A Poisson Model for Entanglement Optimization in the Quantum Internet
AA Poisson Model for Entanglement Optimization in the QuantumInternet
Laszlo Gyongyosi ∗† Sandor Imre ‡ Abstract
We define a nature-inspired model for entanglement optimization in the quantum Internet.The optimization model aims to maximize the entanglement fidelity and relative entropy ofentanglement for the entangled connections of the entangled network structure of the quan-tum Internet. The cost functions are subject of a minimization defined to cover and integratethe physical attributes of entanglement transmission, purification, and storage of entanglementin quantum memories. The method can be implemented with low complexity that allows astraightforward application in the quantum Internet and quantum networking scenarios.
Quantum entanglement and the entangled network structure serve as fundamental concepts of thequantum Internet [1, 3–6], long-distance quantum networks and future quantum communications[29–44]. Since the no-cloning theorem makes it impossible to use the “copy-and-resend” mechanismsof traditional repeaters [7, 45], in a quantum Internet scenario the quantum repeaters have totransmit correlations in a different way [1–5]. In the entangled network structure of the quantumInternet, the main task of quantum repeaters is to distribute quantum entanglement between distantpoints that will then serve as a fundamental base resource for quantum teleportation and otherquantum protocols [1]. Since in an experimental scenario [15–21] the quantum links between nodesare noisy and entanglement fidelity decreases as hop distance increases, entanglement purification isapplied to improve the entanglement fidelity between nodes [1, 3–6]. Quantum nodes also performinternal quantum error correction that is a requirement for reliability and storage in quantummemories [1, 5, 6, 8, 50–60]. Both entanglement purification and quantum error correction steps inlocal nodes are high-cost tasks that require significant minimization [1, 3–6, 15, 16, 19–32, 47–49, 61–84].The shared entangled states between nodes form entangled connections. Significant attributesof these entangled connections are entanglement fidelity [1,5,6], and correlation in terms of relativeentropy entanglement [76, 77]. Entanglement fidelity is a crucial parameter. It serves as the pri-mary objective function in our model, which is a subject of maximization. Maximizing the relative ∗ School of Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ, U.K., andDepartment of Networked Systems and Services, Budapest University of Technology and Economics, 1117 Budapest,Hungary, and MTA-BME Information Systems Research Group, Hungarian Academy of Sciences, 1051 Budapest,Hungary. † Parts of this work were presented in conference proceedings [8]. ‡ Department of Networked Systems and Services, Budapest University of Technology and Economics, 1117 Bu-dapest, Hungary. a r X i v : . [ qu a n t - ph ] S e p ntropy of entanglement is the secondary objective function. Minimizing the cost of classical com-munications, which is required by the entanglement optimization method as an auxiliary objectivefunction, is also considered.Besides these attributes, the entangled connections are characterized by the entanglementthroughput that identifies the number of transmittable entangled systems per sec at a particu-lar fidelity. In our model, the nodes are associated with an incoming entanglement throughput [1],that serves as a resource for the nodes to maximize the entanglement fidelity and the relative en-tropy of entanglement. The nodes receive and process the incoming entangled states. Each nodeperforms purification and internal quantum error correction, and it stores the entangled systems inlocal quantum memories. The amount of input entangled systems in a node is therefore connectedto the achievable maximal entanglement fidelity and correlation in the entangled states associatedwith that node. The objective of the proposed model is to reveal this connection and to define aframework for entanglement optimization in the quantum nodes of an arbitrary quantum network.The required input information for the optimization without loss of generality are the number ofnodes, the number of fidelity types of the received entangled states, and the node characteristics.In a realistic setting, these cover the incoming entanglement throughput in a node and the costs ofinternal entanglement purification steps, internal quantum error corrections, and quantum memoryusage.In this work, an optimization framework for quantum networks is defined. The method aimsto maximize the achievable entanglement fidelity and correlation of entangled systems, in parallelwith the minimization of the cost of entanglement purification and quantum error correction stepsin the quantum nodes of the network. The problem model is therefore defined as a multiobjectiveoptimization. This paper aims to provide a model that utilizes the realistic parameters of theinternal mechanisms of the nodes and the physical attributes of entanglement transmission. Theproposed framework integrates the results of quantum Shannon theory, the theory of evolutionarymultiobjective optimization algorithms [9, 10], and the mathematical modeling of seismic wavepropagation [9–14].Inspired by the statistical distribution of seismic events and the modeling of wave propagationsin nature, the model utilizes a Poisson distribution framework to find optimal solutions in theobjective space. In the theory of earthquake analysis and spatial connection theory [9–14], Poissondistributions are crucial in finding new epicenters. Motivated by these findings, a Poisson modelis proposed to find new solutions in the objective space that is defined by the multiobjective opti-mization problem. The solutions in the objective space are represented by epicenters with severallocations around them that also represent solutions in the feasible space [9, 10]. The epicentershave a magnitude and seismic power operators that determine the distributions of the locationsand fitness [9, 10] of locations around the epicenters. Epicenters with low magnitude generate highseismic power in the locations, whereas epicenters with high magnitude generate low seismic powerin the locations. Epicenters are generated randomly in the feasible space, and each epicenter isweighted from which the magnitude and power are derived. By a general assumption, epicenterswith lower magnitude produce more locations because the locations are closer to the epicenter.The locations are placed within a certain magnitude around the epicenters in the feasible space.The optimization framework involves a set of solutions to the Pareto optimal front [9, 10] by com-bining the concept of Pareto dominance and seismic wave propagations. The new epicenters aredetermined by a Poisson distribution in analogue to prediction theory in earthquake models. Themathematical model of epicenters allows us to find new solutions iteratively and to find a global2ptimum. The framework has low complexity that allows an efficient practical implementation tosolve the defined multiobjective optimization problem.The multiobjective optimization problem model considers the fidelity and correlation of entan-glement of entangled states available in the quantum nodes. The resources for the nodes are theincoming entangled states from the quantum links, and the already stored entangled quantum sys-tems in the local quantum memories. In the optimization procedure, both memory consumptionsand environmental effects, such as entanglement purification and quantum error correction steps,are considered to develop the cost functions. In particular, the amount of resource, in terms ofnumber of available entangled systems, is a coefficient that can be improved by increasing the in-coming number of entangled systems, such as the incoming entanglement throughput in a node. Inthe proposed model, the incoming entanglement fidelity is further divided into some classes, whichallows us to differentiate the resources in the nodes with respect to their fidelity types. Therefore,the fidelity type serves as a quality index for the optimization procedure. The optimization aimsto find the optimal incoming entanglement throughput for all nodes that leads to a maximizationof entanglement fidelity and correlation of entangled states with respect to the relative entropy ofentanglement, for all entangled connections in the quantum network.The novel contributions of our manuscript are as follows:1. A nature-inspired, multiobjective optimization framework is conceived for the quantum Inter-net. The model considers the physical attributes of entanglement transmission and quantum mem-ories to provide a realistic setting (realistic objective functions and cost functions). The method fuses the results of quantum Shannon theory and theory of evolutionary multiob-jective optimization algorithms. The model maximizes the entanglement fidelity and relative entropy of entanglement for allentangled connections of the network. It minimizes the cost functions to reduce the costs ofentanglement purification, error correction, and quantum memory usage. The optimization framework allows a low-complexity implementation.
This paper is organized as follows. Section 2 presents the problem statement. Section 3 detailsthe optimization method. Section 4 provides the problem resolution. Section 5 proposes numer-ical evidence. Finally, Section 6 concludes the paper. Supplemental material is included in theAppendix.
The problem to be solved is summarized in Problem 1.
