In the quantum Internet [1, 2, 3, 4], noise, eavesdropping and other random failures on the entangled links lead to a time-varying link stability, which makes topology adaption of quantum repeater networks an emerging issue [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. Because the failures of the entangled links are random, the actual shortest paths must be updated dynamically, which is particularly important for a seamless transmission of quantum information. In a quantum Internet scenario, the quantum repeater nodes are connected through several different entanglement levels [1, 2, 3, 4, 8, 9, 10, 11, 12, 20, 21, 22, 23, 24, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45] making the problem harder. In this multi-level quantum network, the entanglement level of a link between the quantum nodes refers to the level of entanglement shared between the source and the target nodes incident to that link. The level of an entangled link not only identifies the number of spanned quantum nodes (hop-distance), but also provides a base-point on the stability of that link between the nodes.
Here, we define a dynamic topology adapting method to efficiently manage the random link errors of the quantum links of the quantum Internet. The vulnerability of the entangled links requires topology adaption to provide seamless communication between the quantum nodes. The quantum links are characterized by a given probability of existence (stability) in an overlay quantum network, which describes the probability of a given level of entanglement shared between nodes. The proposed adapting algorithm determines a set of entangled links in a proactive way for the formulation of a new shortest path in the updated topology of the overlay entangled quantum network. Specifically, to minimize the risk of a possible failure, we introduce a threshold parameter for each entanglement level to construct alternative paths in the network before a given entangled link becomes critical. In particular, the shortest path is determined in a base-graph, which allows us to perform efficient decentralized routing in the quantum repeater network . The base-graph is a -dimensional -size lattice graph that contains the mapped entangled overlay quantum network so that all nodes and connections are preserved. The base-graph allows us to model the entangled link in a distance-based manner and to perform efficient decentralized routing in the network.
The routing metric in the base-graph is the number of links between a source and the target (diameter), while the set of new entangled links is determined with respect to their probability of existence. Because the probability of the stability of a given quantum link depends on the level of entanglement shared through that link, the metric of dynamic adaption depends on the actual overlay quantum network setting. Therefore, the algorithm provides a consecutive update of quantum links and prepares a network for changing link conditions to improve the reliability.
The updated configuration of links provides an input for the decentralized routing scheme, which is performed purely in a base-graph . We assume a multiuser scenario with multiple parallel demands in the network; therefore, the task of the dynamic topology adaption algorithm is to maximize the aggregate link probability over all the demanded paths of the users.
We also study the problem of entanglement assignment and define an optimization procedure. The problem with entanglement assignment is that an intermediate quantum repeater node occurs if the demands of two or more source nodes with respect to a given target node are interfering; i.e., several source nodes would like to reserve a given resource entangled state in an intermediate quantum node to establish entanglement with a target node through the intermediate node by the operation of entanglement swapping. The solution to this problem requires the definition of disjoint sets in an intermediate quantum node, a set of entangled resource states (resource set), and a set of interfering entangled states (interference set). With the utilization of our base-graph approach, we characterize the computational complexity of the problem and prove that efficient optimized solutions exist.
This paper is organized as follows. In Section 2, we discuss the proposed topology adaption algorithm for the quantum Internet. In Section 3, we study the problem of entanglement assignment and provide an optimization for managing interfering demands. Finally, Section 4 concludes the results. Supplemental information is included in the Appendix.
2 Topology Adaption
Let be an entangled overlay quantum network, , where is an initial set of entangled links , , where is -level entangled link, , while is the set of quantum nodes of the overlay quantum network . Assuming a quantum repeater network with the doubling architecture [1, 2, 3, 4], for an -level entangled link the hop distance between and , where , is [1, 8], and each -level entangled link can be established only with a given probability, , that is defined without loss of generality as
where is the success probability of the entanglement swapping procedure between and , is the photon loss probability of the quantum link between nodes and , while is the fidelity of the -dimensional maximally entangled system associated to link , respectively.
The aim of the link configuration algorithm is to determine the shortest path between a transmitter node and a target node via the updated link set , of -level entangled links, where set identifies those links which have the highest probability in the network. Each link of is characterized by a metric called entanglement throughput, which specifically is the number of -dimensional maximally entangled states per second of a particular fidelity , and it is denoted by for a given in the modified entangled overlay quantum network .
To describe the entangled connections between the quantum nodes, we characterize a base-graph. The base-graph is an abstracted graph that contains all information about the overlay (physical) quantum network (node positions, hop-distances, established entangled links, link properties). As it has been shown in , the base-graph approach allows us to perform efficient decentralized routing in a quantum Internet scenario.
Let be a -dimensional -size base-graph of an entangled overlay quantum network . Let two nodes in , and let and be the maps of in , where is a mapping function  that maps from onto . As it can be verified , the probability that and are connected through an -level entanglement in is as
where is the probability that nodes are connected through an -level entangled link in the overlay quantum network .
