we need to consider that the network now has - The routers that arrange traffic across the Internet. the same number of nodes, but the edges that join the nodes are doubled, see Fig.(2), where each edge corre- Connections are undirected, and traffic (including sponds to the partially entanglement state jΨi. A better its direction) changes constantly [10]. In table I we way to understand this, is to perceive it with the same summarize the main characteristics of the network, number of nodes and edges, but now the probability that which will be studied below. a link is open is no longer p but p2 as the links are two copies of jΨi.We can see this in Fig.(2). The process of Treball de Fi de Grau 2 Barcelona, June 2018 Quantum percolation in complex networks Andr´esLaverde Mar´ın N L hci hki Vertex type Format tendecy of nodes to connect to peers in terms of the de- Internet AS 23752 65048 0.20 4.92 AS Undirect gree. On the right of Fig.(5) we can see knn(k) which represents the average nearest neighbours degree of the TABLE I: Main characteristics of the internet network, Num- internet network, which decreases in function of the de- ber of nodes N, number of edges L, Clustering coefficient hci, grees. This presents a disassortative pattern of degree, Average degree hki , Vertex type, Network format. i.e the internet network connects in such a way that small degree nodes tend to connect with large degree nodes, we To understand the topology of this network, we can can see this in Fig.(4), we see that the smaller nodes (the start by studying its cumulative degree distribution Eq.