Problem 1
For a given quantum network with N nodes, for all nodes x i , i = 1 , . . . , N , the entan-glement fidelity and relative entropy of entanglement for all entangled connections are maximized,and the cost of optimal purification and quantum error correction and the cost of memory usagefor all nodes are minimized. B F ( x ) be the incoming number of received entangledstates (incoming entanglement throughput) in a given quantum node x , measured in the numberof d -dimensional entangled states per sec at a particular entanglement fidelity F [1, 3, 4].Let N be the number of nodes in the network, and let T be the number of fidelity types F j , j = 1 , . . . , T of the entangled states in the quantum network.Let B jF ( x i ) be the number of incoming entangled states in an i th node x i , i = 1 , . . . , N , fromfidelity type j . In our model, B jF ( x i ) represents the utilizable resources in a particular node x i .Thus, the task is to determine this value for all nodes in the quantum network to maximize thefidelity and relative entropy of shared entanglement for all entangled connections.Let X be an N × T matrix X = (cid:16) B jF ( x i ) (cid:17) N × T . (1)The matrix describes the number of entangled states of each fidelity type for all nodes in thenetwork, B jF ( x i ) ≥ i and j . For a given node x i , let F ( x i ) be the primary objective function that identifies the cumulativeentanglement fidelity (a sum of entanglement fidelities in x i ) after an entanglement purificationP ( x i ) and an optimal quantum error correction C ( x i ) in x i . In our framework, F i ( X ) for a node x i is defined as F i ( X ) = T (cid:88) j =1 T (cid:88) k =1 A ijk ˜ B jF ( x i ) ˜ B kF ( x i ) + T (cid:88) j =1 R ij ˜ B jF ( x i ) + c i , (2)where A ijk is the quadratic regression coefficient, R ij is the simple regression coefficient, c i is aconstant, and ˜ B jF ( x i ) is defined as˜ B jF ( x i ) = B jF ( x i ) + (cid:104) B (cid:105) jF ( x i ) , (3)where (cid:104) B (cid:105) jF ( x i ) is an initialization value for B jF ( x i ) in a particular node x i .Then let E ( D i ( X )) be the secondary objective function that refers to the expected amount ofcumulative relative entropy of entanglement (a sum of relative entropy of entanglement) in node x i , defined as E ( D i ( X )) = T (cid:88) j =1 T (cid:88) k =1 A ∗ ijk ˜ B jF ( x i ) ˜ B kF ( x i )+ T (cid:88) j =1 R ∗ ij ˜ B jF ( x i ) + c ∗ i , (4)where A ∗ ijk , R ∗ ij , and c ∗ i are some regression coefficients, by definition.Therefore, the aim is to find the values of B jF ( x i ) for all i and j in (1), such that F i ( X ) and E ( D i ( X )) are maximized for all i .Assuming that the fidelity of entanglement is dynamically changing and evolves over time, the w j ( x i ) quantum memory coefficient is introduced for the storage of entangled states from the j th fidelity type in a node x i as follows: w j ( x i ) = η j B jF ( x i ) + κ j (cid:104) B (cid:105) jF ( x i ) , (5)4here η j and κ j are coefficients that describe the storage characteristic of entangled states withthe j th fidelity type. The cumulative entanglement fidelity (2) and cumulative relative entropy of entanglement (4) ina particular node x i are associated with a f C (P ( x i )) cost entanglement purification P ( x i ) and a f C (C ( x i )) cost of optimal quantum error correction C ( x i ) in x i , where f C ( · ) is the cost function.Then let C ( X ) be the total cost function for all of the T fidelity types and for all of the N nodes, as follows: C ( X ) = N (cid:88) i =1 f C (P (x i )) + f C (C ( x i ))= N (cid:88) i =1 T (cid:88) i =1 f j B jF ( x i ) , (6)where f j is a total cost of purification and error correction associated with the j th fidelity type ofentangled states.Let F ∗ be a critical fidelity on the received quantum states. The entangled states are thendecomposable into two sets S low and S high with fidelity bounds S low ( F ) and S high ( F ) as S low ( F ) : max ∀ i F i < F ∗ , (7)and S high ( F ) : min ∀ i F i ≥ F ∗ . (8)For the quantum systems of S low , the highest fidelity is below the critical amount F ∗ , and for set S high , the lowest fidelity is at least F ∗ . Then let X S low and X S high identify the set of nodes forwhich condition (7) or (8) holds, respectively.Let S i ( X ) be the cost of quantum memory usage in node x i , defined as S i ( X ) = λ T (cid:88) j =1 α i i B jF ( x i ) , (9)where λ is a constant, α i is a quality coefficient that identifies set (7) or (8) for a given node x i ,and Υ i is the capacity coefficient of the quantum memory.The main components of the network model are depicted in Fig. 1. The optimization problem is as follows. The entanglement fidelity and the relative entropy ofentanglement for all types of fidelity of stored entanglement for all nodes are maximized, whilethe cost of entanglement purification and quantum error correction is minimal, and the memoryusage cost (required storage time) is also minimal. These requirements define a multiobjectiveoptimization problem [9, 10]. 5 (cid:10) jF i
B x high X (cid:43) (cid:35) i x (cid:9) (cid:10) kF j B x (cid:33) low X (cid:43) (cid:35) j x (cid:9) (cid:10) i X (cid:30) (cid:9) (cid:10)(cid:9) (cid:10) i D X (cid:18) (cid:9) (cid:10) j X (cid:30) (cid:9) (cid:10) (cid:9) (cid:10) j D X (cid:18) (cid:35) (cid:35) Figure 1: Illustration of the network model components. The quantum nodes x i and x j are as-sociated with current input values B jF ( x i ) and B lF ( x j ) (blue and green arrows), where j and l identify the fidelity types of received entangled states. The nodes have several entangled connec-tions (depicted by gray lines) in the network. The nodes are associated with subject functions F i ( X ), E ( D i ( X )), and F j ( X ), E ( D j ( X )). The maximum of the received entanglement fidelityin the nodes allows the classification of the nodes to sets X S low and X S high : node x i belongs to set X S low , whereas node x j belongs to set X S high (depicted by dashed frames).Utilizing functions (2) and (4), the function subject of a maximization to yield maximal entan-glement fidelity and maximal relative entropy of entanglement in all nodes of the network is definedvia main objective function G ( X ): G ( X ) = max N (cid:88) i =1 F i ( X ) E ( D i ( X )) . (10)Function G ( X ) should be maximized while cost functions (6) and (9) are minimized via functions F ( N ) and F ( N ): F ( N ) = min C ( X ) = N (cid:88) i =1 T (cid:88) i =1 f j B jF ( x i ) , (11)and F ( N ) = min S ( X ) = N (cid:88) i =1 S i ( X ) , (12)6ith the problem constraints [9, 10] C , C , and C for all i and j . Constraint C is defined as C : ζ ( X ) ≥ γ, (13)where γ is a cumulative lower bound on the required entanglement fidelity for all nodes, while ζ ( X )is ζ ( X ) = N (cid:88) i =1 F i ( X ) , (14)and constraint C is C : X ≤ Λ , (15)where Λ is an upper bound on the total cost function C ( X ), while X isX = N (cid:88) i =1 T (cid:88) i =1 f j B jF ( x i ) . (16)For constraint C , let τ j ( X ) be a differentiation of storage characteristic of entangled states fromthe j th fidelity type: τ j ( X ) = N (cid:88) i =1 ( w j ( x i ) − Ω) , (17)where Ω = (cid:80) Ni =1 w j ( x i ) N . (18)Then, C is defined as C : ν ( X ) ≤ Π , (19)where Π is an upper bound on the storage characteristic of entangled states from the j th fidelitytype, while ν is evaluated via (17) as ν = N (cid:88) j =1 τ j ( X ) . (20) This section defines the Poisson entanglement optimization method, and it is applied to the solutionof the multiobjective optimization problem of Section 2.
The quantum Internet is defined as a complex network model with quantum and classical layersthat involve several optimization criteria and objectives. An optimization problem model of thequantum Internet therefore induces a multiobjective optimization problem model that considersthe special requirements of the environment of the quantum Internet. These requirements coverthe entanglement transmission procedure, processing of quantum entanglement in the quantumnodes, and auxiliary communication through the classical links that support the entangled network7tructure. The quantum transmission procedure models the generation of the entangled quantumnetwork with quantitative and qualitative measures. In this manner, a quantitative measure is therelative entropy of entanglement between the quantum nodes, while the entanglement fidelity is aqualitative measure. Classical communication could also cause an overhead in the entanglementdistribution mechanism of the quantum Internet. Thus, a multiobjective optimization frameworkshould consider the attributes of both quantum and classical layers.To address the multiple criteria and several objectives of the quantum Internet, a multiobjectiveoptimization framework is defined. The multiple criteria of the quantum Internet are defined asdiverse objective functions that should be satisfied in parallel. The problem is therefore analogousto finding solutions in an objective space such that the objective space is defined and spannedby the input problems induced by the environment of the quantum Internet. The multiobjectiveoptimization framework should evolve a set of solutions to the Pareto optimal front. In our model,these solutions are evolved via the mathematical model of epicenters that provide a naturallyinspired answer to the multiobjective problem defined via the environment of the quantum Internet.The mathematical model of epicenters utilizes the theory of Pareto dominance in the problemresolution such that the selection and evaluation processes in the objective space that are requiredto identify a global optimal solution are controlled via our nature-inspired model. The proposedPoisson model ensures a robust randomization and efficient convergence in the objective space suchthat the solutions determined by utilizing the epicenters in the objective space will converge to aglobal optimal solution. The global optimal solution in the objective space represents the parallelsatisfaction of the multiple criteria and objective functions defined by the quantum Internet. Therandomness injected by the Poisson distribution not just avoids early convergence to a local optimalsolution but also induces a fast convergence for the global optimal solution in the objective space.Since the multiple objectives and optimization criteria of the mathematical framework aremotivated by practical assumptions and considerations of the quantum Internet, the proposedmathematical model of epicenters is strongly connected with a quantum Internet scenario. Asfollows, the utility of the proposed multiobjective optimization framework represents an effectivesolution for the practical optimization problems induced by the quantum Internet.
The attributes of the Poisson operator are as follows.