Due to noise or other random link failure, the probability of a given -level entangled link is not constant in the overlay quantum network . This fact has the consequence that an actual shortest path determined for set cannot be used further, and a new shortest path needs to be determined. The aim of the adaption algorithm is to update the probabilities in the base-graph according to a threshold for a given -level entangled link to determine the actual shortest path for . Only that link will be included in the updated set of links for which the condition holds. The algorithm is performed for all entangled contacts of a given node for . Then, we apply the decentralized routing method defined in  to determine the shortest path by at most steps in a -dimensional -size base-graph .
The topology adaption algorithm in is given in Algorithm 1.
, estimateof an -level entangled link , where and are nodes in the overlay network , while is the level of entanglement between and . Let refer to the initial set of entangled links. Step 2. Using a threshold probability of an -level entangled link, update for all node pairs via the following rule:
3 Entanglement Assignment
In this section, we formulate the problem of entanglement assignment for an entangled overlay quantum network , where is determined by the topology adaption algorithm in the , the -dimensional base-graph of the entangled overlay quantum network .
Let and refer to the source and target nodes associated with the demand of a user in , . Let refer to the entangled link , and let be the set (resource set) of -dimensional maximally entangled states shared between through link , as [1, 8]
where is the number of entangled states established between nodes and through .
Let the set of entangled links determined in by Step 2 of . The problem of the entanglement assignment between and for all is subject to the following minimization, precisely:
where is the number of users, is the resource set of -dimensional maximally entangled states available through the entangled link , is a set of edges, is the probability of existence of the -level entangled link in the overlay network , is the corresponding probability of the -level entangled link in the base-graph , and is a variable which equals 1 if the maximally entangled resource state is assigned to the -th user through link or 0 otherwise.
Let refer to the number of -dimensional maximally entangled states per second of a particular fidelity F available through the entangled link , and let be the demand of user with respect to the number of -dimensional maximally entangled states per second of a particular fidelity F through link . Specifically, for a given , the following relation holds :
As it can be verified via the flow conservation rules (Kirchhoff’s law) , for all users , a quantity can be defined with respect to , without loss of generality, as
where is a node associated to a given user , such that is a link incident out of , while is a link incident onto node in the base-graph , while
where and are the source and target nodes of .
Let’s assume the situation of a source node of user , and a source node of user would like to share entanglement with a target node through an intermediate node. Let refer to the resource set of entangled states shared between and through an entangled link . Let each source node be associated to a given entangled state from (resource state) in .
Let’s index the interfering queries of , as , where refers to the demand of , is the interfering demand of , and is a set of interfering demands (interference set) with respect to .
The next constraint assures that for a given maximally entangled resource state , at most one interfering entangled state (i.e., or ) is assigned from for each as
where is a variable equal to 1 if is assigned to resource state or 0 otherwise. The parallel serving of the interfering demands is trivial if . On the other hand, if , the problem requires an optimization procedure to achieve an optimal assignment.
The optimal assignment of the interfering entangled states of at a given set of is characterized by the following constraints.
The constraints of the optimization method is defined in Method 1.
The optimization problem for a quantum node is illustrated in Fig. 2. Source nodes , are associated with the resource entangled state from set via demands in an intermediate node to establish entanglement with a distant target node .
Set consists of the stored entangled resource states of , which are established between and through link , while set refers to the stored entangled resource states of and established via . The entangled resource states between and form the set of interfering entangled states. The link configuration is determined by the link selection algorithm . As the allocation of the resource entanglement is achieved, the entanglement is extended from the source nodes , to the target node through the intermediate node by the entanglement swapping operation (not shown). The operation is applied on the entangled states of and and and , respectively.
3.2 Computational Complexity
Let’s assume an entangled overlay quantum network with nodes. At a given interfering set and resource set of , the aim is to determine the optimal assignment of the entangled states of . Let , where is a minimized value.
An assignment of the states of to a resource set of link is referred as valid if consists of exactly entangled states, . If , then , in which case the assignment is referred to as not valid. Assignments for all sets can be performed in at most steps by some fundamental theory. By a similar assumption, whether different states from are assigned to different states of given set can also be determined in at most steps. In fact, these complexities prove that the problem of assignment of the entangled states belongs to the class of NP problems .
In particular, it can also be verified that there exists a known NP-complete problem, which polynomially reduces to the entanglement assignment problem. Precisely, this problem is the optimal vertex-coloring of a graph of conflicts (graph-coloring)  and is discussed as follows.
3.3 Polynomial Reduction
Let a graph of conflicts, where is a set of vertices, while is a set of edges, . If two vertices are connected by an edge , the vertices are referred to as conflicting vertices. The problem is therefore to find the optimal assignment of colors to vertices from using exactly colors such that any two conflicting vertices of receive different colors.
As it can be verified, identifying the vertices of by the entangled states of via set , , where a set consist of all entangled states of node , an edge between represents a situation when two states are interfering with respect to a given state in , thus vertices require different colors.
and a set
of all interfering entangled states, where
Specifically, the problem for a given is therefore that the conflicting vertices of have to be assigned to different states from a corresponding set .
