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Resource Prioritization and Balancing for the Quantum Internet Laszlo Gyongyosi1,2* & Sandor Imre1

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Resource Prioritization and Balancing for the Quantum Internet Laszlo Gyongyosi1,2* & Sandor Imre1 www.nature.com/scientificreports OPEN Resource prioritization and balancing for the quantum internet Laszlo Gyongyosi1,2* & Sandor Imre1 The quantum Internet enables networking based on the fundamentals of quantum mechanics. Here, methods and procedures of resource prioritization and resource balancing are defned for the quantum Internet. We defne a model for resource consumption optimization in quantum repeaters, and a strongly-entangled network structure for resource balancing. We study the resource-balancing efciency of the strongly-entangled structure. We prove that a strongly-entangled quantum network is two times more efcient in a resource balancing problem than a full-mesh network of the traditional Internet. Te quantum Internet1–30 aims to provide an adequate answer to the computational power that becomes available with quantum computers31–60. To provide a seamless transition to the legal users from the traditional Internet to the quantum Internet, the creation of advanced services and methods for the quantum Internet are emerging tasks52–54,61–67. Te quantum Internet is modeled as a quantum network consisting of quantum repeaters and entangled connections between the quantum repeaters2,66–129. Tis entangled quantum network forms a general framework for the quantum Internet, enabling long-distance quantum communications, multi-hop entangle- ment and multi-hop QKD (quantum key distribution)25, utilization of quantum protocols, advanced distributed computing, high-precision sensor networks, and the establishment of a global-scale quantum Internet. A crucial problem related to the quantum Internet is the resource optimization of the quantum repeaters and the handling of resource requirement issues such as non-servable resource requests in the quantum repeaters 18–26. Tese fundamental questions are still open and have not been addressed for the quantum Internet. Here, we defne methods for resource prioritization and resource balancing for the quantum Internet. Te aim of the proposed solutions is to optimize the resource allocation mechanisms and to reduce the resource consumption of the network entities of the quantum Internet. A model of resource consumption130–134 of quantum repeaters is proposed, and its optimization is realized through the weightings of the entanglement throughputs of the entangled connections of the quantum repeaters. We also propose a method for optimizing the entanglement swapping procedure and determine the conditions of deadlock-free entanglement swapping. For resource balanc- ing, a strongly-entangled network structure is defned. Tis network is modeled as an independent entity in the quantum Internet, composed of an arbitrary number of quantum repeaters such that all quantum repeaters are entangled with each other. Te primary aim of the strongly-entangled structure is to serve those quantum nodes that have non-servable resource requests due to resource issues or an arbitrary network issue; these quantum nodes are referred to as low-priority quantum nodes. Te strongly-entangled structure injects additional resources into the quantum network to manage the resource issues of an arbitrary number of low-priority quantum nodes. Te structure also provides optimized resource balancing for the low-priority quantum nodes. We prove the resource-balancing efciency of the strongly-entangled structure and study its fault tolerance. We show that a strongly-entangled quantum network structure, due to the advanced attributes of quantum networking, is two times more efcient in resource balanc- ing than a classical full-mesh135,136 network structure. Te novel contributions of our manuscript are as follows: 1. We defne methods and procedures for resource prioritization and resource balancing in the quantum Inter- net. 1Department of Networked Systems and Services, Budapest University of Technology and Economics, Budapest 1117, Hungary. 2MTA-BME Information Systems Research Group, Hungarian Academy of Sciences, Budapest 1051, Hungary. *email: [email protected] Scientifc Reports | (2020) 10:22390 | https://doi.org/10.1038/s41598-020-78960-5 1 Vol.:(0123456789) www.nature.com/scientificreports/ 2. Te resource prioritization covers the resource consumption optimization of the quantum repeaters via the entanglement throughput weightings, prioritization of entanglement swapping in the quantum repeaters, and deadlock-free entanglement swapping. 