The D ( E ) dispersion coefficient of an epicenter E (solution in the feasible space S F ) determines thenumber of affected L j , j = 1 , . . . , D ( E ), locations around an epicenter E . The random locationsaround an epicenter also represent solutions in S F that help in increasing the diversity of population P (a set of possible solutions) to find a global optimum. The diversity increment is therefore a toolto avoid an early convergence to a local optimum [9, 10].The dispersion D ( E i ) operator for an i th epicenter E i is defined as D ( E i ) = m (cid:16) ˜ f ( (cid:104)E(cid:105) ) − ˜ f ( E i ) (cid:17) + ϑ |P| (cid:80) i =1 (cid:16) ˜ f ( (cid:104)E(cid:105) ) − ˜ f ( E i ) (cid:17) + ϑ , (21)8here m is a control parameter, E i is an i th individual (epicenter) from the |P| individuals (epi-centers) in population P , |P| is the size of population P , function ˜ f ( · ) is the fitness value (seeSection A.2.1), ˜ f ( (cid:104)E(cid:105) ) is a maximum objective value among the |P| individuals, and ϑ is a residualquantity.Without loss of generality, assuming |P| epicenters, the q total number of locations is as q = |P| (cid:88) i =1 D ( E i ) . (22) Assume that L j is a random location around E i . For L j , the Euclidean distance d ( E i , l j ) betweenan i th epicenter E i and the projection point l j of a j th location point L j , j = 1 , . . . , D ( E ) on theellipsoid around E i is as follows: d ( E i , l j ) = (cid:113) (dim ( l j )) + (dim ( l j )) = (cid:115) tg α E i ( l j ) a − + tg α E i ( l j ) , (23)where dim i ( · ) is the i th dimension of l j , and(dim ( l j )) a + (dim ( l j )) b = 1 , (24)where coefficients a and b define the shape of the ellipse around epicenter E i (see Fig. 2), while α E i ( l j ) is an angle: tgα E i ( l j ) = dim ( l j )dim ( l j ) . (25)The seismic power P ( E i , L j ) operator for an i th epicenter E i in a j th location point L j , j =1 , . . . , D ( E i ) is defined as P ( E i , L j ) = (cid:18) d ( E i , l j ) M ( E i , L j ) (cid:19) b b e σ ln P ( E i,Lj ) , (26)where b and b are regression coefficients, σ ln P ( E j ) is the standard deviation [14], M ( E i , L j ) is theseismic magnitude in a location L j , and l j is the projection of L j onto the ellipsoid around E i [14].Thus, at a given L j with d ( E i , l j ) ((23)), from P ( E i , L j ) (see (26)), the magnitude M ( E i , L j )between epicenter E i and location L j is evaluated as M ( E i , L j ) = (cid:32) P ( E i , L j ) 1 b e σ ln P ( E i,Lj ) (cid:33) b d ( E i , l j ) . (27)9 .2.3 Cumulative Magnitude Let L E i j be the location point where the seismic power P (cid:16) E i , L E i j (cid:17) is maximal for a given epicenter E i . Let P ∗ ( E i ) be the maximal seismic power, P ∗ ( E i ) = max ∀ j P (cid:16) E i , L E i j (cid:17) . (28)Assuming that |P| epicenters, E ,..., |P| exist in the system, let identify by P max ( E (cid:48) ) the epicenter E (cid:48) with a maximal seismic power among as P max (cid:0) E (cid:48) (cid:1) = max ∀ i ( P ∗ ( E i )) , (29)with magnitude M (cid:16) E (cid:48) , L E (cid:48) j (cid:17) , where L E (cid:48) j is the location point where the seismic power P max ( E (cid:48) ) ismaximal yielded for E (cid:48) .Then the C ( E i ) cumulative magnitude for an epicenter E i is defined as C ( E i ) = M (cid:16) ˜ f ( E i ) − ˜ f ( E (cid:48) ) (cid:17) + ϑ (cid:80) |P| i =1 (cid:16) ˜ f ( E i ) − ˜ f ( E (cid:48) ) (cid:17) + ϑ , (30)where E (cid:48) is the highest seismic power epicenter with magnitude M (cid:16) E (cid:48) , L E (cid:48) j (cid:17) , ˜ f ( E (cid:48) ) is the minimumobjective value among the |P| epicenters, and M is a control parameter defined as M = |P| (cid:88) i =1 M (cid:16) E i , L E i j (cid:17) , (31)where L E i j provides the maximal seismic power for an i th epicenter E i , functions ˜ f ( E i ) and ˜ f ( E (cid:48) )are the fitness values (see Section A.2.1) for the current epicenter E i and for the highest seismicpower epicenter E (cid:48) , and ϑ is a residual quantity. Assume that E i is a current epicenter (solution) and R k and R l are two random reference pointsaround E i . Using the C ( E i ) cumulative seismic magnitude (see (30)) of an epicenter E i , the gener-ation of a new epicenter E p is as follows:Let Φ ( E i , R k , R l ) be a Poisson range identifier function [12,13] for E i using R k and R l as randomreference points: Φ ( E i , R k , R l )= d ( E i , R k ) c w ( R k , R l )cos ( θ ( (cid:96) E i , R k , (cid:96) R k , R l )) · d ( R k , R l ) c w ( E i , R k ) , (32)where E i is a current epicenter, R k and R l are random reference points, d ( · ) is the Euclideandistance function, c w ( E i , R k ) and c w ( R k , R l ) are weighting coefficients between epicenters E i and10 k and between R k and R l , and θ ( (cid:96) E i , R k , (cid:96) R k , R l ) is the angle between lines (cid:96) E i , R k and (cid:96) R k , R l : θ ( (cid:96) E i , R k , (cid:96) R k , R l )= cos − (cid:32) d ( E i , R k ) + d ( E k , R l ) − d ( E i , R l ) d ( E i , R k ) d ( R k , R l ) (cid:33) . (33)Without loss of generality, using (32), a Poissonian distance function D ( E p ) for the finding of newepicenter E p is defined via a P Poisson distribution [12, 13] as follows: D ( E p ) = P ( k, λ ) , (34)where k = Φ ( E i , R k , R l ) , (35)with mean λ = E [Φ ( E i , R k , R l )] . (36)Therefore, the resulting new epicenter E p is a Poisson random epicenter E p with a Poisson rangeidentifier D ( E p ).For a large set of reference points, only those reference points that are within the r ( E i ) radiusaround the current solution E i are selected for the determination of the new solution E p . This radiusis defined as r ( E i ) = χ Q ( M ) − Q , (37)where ˜ M is the average magnitude,˜ M = 1 |P| M = 1 |P| |P| (cid:88) i =1 M (cid:16) E i , L E i j (cid:17) , (38) Q and Q are constants, and χ is a normalization term. Motivated by the corresponding seismologicrelations of the Dobrovolsky-Megathrust radius formula [13], the constants in (37) are selected as Q = 0 .
414 and Q = 1 . r ( E i ) of (37), the weights of reference points are determined by theseismic power function (26). The hypocentral of an epicenter is aimed to increase the diversity of population by a randomization.Let dim k ( E i ) be an actual randomly selected k th dimension and k = 1 , . . . , dim ( E i ) be a currentepicenter E i , i = 1 , . . . , |P| . The H (dim k ( E i )) hypocentral provides a random displacement [12, 13]of dim k ( E i ) using C ( E i ) (see (30)): H (dim k ( E i )) = dim k ( E i (cid:48) )= (cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116) M (cid:18) dim k ( E i ) ,L dim k ( E i ) j (cid:19) dim k ( E i ) + ( U ( − C ( E i ) , C ( E i ))) (39)11here U ( − C ( E i ) , C ( E i )) is a uniform random number from the range of [ − C ( E i ) , C ( E i )] to yieldthe displacement dim k ( E i (cid:48) ), M (cid:16) dim k ( E i ) , L dim k ( E i ) j (cid:17) is the magnitude, and L dim k ( E i ) j is a locationpoint where P (cid:16) dim k ( E i ) , L dim k ( E i ) j (cid:17) is maximal for dim k ( E i ).The D ( E i ) locations around the cumulative magnitude C ( E i ) of E i are generated by (39) throughall the randomly selected Y dimensions, where Y is as follows [9, 10]: Y = U (1 , dim ( E i )) . (40)The process is repeated for all E i . To generate random locations around dim k ( E i ), a Poisson distribution is also used to increase thediversity of the population. A random location in the k th dimension L dim k ( E i ) r around dim k ( E i ) isgenerated as follows: L dim k ( E i ) r = dim k ( E i ) w, (41)where w ∈ P ( X = k, λ ) (42)is a Poisson random number with distribution coefficients k and λ . Given that it is possible thatusing (41) some randomly generated locations will be out of the feasible space S F , a normalizationoperator N ( · ) of L dim k ( E i ) r is defined to keep the new locations around dim k ( E i ) in S F , as follows[9, 10]: L dim k ( E i ) r = L dim k ( E i ) r (cid:16) mod (cid:16) B kup − B klow (cid:17)(cid:17) + B klow , (43)where B klow and B kup are lower and upper bounds on the boundaries of locations in a k th dimension,and mod( · ) is to a modular arithmetic function. The procedure is repeated for the randomlyselected t = U (1 , dim ( E i )) dimensions of E i , for ∀ i . The method of convergence of solutions in the Poisson optimization is summarized in Method 1.
Method 1
Convergence of Solutions
Step 1.
Generate |P| epicenters, E , . . . , E |P| , with D ( E i ) random locations around a given i th epicenter E i . Step 2.
Select an epicenter E i , and determine the seismic operators D ( E i ), P ( E i , L j ), M ( E i , L j ). Step 3.
Determine the D ( E p ) Poisson distance function using references R k and R l to yielda new solution E p . Step 4.
Repeat steps 1–3, until the closest epicenter to the E (cid:48) optimal epicenter is notfound or other stopping criteria are not met.An epicenter E i and the generation of a new solution E p with an in the objective space S O aredepicted in Fig. 2. The ellipsoid around E i and the projection point l k of the reference location R k are serving the determination of power function P ( E i , R k ) in the reference location R k .12 new epicenter E p is determined via the Poisson function D ( E p ). Locations with low powerfunction (26) values have high magnitudes (27) from the epicenter, whereas locations with highpower function values have low magnitudes from the epicenter. Figure 2: Iteration step of the Poisson optimization model in the objective space S O . An i th epicenter, E i (depicted by the red dot), with a projected point l k of random reference location R k .Reference locations R k and R l (blue dots) identify locations L k and L l , respectively. The powerin R k is P ( E i , R k ) (see (26)), while the magnitude is M ( E i , R k ) (see (27)). Notation dim i ( · )refers to the i th dimension of l k , and coefficients a and b define the shape of the ellipse (yellow)around epicenter E i . The H (dim k ( E i )) hypocentral of E i is determined via the range of the C ( E i )cumulative magnitude (depicted by the green circle). The new epicenter E p (depicted by thegreen dot) is determined by the D ( E p ) Poisson distance function using R k and R l , with angle θ ( (cid:96) E i , R k , (cid:96) R k , R l ) between lines (cid:96) E i , R k and (cid:96) R k , R l . The algorithmical framework that utilizes the Poisson entanglement optimization method for theproblem statement presented in Section 2 is defined in Algorithm 1.13 lgorithm 1
Poisson Optimization in the Quantum Internet
Step 0.
In an initial phase, a random population P of |P| feasible solutions isgenerated [9, 10] Let G be an upper bound on the number of generations, n G . Step 1.
For each epicenter x i = E i in P , define D ( E i ) random locations around E i . For adiversity increment, determine the H (dim k ( E i )) hypocentral displacement function (39) fordim k ( E i ), for k = 1 , . . . , dim ( E i ). Step 2.