Without loss of generality, since the sets depends only on the determination of the link set , using a -dimensional -size base-graph , the entanglement assignation problem for a given can be solved in at most steps by our algorithm. As follows, if it is feasible to color vertices from using different colors, then any valid coloring proper the assignment of the different states from for the interfering states of .
In particular, these prove that the known NP-complete problem of vertex-coloring of a graph of conflicts can be transformed in polynomial time to the entanglement assignment problem, thus it polynomially reduces to our problem.
These facts prove that the entanglement assignment problem is NP-complete; therefore, to provide a time-efficient solution, an optimized approach is required.
In our case, the time complexity is determined only by , as the problem is to find a shortest path between a neighbor of the intermediate node, which is not the target node, and the target node. As follows, the overall complexity to solve (5) is bounded from above by . If the intermediate node has no other neighbor besides the target node, then no other path exists between the intermediate node and the target node, requiring the establishment of a new entangled state between the intermediate node and the target node.
We defined a dynamic topology adaption algorithm for the quantum Internet. The method determines a set of entangled links to establish a new shortest path between the nodes. We utilized a base-graph construction for the efficient determination of the shortest paths required by the parallel demands of a multiuser network scenario. We studied the problem of entanglement assignment and defined an optimization procedure. The results can be applied directly for routing optimization problems in entangled quantum networks and allow for the provision of reliable quantum communications in the presence of time-varying stability quantum links of the quantum Internet.
This work was partially supported by the National Research Development and Innovation Office of Hungary (Project No. 2017-1.2.1-NKP-2017-00001), by the Hungarian Scientific Research Fund - OTKA K-112125 and in part by the BME Artificial Intelligence FIKP grant of EMMI (BME FIKP-MI/SC).
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Appendix A Appendix
The notations of the manuscript are summarized in Table A.1.
|L1||Manhattan distance (L1 metric).|
|Level of entanglement.|
|An -level entangled link. For an link, the hop-distance is .|
|Hop-distance of an -level entangled link between nodes and .|
|Fidelity of entanglement.|
|-level (direct) entanglement, .|
|-level entanglement, .|
|-level entanglement, .|
|An edge between quantum nodes and , refers to an -level entangled link.|
|Probability of existence of , .|
|Overlay quantum network, , where is the set of nodes, is the set of edges.|
|Set of nodes of .|
|Set of edges of .|
|An -size, -dimensional base-graph.|
|Size of base-graph .|
|Dimension of base-graph .|
|Transmitter node, .|
|Receiver node, .|
|A repeater node in , .|
|Identifies an -level entanglement, , between quantum nodes and .|
|Let , refer to a set of edges between the nodes of .|
|Position assigned to an overlay quantum network node in a -dimensional, -sized finite square-lattice base-graph .|
|Mapping function that achieves the mapping from onto .|
|L1 distance between and in . For , evaluated as
|The probability that and are connected through an -level entanglement in .|
|Normalizing term, defined as .|
|Constant defined as
where is the probability that nodes are connected through an -level entanglement in the overlay quantum network .
|Topology adaption algorithm in base-graph .|
|Updated set of links, , of -level entangled links, . Set of entangled links that have the highest probability in the network.|
|Entanglement throughput. Number of -dimensional maximally entangled states per second of a particular fidelity .|
|Threshold for a given -level entangled link from set to determine the actual shortest path, .|
|Diameter between source node and target node . Refers to the maximum value of the shortest path (total number of edges on a path) between and .|
|Resource set, a set of -dimensional maximally entangled states shared between through link ,
where is the number of entangled states established between nodes and through .
|Parameter subject to a minimization.|
|Number of users.|
|Probability of existence of an -level entangled link in the overlay network .|
|Probability of the -level entangled link in the base-graph .|
|Variable, equals 1 if the maximally entangled resource state is assigned to the -th user through link , 0 otherwise.|
|A demand of user with respect to the number of -dimensional maximally entangled states per second of a particular fidelity F through link .|
|A node, associated to a given user such that is a link incident out of , while is a link incident onto node in the base-graph.|
|Parameter for flow conservation rules (from Kirchhoff’s law).|
|Parameter for flow conservation rules (from Kirchhoff’s law).|
|Parameter for flow conservation rules (from Kirchhoff’s law).|
|A source node of .|
|A target node of .|
|Source node of user .|
|Source node of user .|
|An intermediate node.|
|Resource state, an entangled state from the resource set.|
|A demand from a source node .|
|An interfering demand from a source node .|
|Interference set, a set of interfering demands with respect to resource state .|
|A variable, equal to 1 if is assigned to resource state , 0 otherwise.|
|Cardinality of interference set , .|
|A graph of conflicts, where is a set of vertices, while is a set of edges, .|
|An edge in a graph of conflicts , the vertices are conflicting vertices.|
|A set of entangled states of node .|
|A set of vertices of in a graph of conflicts , , .|
|Set of interfering entangled states, , .|
|Variable, identifies the number of interfering states in set .|