3. A strongly-entangled structure is defned for an optimal resource balancing. We prove the resource-balancing efciency of the proposed structure and prove its fault tolerance. We show that a strongly-entangled quan- tum network structure is two times more efcient in resource balancing than a classical full-mesh network structure. Tis paper is organized as follows. In “Preliminaries” section, preliminaries are summarized. In “Method” sec- tion, methods for resource consumption optimization are defned. “Strongly-entangled structure for resource balancing in the quantum internet” section proposes a solution for optimal resource balancing. A performance analysis is given in “Performance evaluation” section. Finally, “Conclusions” section provides the conclusions. Supplementary information is included in the Appendix. Preliminaries Basic terms. Entanglement fdelity. Te aim of the entanglement distribution procedure is to establish a d- dimensional entangled system between the distant points A and B, through the intermediate quantum repeater 1 nodes. Let d 2 , and let β00 ( 00 11 ) be the entangled state subject to be established between distant = | � = √2 | � + | � points A and B. At a particular two-partite state σ established between A and B, the fdelity of σ is evaluated as F β00 σ β00 . = � | | � (1) Without loss of generality, an aim of a practical entanglement distribution is to reach F 0.982–4,12,68,69,137,138. ≥ Entangled network structure. Let V refer to the nodes of an entangled quantum network N, which consists of a transmitter node A V , a receiver node B V , and quantum repeater nodes Ri V , i 1, ..., q . Let ∈ ∈ ∈ = E Ej , j 1, ..., m refer to a set of edges (an edge refers to an entangled connection in a graph representa- = = tion) between the nodes of V, where each Ej identifes an Ll-level entanglement, l 1, ..., r , between quantum = nodes xj and yj of edge Ej , respectively. Let N V, S be an actual quantum network with V nodes and a = | | set S of entangled connections. An L -level, l 1, ..., r , entangled connection EL x, y , refers to the shared l = l entanglement between a source node x and a target node y, with hop-distance l 1 d x, y 2 − , (2) Ll = since the entanglement swapping (extension) procedure doubles the span of the entangled pair in each step. Tis architecture is also referred to as the doubling architecture 2,68,69,138. For a particular Ll-level entangled connection EL x, y with hop-distance (2), there are d x, y 1 inter- l Ll − mediate nodes between the quantum nodes x and y. Entanglement throughput. B (Ei ) L Let F Ll refer to the entanglement throughput of a given l entangled Ei Ei connection Ll measured in the number of d-dimensional entangled states established over Ll per sec at a par- ticular fdelity F (dimension of a qubit system is d 2)2–4,12,68,69,137,138. = Ei For any entangled connection Ll , a condition c should be satisfed, as i * i c BF (E ) B (E ), for i, : Ll ≥ F Ll ∀ (3) B* (Ei ) Ei where F Ll is a critical lower bound on the entanglement throughput at a particular fdelity F of a given Ll , B (Ei ) Ei B* (Ei ) i.e., F Ll of a particular Ll has to be at least F Ll . Oscillator cycles. To quantify the entanglement throughput of the entangled connections, time is measured in number of cycles C. Te time tC of a cycle C is determined by an oscillator unit OC that is available for all the entities of the quantum network, such that tC 1/fC , where fC is the frequency of OC , with fC 1/tC. = = Defnitions. Resource consumption of a quantum repeater. Let α(Ri,Ll(k)) be the resource consumption of quantum repeater Ri associated with a k-th entangled connection L (k) , k 1, ..., z , where l is the level of l = entanglement of the connection and z is the total number of entangled connections of Ri. Let ϒ(Ri,Ll(k)) be the resource consumption of quantum repeater Ri associated with the quantum memory usage at Ll(k) ; let φ(Ri,Ll(k)) be the resource consumption of quantum repeater Ri associated with the entangle- ment purifcation of Ll(k) ; let τ(Ri,Ll(k)) be the resource consumption of quantum repeater Ri associated with the entanglement distribution to a target node B; and let ν(Ri,Ll(k)) be the resource consumption of quantum repeater Ri associated with the entanglement swapping US of Ll(k) . Ten, α(Ri,Ll(k)) can be defned as α(R ,L(k)) B (L (k))(∂(R ,L(k))) ζ (R ,L(k)) C(R ,L(k)), i l := F l i l + i l + i l (4) where the term ∂(Ri,Ll(k)) is defned as ∂(R ,L(k)) ϒ(R ,L(k)) φ(R ,L(k)) τ(R ,L(k)) ν(R ,L(k)), i l := i l + i l + i l + i l (5) Scientifc Reports | (2020) 10:22390 | https://doi.org/10.1038/s41598-020-78960-5 2 Vol:.(1234567890) www.nature.com/scientificreports/ where BF (Ll(k)) is the entanglement throughput (Bell states per C) of the entangled connection Ll(k) , while ζ(Ri,Ll(k)) identifes the resource consumption of quantum repeater Ri associated with the path selection, and C(Ri,Ll(k)) refers to the cost of auxiliary classical communications. Set of outcoming entangled states. Let ρA be an input entangled density matrix (i.e., half pair of an entangled state) in quantum repeater Ri , and let A (ρA) be the set of possible r outcoming entangled states in Ri, A (ρ ) σ ,1, ..., σ , , A := B B r (6) where σB,i is the i-th possible outcoming density matrix.
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