Determine the seismic power P ( E i , L j ) operator via (26) for an i th epicenter E i in a j th location point L j , j = 1 , . . . , D ( E i ). Determine the L E i j , the location point where theseismic power P (cid:16) E i , L E i j (cid:17) is maximal for a given epicenter E i , via (28). Step 3.
Determine epicenter E (cid:48) with a maximal seismic power P max ( E (cid:48) ) via (29). Computeseismic magnitude M (cid:16) E (cid:48) , L E (cid:48) j (cid:17) via (27), and determine the sum of all N seismicmagnitudes M via (31). Step 4.
Compute the D ( E i ) dispersion via (21) and the C ( E i ) cumulative seismicmagnitude via (30). Select non-dominated solutions from the P population set to the set N P of non-dominated solutions. Identify ϕ k as ϕ k = L E i k , where L E k k is a k th locationaround E i . Update N P with the non-dominated solutions.
Step 5.
Create set P (cid:48) of epicenters by selecting p feasible solutions from P using the Pr ( ϕ i )selection probability as Pr ( ϕ i ) = ˜ f ( ϕ i ) (cid:46) (cid:80) r ∈P ˜ f ( ϕ r ). Apply Sub-procedure 1. Step 6. If n G ≥ G , then stop the iteration; otherwise, repeat steps 1–4.Sub-procedure 1 of step 5 is discussed in the Appendix. To achieve the minimization of classical communications required by the entanglement optimization,the S -metric (or hypervolume indicator) is integrated, which is a quality measure for the solutionsor a contribution of a single solution in a solution set [9, 10] By definition, this metric identifies thesize of dominated space (size of space covered).By theory, the S ( R ) S -metric for a solution set R = { r , . . . , r n } is as follows: S ( R ) = L (cid:32) (cid:91) r ∈R { x ref ∠ x ∠ x | r } (cid:33) , (44)where L is a Lebesgue measure, notation b ∠ a means a dominates b (or b is dominated by a ), and x ref is a reference point dominated by all valid solutions in the solution set [9, 10].For a given solution r i , the S -metric identifies the size of space dominated by r i but not domi-nated by any other solution, without loss of generality as: S ( r i ) = ∆ S ( R , r i ) = S ( R ) − S ( R\ { r i } ) . (45)In the optimization of classical communications, the existence of two objective functions is assumed.The first objective function, f , is associated with the minimization of the cost of the first type ofclassical communications related to the reception and storage of entangled systems in the quantumnodes (it covers the classical communications related to the required entanglement throughput14y the nodes, fidelity of received entanglement, number of stored entangled states, and fidelityparameters). Thus, f : min ∀ i C ( x i ) , (46)where C ( x i ) is the cost associated with the first type of classical communications related to a x i .The second objective function, f , is associated with the cost of the second type of classicalcommunications that is related to entanglement purification: f : min ∀ i C ( x i ) , (47)where C ( x i ) is the cost associated with the second type of classical communications with respectto x i .Assuming objective functions f and f , the S ( r i ) of a particular solution r i is as follows: S ( r i ) = ( f ( r i ) − f ( r i − )) ( f ( r i ) − f ( r i +1 )) . (48)Given that the S -metric is calculated for the solutions, a set of nearest neighbors that restrict thespace can be determined. Since the volume of this space can be quantified by the hypervolume,the solutions that satisfy objectives f and f can be found by utilizing (48). The computational complexity of the Poissonian optimization method is derived as follows. Giventhat |P| epicenters are generated in the search space and that the number of locations for an i th epicenter E i is determined by the dispersion operator D ( E i ), the resulting computational complexityat a total number of locations q = (cid:80) |P| i =1 D ( E i ) (see (22)) is O (cid:16) ( |P| + q ) d /2 log ( |P| + q ) (cid:17) , (49)since after a sorting process the locations for a given epicenter E i can be calculated with complexity O ( D ( E i )), where d is the number of objectives.Considering that in our setting d = 2, the total complexity is O (( |P| + q ) log ( |P| + q )) . (50) The resolution of the problem shown in Section 2 using the Poissonian entanglement optimizationframework of Section 3 is as follows.Let X S low be a set of nodes for which condition (7) holds for the fidelity of the received entangledstates in the nodes, and let X S high be a set of nodes for which condition (8) holds for the receivedfidelity entanglement.Then let | X S low | and (cid:12)(cid:12) X S high (cid:12)(cid:12) be the cardinality of X S low and X S high , respectively.Specifically, function (10) for the X S low -type nodes is rewritten as G X S low ( X ) = max | X S low | (cid:88) i =1 F X S low i ( X ) E (cid:16) D X S low i ( X ) (cid:17) , (51)15here F X S low i ( X ) is the entanglement fidelity function for an i th X S low -type node x i , x i ∈ X S low ,and E (cid:16) D X S low i ( X ) (cid:17) is the expected relative entropy of entanglement in an i th X S low -type x i .Similarly, for the X S low -type nodes, function (10) is as follows: G X S high ( X ) = max (cid:12)(cid:12)(cid:12) X S high (cid:12)(cid:12)(cid:12) (cid:88) i =1 F X S high i ( X ) E (cid:18) D X S high i ( X ) (cid:19) . (52)From (51) and (52), a cumulative G X S high ⊗ X S high ( X ) is defined as G X S low ⊗ X S high ( X ) = (cid:12)(cid:12)(cid:12) X S high (cid:12)(cid:12)(cid:12) (cid:88) i =1 A i F X S high i ( X ) E (cid:18) D X S high i ( X ) (cid:19) + | X S low | + (cid:12)(cid:12)(cid:12) X S high (cid:12)(cid:12)(cid:12) (cid:88) i = (cid:12)(cid:12)(cid:12) X S high (cid:12)(cid:12)(cid:12) +1 A i F X S low i ( X ) E (cid:16) D X S low i ( X ) (cid:17) F ( X ) , (53)where A i refers to the number of received entangled systems in an i th node, while F ( X ) = min C ( X ) = N (cid:88) i =1 T (cid:88) i =1 f j B jF ( x i ) . (54)The fidelity types of the available resource states in the nodes should be further divided into T classes. The final function is then evaluated as G X S low ⊗ X S high ( X ) = F ( X ) F ( X )= | X S low | + (cid:12)(cid:12)(cid:12) X S high (cid:12)(cid:12)(cid:12) (cid:88) i =1 T (cid:88) j =1 f j B jF ( x i ) F ( X ) , (55)where F ( X ) = min S ( X ) = | X S low | + (cid:12)(cid:12)(cid:12) X S high (cid:12)(cid:12)(cid:12) (cid:88) i =1 S i ( X ) . (56)Thus, G X S low ⊗ X S high ( X ) = | X S low | + (cid:12)(cid:12)(cid:12) X S high (cid:12)(cid:12)(cid:12) (cid:88) i =1 S i ( X ) , (57)such that [9, 10] | X S low | + (cid:12)(cid:12)(cid:12) X S high (cid:12)(cid:12)(cid:12) (cid:88) i =1 F i ( X ) ≥ γF ( N )= γ | X S low | + (cid:12)(cid:12)(cid:12) X S high (cid:12)(cid:12)(cid:12) (cid:88) i =1 T (cid:88) j =1 f j B jF ( x i ) ≤ ν X ( ϕ i ) Λ ≤ Π B jF ( x i ) , (58)16here ν X ( ϕ i ) = Z (cid:80) j =1 τ j ( ϕ i ), γ is given by the constraint of (13), while Π is given by the constraintof (19). Let F i ( X ) ∈ [0 ,
1] be the objective function that refers to the resulting entanglement fidelity in aparticular node x i , after purification and quantum error correction with per-node cost functions F i ( X ), and F i ( X ), respectively.Precisely, a current i th epicenter E i identifies a solution in the objective space S O ,S O : (cid:8) F i ( N ) , F i ( N ) , F i ( X ) (cid:9) . (59)The random locations around E i also represent possible solutions. Let E ∗ be an optimal solutionin the S O subject space, which maximizes F i ( X ) and minimizes F i ( X ) and F i ( X ). From E i , thealgorithm determines a new solution (epicenter) E p via the D ( E p ) Poisson distance function, usingthe connection model between the locations around E i . To improve the diversity, locations around E p are generated. The new epicenter E p converges to an optimal solution E ∗ . The iterations arerepeated until E ∗ is not found or until a stopping criterion is met.The iteration from a current solution E i to a new solution E p toward a global optimal E ∗ in S O is illustrated in Fig. 3. In this section, a numerical evidence is proposed to demonstrate the Poisson entanglement opti-mization method.
To demonstrate the results of Section 4, let F i ( X ) be the object function subject to maximize.The problem is to determine a matrix X that maximizes F i ( X ), and also E (cid:16) D N S low i ( X ) (cid:17) , andminimizes the cost functions F i ( N ) and F i ( N ). Thus, for each node N , the optimal number ofreceived and stored entangled systems should be determined, with high and low fidelity classes.Particularly, finding an optimal solution E ∗ in S O with the assumptions given Section 4, istherefore means the selection of the optimal objective function (e.g., maximizing the entanglementfidelity F i ( X ) or maximizing the relative entropy of entanglement E (cid:16) D N S low i ( X ) (cid:17) ), in particularnode types X S low and X S high , while all cost functions are minimized in the quantum network.A solution set in S O is depicted in Fig. 4.An optimal solution E ∗ in S O therefore yields the maximization of entanglement fidelity F N ( X )if a particular node N belongs to the class N S high , whereas it maximizes the relative entropy ofentanglement E (cid:16) D N S low i ( X ) (cid:17) if N belongs to the class N S low . Increasing B jF ( x i ) for a N S high -classnode and then performing an optimal purification and quantum error correction could significantly17 (cid:10) i F N (cid:9) (cid:10) i X (cid:30) (cid:9) (cid:10) i F N (cid:9) (cid:10) i X (cid:30) (cid:9) (cid:10) i F N (cid:9) (cid:10) i F N i (cid:29) j (cid:29) (a) (b) * (cid:29) Figure 3: Distribution of solutions for entanglement fidelity maximization in the objective spaceS O : (cid:8) F i ( N ) , F i ( N ) , F i ( X ) (cid:9) of the entanglement optimization problem (cost functions F ( N ) and F ( N ) and the objective function F i ( X ) are normalized onto the range of [0 , E i refers to a current solution (depicted by the red dot) with the Poisson distributedreference locations (the reference points are not real solutions). The random reference locations areclustered into two classes: (1) reference locations within radius r ( E i ) around E i and (2) referencelocations outside the radius (depicted by the gray dots). Reference locations outside the range areneglected in the iteration. (b): A new epicenter E p (depicted by the green dot) is determined viathe connection model of relevant reference points (e.g., lie inside the range of r ( E i )), which yieldsthe D ( E p ) Poisson distance function. The new solution, E p , converges toward an optimal solution E ∗ (depicted by the purple dot). The reference locations inside the relevance region are weightedby the seismic power function.improve the fidelity of entanglement. On the other hand, for a N S low -class node, the fidelityimprovement at an optimal purification and quantum error correction is insignificant. Thus, incre-menting B jF ( x i ) does not lead to a significant improvement in the fidelity. The optimal solutionfor these nodes is to focus on improving the relative entropy of entanglement, which requires lowercost function values.This decision strategy provides a global optimal with respect to all quantum nodes of thequantum network.The decision making is illustrated in Fig. 5. In Fig. 5(a), the F entanglement fidelity is de-picted in function of F i ( N ) for N S low and N S high nodes. In Fig. 5(b), the D relative entropy of18 (cid:10) i X (cid:30) (cid:9) (cid:10) i F N (cid:9) (cid:10) i F N * (cid:29) Figure 4: Solution set in S O , with an optimal epicenter E ∗ , F i ( N ) ∈ [0 . , . F i ( N ) ∈ [0 . , . F i ( X ) ∈ [0 . , F i ( N ) for N S low and N S high nodes. The initial values of F and D are assumed to be equal for a given class, while the value of F i ( N ) is set to constant forillustration purposes.For an N S high node, the increment of F i ( N ) leads to significant improvement in F , whilethe increment in D is moderate. For an N S low node, the increment of F i ( N ) leads to moderateimprovement in F , while the improvement in D is significant. As a corollary, the increment of theentanglement throughput is a useful approach to increase the entanglement fidelity for the X S high set, and to boost the relative entropy of entanglement in the X S low set. First, we analyze the distribution of solutions in the feasible space S F focusing on the magnitudesassociated to the locations around epicenters.Let assume that the total number of q locations (see (22)) can be divided into m magnituderanges [11], such that m (cid:88) i =1 n i = q = |P| (cid:88) i =1 D ( E i ) , (60)19igure 5: Illustration of the decision making. (a): The F entanglement fidelity values in functionof F i ( N ) for N S low (red line) and N S high (blue line) nodes. The value of F i ( N ) is set to constant.(b): The D relative entropy of entanglement values in function of F i ( N ) for N S low (red line) and N S high (blue line) nodes. The value of F i ( N ) is set to constant.where n i is the number of locations belonging to an i th magnitude range, |P| is the population size.Then let M i be the magnitude associated to the i th magnitude range. Then a ˜ n i approximation of n i is evaluated as ˜ n i = f ( M i ) , (61)where f ( · ) is a fitting function. To give an estimate on n i at a particular magnitude M i , we utilizea power law distribution [11] function B ( n i ) for a log-scaled n i , as B ( n i ) : log ( n i ) = a − b ˜ M i , (62)where ˜ M i is a log scaled M i , while a and b are constants [11].Then, the ˜ n i Poisson estimate is yielded as˜ n i = σ i = λ i , (63)where σ i is the observational variance, while λ i is the mean of a Poisson distribution. Since thesum of independent Poisson variables is also a Poisson variable with mean equals to the sum of thecomponents means, λ ( q ) = m (cid:88) i =1 λ i ≈ q, (64)where λ ( q ) is the mean total number, while λ i is an i th component mean. Using the Poissonproperty σ = λ , the σ q estimated uncertainty is yielded as [11] σ q = λ ( q ) = (cid:80) mi =1 f (cid:16) ˜ M i (cid:17) . Thususing a corresponding fitting function f ( · ), the mean and the variance of the total number of eventsare equal to the sum of the fitted values. 20n our model the distribution of the log scaled ˜ n i = λ i values in function of M i are wellapproachable by the power law distribution B ( λ i ) : log ( λ i ) = a − b ˜ M i , while the distribution ofthe λ ( q ) total number (64) of locations are approachable by a N (cid:0) λ ( q ) , σ N (cid:1) , Gaussian distributionwith variance σ N = λ ( q ) as λ ( q ) → ∞ , by theory.A distributions of B ( λ i ) in function of the magnitude M i and coefficient b are illustrated inFig. 6. (cid:9) (cid:10) i (cid:77) (cid:26) (cid:9) (cid:10) i (cid:77) (cid:26) i M i M
10 10 b b (a) (b) Figure 6: Distribution of B ( λ i ) for different magnitudes M i , M i = 1 , . . . ,
10 and coefficient b , for(a): a = 10, and (b): a = 20.The distributions of λ ( q ) (see (64)) for k it iterations are depicted in Fig. 7. In Fig. 7(a), λ ( q ) =10 , while Fig. 7(b) illustrated the distribution at λ ( q ) = 10 . As λ ( q ) → ∞ , the distributions of λ ( q ) can be approximated by a N (cid:0) λ ( q ) , σ N (cid:1) , σ N = λ ( q ) Gaussian distribution.The associated distributions of B ( λ i ) for the values of λ ( q ) are depicted in Fig. 8. The maximumvalue of B ( λ i ) is selected to B ( λ i ) ≈
10 in each cases which values are picked up at λ ( q ), where λ ( q ) = 10 in Fig. 8(a), and λ ( q ) = 10 in Fig. 8(b). The B ( λ i ) values approximates to a Gaussiandistribution. The statistical distribution of B ( λ i ) is therefore constitutes a similar pattern forarbitrary λ ( q ). We defined an optimization framework for the transmission and processing of quantum entan-glement in the entangled network structure of the quantum Internet. The proposed Poissonianentanglement optimization framework fuses the fundamental concepts of quantum Shannon theorywith the theory of evolutionary algorithms and seismic wave propagations. Two objective functionsare defined, with primary focus on the entanglement fidelity and secondary focus on the relative21
200 400 600 800 10000 200 400 600 800 1000 (a) (b) (cid:184) (cid:184) (cid:184) it k it k (cid:9) (cid:10) q (cid:77) (cid:9) (cid:10) q (cid:77) Figure 7: Distribution of λ ( q ) for k it iterations, k it = 0 , . . . , λ ( q ) = 10 , and (b): λ ( q ) = 10 . (b) it k
0 200 400 600 800 1000 (a) it k
0 200 400 600 800 1000 (cid:9) (cid:10) i (cid:77) (cid:26) (cid:9) (cid:10) i (cid:77) (cid:26) Figure 8: The distributions of B ( λ i ) for k it iterations, k it = 0 , . . . , λ ( q ) = 10 , and (b): λ ( q ) = 10 .entropy of entanglement. As an additional objective function, the minimization of classical commu-nications required by the entanglement optimization procedure is considered. The cost functionsare defined to cover the physical attributes of entanglement transmission, purification, and stor-age in quantum memories. This method can be implemented with low complexity that allows astraightforward application in future quantum Internet and quantum networking scenarios. Acknowledgements
This work was partially supported by the European Research Council through the Advanced FellowGrant, in part by the Royal Society’s Wolfson Research Merit Award, in part by the Engineeringand Physical Sciences Research Council under Grant EP/L018659/1, by the Hungarian ScientificResearch Fund - OTKA K-112125, and by the National Research Development and Innovation22ffice of Hungary (Project No. 2017-1.2.1-NKP-2017-00001), and in part by the BME ArtificialIntelligence FIKP grant of EMMI (BME FIKP-MI/SC).
References [1] Van Meter, R.
Quantum Networking , John Wiley and Sons Ltd, ISBN 1118648927,9781118648926 (2014).[2] Gyongyosi, L., Imre, S. and Nguyen, H. V. A Survey on Quantum Channel Capacities,
IEEECommunications Surveys and Tutorials , 1, DOI: 10.1109/COMST.2017.2786748 (2018).[3] Lloyd, S., Shapiro, J. H., Wong, F. N. C., Kumar, P., Shahriar, S. M. and Yuen, H. P.Infrastructure for the quantum Internet, ACM SIGCOMM Computer Communication Review ,34(5):9–20, (2004).[4] Kimble, H. J. The quantum internet,
Nature
IEEE/ACM Transactions on Networking
Networking Science , Volume 3, Issue 1–4, pp 82–95, (2013).[7] Imre, S. and Gyongyosi, L.
Advanced Quantum Communications - An Engineering Approach .New Jersey, Wiley-IEEE Press (2013).[8] Gyongyosi, L. and Imre, S. A Poisson model for entanglement optimization in the quantumInternet,
Proceedings of Quantum Information Science, Sensing, and Computation X ; Vol.10660, 106600C (2018) DOI: 10.1117/12.2309416 (2018).[9] Tan, Y.
Fireworks Algorithm, A Novel Swarm Intelligence Optimization Method , Springer(2015).[10] Zheng, Y. J., Song,Q. and Chen, S. Y. Multiobjective fireworks optimization for variable-ratefertilization in oil crop production.
Appl. Soft Comput. arXiv:0807.2396v2 DOI 10.1029/2008GL035353 (2008).[12] Kannan, S. Improving Innovative Mathematical Model for Earthquake Prediction,
J GeolGeosci , DOI: 10.4172/2329-6755.1000168 (2014).[13] Mejia, J., Rojas, K., aleza, N. and Villagracia, A. R. Earthquake prediction through Kannan-Mathematical-Model Analysis and Dobrovolsky-based clustering Technique,
DLSU ResearchCongress 2015 (2015).[14] Stamatovska, S. G. The Latest Mathematical Models of Earthquake Ground Motion,
SeismicWaves - Research and Analysis , InTech (2012).[15] Xiao, Y. F., Gong, Q. Optical microcavity: from fundamental physics to functional photonicsdevices,
Science Bulletin . 61(3): 185-186 (2016).2316] Zhang et al. Quantum Secure Direct Communication with Quantum Memory.
Phys. Rev. Lett .118, 220501 (2017).[17] Biamonte, J. et al. Quantum Machine Learning.
Nature , 549, 195-202 (2017).[18] Lloyd, S. Mohseni, M. and Rebentrost, P. Quantum algorithms for supervised and unsupervisedmachine learning. arXiv:1307.0411 (2013).[19] Chou, C., Laurat, J., Deng, H., Choi, K. S., de Riedmatten, H., Felinto, D. and Kimble, H.J. Functional quantum nodes for entanglement distribution over scalable quantum networks.
Science , 316(5829):1316–1320 (2007).[20] Sheng, Y. B., Zhou, L. Distributed secure quantum machine learning.
Science Bulletin (2017).[21] Kok, P., Munro, W. J., Nemoto, K., Ralph, T. C., Dowling, J. P. and Milburn, G. J. Linearoptical quantum computing with photonic qubits.
Rev. Mod. Phys . 79 , 135-174 (2007).[22] Zou, Z. Z., Yu, X. T. and Zhang, Z. C. Quantum connectivity optimization algorithms forentanglement source deployment in a quantum multi-hop network.
Frontiers of Physics , 13(2):130202. (2018).[23] Xiong, P. Y., Yu, X. T., Zhang, Z. C. et al. Routing protocol for wireless quantum multi-hopmesh backbone network based on partially entangled GHZ state.
Frontiers of Physics , 12(4):120302. (2017).[24] Farouk, A., Batle, J., Elhoseny, M. et al. Robust general N user authentication scheme in acentralized quantum communication network via generalized GHZ states.
Frontiers of Physics ,13(2): 130306. (2018).[25] Wang, Z., Zhang, C., Huang, Y. F. et al. Experimental verification of genuine multipartiteentanglement without shared reference frames.
Science bulletin , 61(9): 714-719 (2016).[26] Deng, F. G., Ren, B. C. and Li, X. H. Quantum hyperentanglement and its applications inquantum information processing.
Science bulletin , 62(1): 46-68. (2017).[27] Zhu, F., Zhang, W., Sheng, Y. et al. Experimental long-distance quantum secure direct com-munication.
Science Bulletin , 62(22): 1519-1524. (2017).[28] Van Meter, R. and Devitt, S. J. Local and Distributed Quantum Computation,
IEEE Computer
Physical Review A , American Physical Society, DOI: 10.1103/PhysRevA.98.022310, DOI:10.1103/PhysRevA.98.022310, (2018).[30] Gyongyosi, L. and Imre, S. Dynamic topology resilience for quantum networks,
Proc. SPIE10547 , Advances in Photonics of Quantum Computing, Memory, and Communication XI,105470Z (22 February 2018); doi: 10.1117/12.2288707. (2018).[31] Gyongyosi, L. and Imre, S. Topology Adaption for the Quantum Internet,
Quantum Informa-tion Processing , Springer Nature, DOI: 10.1007/s11128-018-2064-x, (2018).2432] Pirandola, S., Laurenza, R., Ottaviani, C. and Banchi, L. Fundamental limits of repeaterlessquantum communications,
Nature Communications , 15043, doi:10.1038/ncomms15043 (2017).[33] Pirandola, S., Braunstein, S. L., Laurenza, R., Ottaviani, C., Cope, T. P. W., Spedalieri, G.and Banchi, L. Theory of channel simulation and bounds for private communication,
QuantumSci. Technol . 3, 035009 (2018).[34] Laurenza, R. and Pirandola, S. General bounds for sender-receiver capacities in multipointquantum communications,
Phys. Rev. A
96, 032318 (2017).[35] Pirandola, S. Capacities of repeater-assisted quantum communications, arXiv:1601.00966 (2016).[36] Gyongyosi, L. and Imre, S. Multilayer Optimization for the Quantum Internet,
ScientificReports , Nature, DOI:10.1038/s41598-018-30957-x, (2018).[37] Gyongyosi, L. and Imre, S. Entanglement Availability Differentiation Service forthe Quantum Internet,
Scientific Reports
Scientific Reports
Advanced Quantum Communications - An Engineering Approach .New Jersey, Wiley-IEEE Press (2013).[40] Caleffi, M. End-to-End Entanglement Rate: Toward a Quantum Route Metric, 2017
IEEEGlobecom , DOI: 10.1109/GLOCOMW.2017.8269080, (2018).[41] Caleffi, M. Optimal Routing for Quantum Networks,
IEEE Access , Vol 5, DOI: 10.1109/AC-CESS.2017.2763325 (2017).[42] Caleffi, M., Cacciapuoti, A. S. and Bianchi, G. Quantum Internet: from Communication toDistributed Computing, aXiv:1805.04360 (2018).[43] Castelvecchi, D. The quantum internet has arrived,
Nature arXiv:1810.08421 (2018).[45] Petz, D.
Quantum Information Theory and Quantum Statistics , Springer-Verlag, Heidelberg,Hiv: 6. (2008).[46] Bacsardi, L. On the Way to Quantum-Based Satellite Communication,
IEEE Comm. Mag.
NaturePhysics , 10, 631 (2014). 2548] Lloyd, S. Capacity of the noisy quantum channel.
Physical Rev. A , 55:1613–1622 (1997).[49] Lloyd, S. The Universe as Quantum Computer,
A Computable Universe: Understanding andexploring Nature as computation , Zenil, H. ed., World Scientific, Singapore, arXiv:1312.4455v1 (2013).[50] Shor, P. W. Scheme for reducing decoherence in quantum computer memory.
Phys. Rev. A ,52, R2493-R2496 (1995).[51] Chou, C., Laurat, J., Deng, H., Choi, K. S., de Riedmatten, H., Felinto, D. and Kimble, H.J. Functional quantum nodes for entanglement distribution over scalable quantum networks.
Science , 316(5829):1316–1320 (2007).[52] Muralidharan, S., Kim, J., Lutkenhaus, N., Lukin, M. D. and Jiang. L. Ultrafast and Fault-Tolerant Quantum Communication across Long Distances,
Phys. Rev. Lett . 112, 250501 (2014).[53] Yuan, Z., Chen, Y., Zhao, B., Chen, S., Schmiedmayer, J. and Pan, J. W.
Nature
Lecture Notes in Computer Science (Automata, Languages and Programming SE-52 vol. 5555), Springer) pp 622-633 (2009).[55] Hayashi, M. Prior entanglement between senders enables perfect quantum network coding withmodification,
Physical Review A , Vol.76, 040301(R) (2007).[56] Hayashi, M., Iwama, K., Nishimura, H., Raymond, R. and Yamashita, S, Quantum networkcoding,
Lecture Notes in Computer Science (STACS 2007 SE52 vol. 4393) ed Thomas, W. andWeil, P. (Berlin Heidelberg: Springer) (2007).[57] Chen, L. and Hayashi, M. Multicopy and stochastic transformation of multipartite pure states,
Physical Review A , Vol.83, No.2, 022331, (2011).[58] Schoute, E., Mancinska, L., Islam, T., Kerenidis, I. and Wehner, S. Shortcuts to quantumnetwork routing, arXiv:1610.05238 (2016).[59] Lloyd, S. and Weedbrook, C. Quantum generative adversarial learning.
Phys. Rev. Lett ., 121,arXiv:1804.09139 (2018).[60] Gisin, N. and Thew, R. Quantum Communication.
Nature Photon.
1, 165-171 (2007).[61] Zhang, W. et al. Quantum Secure Direct Communication with Quantum Memory.
Phys. Rev.Lett.
Science ,279, 205-208 (1998).[63] Briegel, H. J., Dur, W., Cirac, J. I. and Zoller, P. Quantum repeaters: the role of imperfectlocal operations in quantum communication.
Phys. Rev. Lett.
81, 5932-5935 (1998).[64] Dur, W., Briegel, H. J., Cirac, J. I. and Zoller, P. Quantum repeaters based on entanglementpurification.
Phys. Rev. A , 59, 169-181 (1999).2665] Duan, L. M., Lukin, M. D., Cirac, J. I. and Zoller, P. Long-distance quantum communicationwith atomic ensembles and linear optics.
Nature , 414, 413-418 (2001).[66] Van Loock, P., Ladd, T. D., Sanaka, K., Yamaguchi, F., Nemoto, K., Munro, W. J. andYamamoto, Y. Hybrid quantum repeater using bright coherent light.
Phys. Rev. Lett. , 96,240501 (2006).[67] Zhao, B., Chen, Z. B., Chen, Y. A., Schmiedmayer, J. and Pan, J. W. Robust creation ofentanglement between remote memory qubits.
Phys. Rev. Lett.
98, 240502 (2007).[68] Goebel, A. M., Wagenknecht, G., Zhang, Q., Chen, Y., Chen, K., Schmiedmayer, J. and Pan,J. W. Multistage Entanglement Swapping.
Phys. Rev. Lett.
Phys. Rev. Lett . 98, 190503(2007).[70] Tittel, W., Afzelius, M., Chaneliere, T., Cone, R. L., Kroll, S., Moiseev, S. A. and Sellars, M.Photon-echo quantum memory in solid state systems.
Laser Photon. Rev.
4, 244-267 (2009).[71] Sangouard, N., Dubessy, R. and Simon, C. Quantum repeaters based on single trapped ions.
Phys. Rev. A , 79, 042340 (2009).[72] Dur, W. and Briegel, H. J. Entanglement purification and quantum error correction.
Rep.Prog. Phys , 70, 1381-1424 (2007).[73] Sheng, Y. B., Zhou, L. Distributed secure quantum machine learning.
Science Bulletin , 62,1025-1019 (2017).[74] Leung, D., Oppenheim, J. and Winter, A. Quantum network communication—the butterflyand beyond.
IEEE Trans. Inf. Theory
56, 3478-90. (2010).[75] Kobayashi, H., Le Gall, F., Nishimura, H. and Rotteler, M. Perfect quantum network commu-nication protocol based on classical network coding,
Proceedings of 2010 IEEE InternationalSymposium on Information Theory (ISIT) pp 2686-90. (2010).[76] Vedral, V., Plenio, M. B., Rippin, M. A. and Knight, P. L. Quantifying Entanglement,
Phys.Rev. Lett . 78, 2275-2279 (1997).[77] Vedral, V. The role of relative entropy in quantum information theory,
Rev. Mod. Phys . 74,197–234 (2002).[78] Long, G-L. and Liu, X-S. Theoretically efficient high-capacity quantum-key-distributionscheme.
Physical Review A
Science China Physics, Mechanics and Astronomy , 61(7): 070321. (2018).[80] Zhang, K. J., Zhang, L., Song, T. T. et al. A potential application in quantum net-works—Deterministic quantum operation sharing schemes with Bell states.
Science ChinaPhysics, Mechanics and Astronomy , 2016, 59(6): 660302. (2018).2781] Ye, T. Y. and Ji, Z. X. Multi-user quantum private comparison with scattered preparationand one-way convergent transmission of quantum states.
Science China Physics, Mechanicsand Astronomy , 60(9): 090312. (2017).[82] Hu, J. Y., Yu, B., Jing, M. Y. et al. Experimental quantum secure direct communication withsingle photons.
Light: Science and Applications , 5(9): e16144. (2016).[83] Wang, K., Yu, X. T., Lu, S. L. et al. Quantum wireless multihop communication based onarbitrary Bell pairs and teleportation,
Physical Review A , 89(2), 022329, (2014).[84] Yu, X. T., Xu, J. and Zhang, Z. C., Distributed wireless quantum communication networks,
Chinese Physics B , 22(9), 090311, (2013). 28
Appendix
A.1 Definitions
A.1.1 Entanglement Fidelity
Let | β (cid:105) = √ ( | (cid:105) + | (cid:105) ) (A.1)be the target Bell state subject to be created at the end of the entanglement distribution procedure.The entanglement fidelity F at an actually created noisy quantum system σ is F ( σ ) = (cid:104) β | σ | β (cid:105) , (A.2)where F is a value between 0 and 1, F = 1 for a perfect Bell state and F < F ( | ϕ (cid:105) , | ψ (cid:105) ) = |(cid:104) ϕ | ψ (cid:105)| . (A.3)The fidelity of quantum states can describe the relation of a pure channel input state | ψ (cid:105) and thereceived mixed quantum system σ = (cid:80) n − i =0 p i ρ i = (cid:80) n − i =0 p i | ψ i (cid:105)(cid:104) ψ i | at the channel output as F ( | ψ (cid:105) , σ ) = (cid:104) ψ | σ | ψ (cid:105) = n − (cid:88) i =0 p i |(cid:104) ψ | ψ i (cid:105)| . (A.4)Fidelity can also be defined for mixed states σ and ρF ( ρ, σ ) = (Tr( (cid:113) √ σρ √ σ )) = (cid:88) i p i (Tr( (cid:113) √ σ i ρ i √ σ i )) . (A.5) A.1.2 Relative Entropy of Entanglement
By definition, the E ( ρ ) relative entropy of entanglement function of a joint state ρ of subsystems A and B is defined by the D ( ·(cid:107)· ) quantum relative entropy function, without loss of generality as E ( ρ ) = min ρ AB D ( ρ (cid:107) ρ AB ) = min ρ AB Tr( ρ log ρ ) − Tr( ρ log( ρ AB )) , (A.6)where ρ AB is the set of separable states ρ AB = (cid:80) ni =1 p i ρ A,i ⊗ ρ B,i . A.2 Evaluation of Solutions
A.2.1 Fitness Function
To evaluate the performance of the epicenters we utilize a mathematical apparatus based on thePareto strength and fitness assignment [9, 10].Let Pr ( E i ) be the probability of selection of an epicenter E i , defined asPr ( E i ) = κ ( E i ) (cid:80) l ∈ K κ ( E l ) , (A.7)29here κ ( E i ) is the sum of d ( · ) Euclidean distances between E i and the other epicenters, as κ ( E i ) = K (cid:88) l =1 d ( E i , E l ) = K (cid:88) l =1 (cid:107)E i − E l (cid:107) , (A.8)where K is a set with cardinality | K | = |P| (cid:88) i =1 D ( E i ) + N (cid:88) i =1 dim( E i ) (cid:88) k =1 R ( i, k ) , (A.9)where D ( E i ) is given in (21), and l ∈ K refers to that the position of E j belongs to set K , and |P| is the population size. Let N P refer to the non-dominated solution archive, and let ϕ i = E i (A.10)refer to the selected epicenter, i.e, to an individual solution in P or in N P .Let Φ ( ϕ i ) be a strength coefficient for solution ϕ i , defined asΦ ( ϕ i ) = (cid:12)(cid:12)(cid:12) ϕ k ∈ P (cid:91) N P (cid:12)(cid:12)(cid:12) ϕ k ∠ ϕ i | , (A.11)where ∠ refers to the Pareto dominance relation between ϕ i and ϕ k = E k . As follows, (A.11)depends on the number of individuals it dominates, by theory [9, 10].By definition, a decision vector A dominates a vector B , i.e., B ∠ A , if f i ( A ) ≤ f i ( B ) (A.12)for ∀ i , i = 1 , . . . , m and for at least one j with i , j = 1 , . . . , n , f j ( A ) ≤ f j ( B ) , (A.13)where f : R m → R n . The set of non-dominated decision vectors in R n is called a Pareto optimal set,while the image under f in the solution space is called the Pareto front [9, 10]. In a multiobjectiveoptimization the aim is to achieve the best Pareto front, by theory.Using (A.11), let α ( ϕ i ) be the raw fitness value of ϕ i evaluated by the Φ ( · ) strength function(see (A.11)) of its dominators as α ( ϕ i ) = (cid:88) ( ϕ k ∈P (cid:83) N P ) ∧ ( ϕ i ∠ ϕ k ) | Φ ( ϕ k ) , (A.14)with an inverse distance function (referred to as the density value of ϕ i ), ρ ( ϕ i ) as ρ ( ϕ i ) = 1 d g ( ϕ i ) , (A.15)where d g ( ϕ i ) is the distance from solution ϕ i to its g th nearest individual, where g is initialized asthe square root of the sample size |P (cid:83) N P| , by theory [9, 10].Using (A.15), a for a random solution r ( ϕ i ) the ˜ f ( · ) fitness function of ϕ i is as˜ f ( ϕ i ) = α ( ϕ i ) + ρ ( ϕ i ) . (A.16)Then let p refer to the number of selected ϕ i solutions in P . Using (A.16), the selection probabilityof each solution is yielded as Pr ( ϕ i ) = ˜ f ( ϕ i ) (cid:80) r ∈P ˜ f ( ϕ r ) . (A.17)30 .2.2 Constraints As a solution ϕ i does not satisfy the problem constraints C , C , C , a H C z ( ϕ i ), z = 1 , , C (see (13)), the H C ( ϕ i ) violation function [9, 10] is as H C ( ϕ i ) = (cid:26) γ − ζ ( ϕ i ) , if ζ ( ϕ i ) ≤ γ , otherwise , (A.18)where ζ ( ϕ i ) = N (cid:88) i =1 F i ( ϕ i ) . (A.19)For constraint C (see (15)), the H C ( ϕ i ) violation function is as followsH C ( ϕ i ) = (cid:26) F ( ϕ i ) − Λ , if F ( ϕ i ) ≥ Λ0 , otherwise , (A.20)where F ( ϕ i ) = N (cid:88) i =1 T (cid:88) i =1 f j B jF ( ϕ i ) . (A.21)For constraint C (see (19)), the H C ( ϕ i ) violation function is asH C ( ϕ i ) = (cid:26) ν ( ϕ i ) − Π , if ν ( ϕ i ) ≥ Π0 , otherwise , (A.22)where ν ( ϕ i ) = N (cid:88) j =1 τ j ( ϕ i ) . (A.23)From (A.18), (A.20) and (A.22) a penalty coefficient ∂ ( ϕ i ) is defined as ∂ ( ϕ i ) = w H C ( ϕ i ) + w H C ( ϕ i ) + w H C ( ϕ i ) , (A.24)where w i -s are weighting coefficients [9, 10]. A.2.3 Selection Condition
Assuming that there are χ number of selected random solutions such that the selection probabilitiesare proportional to their fitness values. The selection of a solution ϕ i is as follows.First from the selected random solutions a mutant solution (cid:126) i is generated as (cid:126) i = ϕ r a + ϑ ( ϕ r b − ϕ r c ) , (A.25)where r i ∈ { a, . . . p } are the random indexes, while ϑ > (cid:126) i a trial solution T i is defined with a j th component T ( j ) i asT ( j ) i = (cid:40) (cid:125) ( j ) i if r (0 , < P cross , or j = r ( i ) ϕ ( j ) i , otherwise , (A.26)31here r (0 ,
1) is a random number from the range [0 , r ( i ) is a random integer within (0 , X ] foreach i , while P cross is the crossover probability ranged in (0 , ϕ i using the trial solution T i is as ϕ i = (cid:26) T i , if ˜ f ( T i ) ≤ ˜ f ( ϕ i ) ϕ i , otherwise , (A.27)where function ˜ f ( · ) is given in (A.16). A.3 Sub-Procedure 1
The Sub-procedure 1 of Algorithm 1 is as follows [9, 10].
Sub-procedure 1
Apply feasible space exploration (41) through the dimensions L dim k ( E i ) r around dim k ( E i ) ofthe epicenters. For i = 1 , . . . , p obtain a T i trial solution (A.26) for ϕ i . Determine the bestsolution between ϕ i and T i via (A.27). If ˜ f (T i ) ≤ ˜ f ( ϕ i ) and T i is a non-dominatedsolution, then update N P with T i . Then, update P with the best solution, and with other p − ϕ q , q = 1 , . . . , p −
1, using the selection probabilityfunction (A.7) as Pr ( ϕ q ) = ˜ f ( ϕ q ) (cid:46) (cid:80) p − i =1 ˜ f ( ϕ i ). A.4 Notations
The notations of the manuscript are summarized in Table A.1.Table A.1: Summary of notations.
Notation Description l Level of entanglement. F Fidelity of entanglement. N Number of nodes in the network. T Number of fidelity types F j , j = 1 , . . . , T of the entangledstates.S O Objective space. S F Feasible space.L l An l -level entangled connection. For an L l link, the hop-distance is 2 l − . d ( x, y ) L l Hop-distance of an l -level entangled connection betweennodes x and y . 32 L l ( x, y ) entangled connection E L l ( x, y ) between nodes x and y . B F ( E L l ( x, y )) Entanglement throughput of an L l -level entangled connec-tion E L l ( x, y ) between nodes ( x, y ). B jF ( x i ) Number of incoming entangled states in an i th node x i , withfidelity-type j , i = 1 , . . . , N . X An N × T matrix, X = (cid:16) B jF ( x i ) (cid:17) N × T , it describes thenumber of resource entangled states injected into the nodesfrom each fidelity-type in the network, B jF ( x i ) ≥ i and j . F ( x i ) A primary objective function. It identifies the cumulativeentanglement fidelity (a sum of entanglement fidelities in x i ) after an entanglement purification P ( x i ) and an optimalquantum error correction C ( x i ) in x i .P ( x i ) Entanglement purification in x i .C ( x i ) Optimal quantum error correction in x i . (cid:104) B (cid:105) jF ( x i ) An initialization value for B jF ( x i ) in a particular node x i . E ( D i ( X )) A secondary objective function. It refers to the expectedamount of cumulative relative entropy of entanglement (asum of relative entropy of entanglement) in node x i , w j ( x i ) Quantum memory coefficient for the storage of entangledstates from the j th fidelity type in a node x i , evaluated as: w j ( x i ) = η j B jF ( x i ) + κ j (cid:104) B (cid:105) jF ( x i ),where η j and κ j are coefficients to describe the storage char-acteristic of entangled states with the j th fidelity type. τ j ( X ) Differentiation of storage characteristic of entangled statesfrom the j th fidelity type, defined as τ j ( X ) = (cid:80) Ni =1 ( w j ( x i ) − Ω) , where Ω = (cid:80) Ni =1 w j ( x i ) (cid:46) N . f C (P ( x i )) Cost of entanglement purification P ( x i ) in x i . f C (C ( x i )) Cost of optimal quantum error correction C ( x i ) in x i . C ( X ) Total cost function, defined as C ( X ) = N (cid:80) i =1 f C ( P ( x i )) + f C (C ( x i )) = N (cid:80) i =1 T (cid:80) i =1 f j B jF ( x i ) , where T is the number of fidelity types, N is the number ofnodes, f j is a total cost of purification and error correctionassociated to the j th fidelity type of entangled states.33 j Total cost of purification and error correction associated tothe j th fidelity type of entanglement fidelity. F ∗ Critical fidelity coefficient. S low , S high Sets with fidelity bounds S low ( F ) and S high ( F ) as S low ( F ) : max ∀ i F i < F ∗ ,and S high ( F ) : min ∀ i F i ≥ F ∗ . X S low Set of nodes for which condition S low ( F ) : max ∀ i F i < F ∗ holds. X S high Set of nodes for which condition S high ( F ) : min ∀ i F i ≥ F ∗ holds. S i ( X ) Cost of quantum memory usage in node x i , defined as S i ( X ) = λ (cid:80) Tj =1 α i i B jF ( x i ),where λ is a constant, α i is a quality coefficient, while Υ i isa capacity coefficient of the quantum memory. G ( X ) Main objective function, G ( X ) = max N (cid:80) i =1 F i ( X ) E ( D i ( X )). F ( N ) Minimization function for cost C ( X ). F ( N ) Minimization function for cost S ( X ). C , C , C Problem constraints. E Epicenter, represents a solution in the feasible space. L j A random location around epicenter E . D ( E ) Dispersion coefficient of an epicenter E (solution in thefeasible space). It determines the number of affected L j , j = 1 , . . . , D ( E ), locations (also represent solutions in thefeasible space) around an epicenter E . P Population P (a set of possible solutions). m Control parameter. E i An i th individual (epicenter) from the |P| individuals (epi-centers) in the population P .˜ f ( · ) Fitness function.˜ f ( (cid:104)E(cid:105) ) A maximum objective value among the |P| individuals. ϑ A residual quantity.34 R ( · ) Rounding function. q Total number of locations, q = (cid:80) |P| i =1 D ( E i ).ˆ D ( E i ) Upper bound on D ( E i ) for a given epicenter E i . d ( E i , l j ) Euclidean distance d ( E i , l j ) between an i th epicenter E i and the projection point l j of a j th location point L j , j =1 , . . . , D ( E ) on the ellipsoid around E i .dim i ( · ) An i th dimension of l j . P ( E i , L j ) Seismic power P ( E i , L j ) operator for an i th epicenter E i . Measures the power in a j th location point L j , j =1 , . . . , D ( E i ), as P ( E i , L j ) = (cid:16) d ( E i ,l j ) M ( E i , L j ) (cid:17) b b e σ ln P ( E i,Lj ),where b and b are regression coefficients, σ ln P ( E j ) is thestandard deviation, while M ( E i , L j ) is the seismic magni-tude in a location L j , while l j is the projection of L j ontothe ellipsoid around E i . M ( E i , L j ) Magnitude between epicenter E i and location L j is evaluatedas M ( E i , L j ) = (cid:18) P ( E i , L j ) b e σ ln P ( E i,Lj ) (cid:19) b d ( E i , l j ). P ∗ ( E i ) Maximal seismic power for a given epicenter E i . C ( E i ) Cumulative magnitude for an epicenter E i . E (cid:48) Highest seismic power epicenter with magnitude M (cid:16) E (cid:48) , L E (cid:48) j (cid:17) .˜ f ( E (cid:48) ) Minimum objective values among the |P| epicenters. M Control parameter, M = (cid:80) |P| i =1 M (cid:16) E i , L E i j (cid:17) ,where L E i j provides the maximal seismic power for an i th epicenter E i Φ ( E i , R k , R l ) Poisson range identifier function of E i , where R k and R l arerandom reference points. c w ( E i , R k ), c w ( R k , R l ) Weighting coefficients between epicenters E i and R k , andbetween R k and R l . D ( E p ) Poissonian distance function D ( E p ), where E p is a new so-lution. 35 ( E i ) Radius around a current solution E i , defined as r ( E i ) = χ Q ( M ) − Q ,where ˜ M is the average magnitude˜ M = |P| M = |P| (cid:80) |P| i =1 M (cid:16) E i , L E i j (cid:17) ,while Q and Q are constants, while χ is a normalizationterm.dim k ( E i ) Randomly selected k th dimension, k = 1 , . . . , dim ( E i ) of acurrent epicenter E i , i = 1 , . . . , |P| . H (dim k ( E i )) Hypocentral, provides a random displacement of dim k ( E i )using C ( E i ). L dim k ( E i ) r A random location in the k th dimension L dim k ( E i ) r arounddim k ( E i ).N ( · ) Normalization operator N ( · ) of L dim k ( E i ) r . It keeps the newlocations around dim k ( E i ) in S F , where B klow and B kup arelower and upper bounds on the boundaries of locations ina k th dimension. S -metric Hypervolume indicator. A quality measure for the solutionsor a contribution of a single solution in a solution set. S ( R ) S -metric for a solution set R = { r , . . . , r n } is as S ( R ) = L (cid:0)(cid:83) r ∈R { x ref ∠ x ∠ x | r } (cid:1) , where L is a Lebesgue measure, notation b ∠ a refers to that a dominates b (or b is dominated by a ), while x ref is a refer-ence point dominated by all valid solutions in the solutionset. f , f Objective functions. C ( x i ) Cost results from the first-type classical communicationsrelated to a x i . C ( x i ) Cost results from the second-type classical communicationswith respect to x i . E ∗ Global optima. m Number of magnitude ranges. n i Number of locations belonging to an i th magnitude range. B ( n i ) Power law distribution function for a log-scaled n i , B ( n i ) : log ( n i ) = a − b ˜ M i ,where ˜ M i is a log scaled M i , while a and b are constants.36 n i Poisson estimate of n i , as˜ n i = σ i = λ i ,where σ i is the observational variance, while λ i is the meanof a Poisson distribution. σ q Estimated uncertainty, σ q = λ ( q ) = (cid:80) mi =1 f (cid:16) ˜ M i (cid:17) , where f ( · ) is a fitting function. λ ( q ) Mean total number, λ ( q ) = (cid:80) mi =1 λ i ≈ q , where λ i is an i th component mean. B ( λ i ) Power law distribution function for λ i = ˜ n i . k itit