1 Introduction
Entangled quantum networks formulated by quantum repeaters are core elements of the quantum Internet [1, 2, 3]. In a quantum repeater network, the quantum nodes share quantum entanglement with each other [4, 5, 6, 7, 8, 9, 10, 11, 12, 13], and the shared entangled states are stored in the local quantum memories of the nodes [19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. In an entangled network structure, the level of the shared entanglement (number of spanned nodes, i.e., the current level of entanglement swapping can be different, which leads to a heterogeneous multilevel entangled quantum network architecture in general [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39]. Because the shared entangled states are stored in the local quantum memories of the quantum nodes, a quantum memory failure in a quantum node can destroy several entangled contacts in the actual shortest main path established between a source and target node [40, 41, 42, 43, 44, 45, 46, 47, 48]. A quantum memory failure event therefore can have serious consequences in the repeater network, for it leads not only to an immediate requirement for an adaptive routing method that reacts to the dynamic network topology changes, but also to the need for an efficient method for the determination of shortest nodedisjoint replacement paths [8, 9, 10, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63].
A quantum memory failure event in a quantum node affects a number of entangled connections in the network, and as a corollary, those quantum nodes that have shared entanglement with the given node are also affected by the quantum memory failure. The aim of the nodedisjoint [14, 15, 17]
replacement paths is to omit the affected quantum nodes to provide immediate and seamless network transmission. Depending on the number of stored entangled contacts, the quantum nodes can be classified as either standard or highdegree nodes. Since a memory failure in a highdegree quantum repeater will result in the loss of the highest number entangled connections, a plausible approach would be omitting these nodes from the actual main path. Specifically, this requirement allows us to utilize the omitted highdegree nodes in the replacement paths, based on the assumption that the replacement paths will serve as merely a temporary solution while the reestablishment of the lost entangled contacts is in progress on the main path.
It is therefore an emerging task to provide seamless transmission in a quantum network during quantum memory failure scenarios, which demands the fast determination of shortest nodedisjoint replacement paths between the quantum nodes. Since most of the currently available quantum routing methods [1, 5, 6, 7] are based on, or a variant of Dijkstra’s shortest path algorithm [49], the efficiency of these routing approaches is limited.
Here, we define a dynamic adaptive routing method for the control of quantum memory failures in the quantum Internet. Our method provides a framework for the discovery of shortest nodedisjoint replacement paths in the entangled network structure of the quantum Internet. The proposed solution minimizes the number of losable entangled contacts during a quantum memory failure in a mainpath node and determines the shortest replacement paths that omit the quantum nodes affected by the memory failure. In particular, all paths are determined in a basegraph [8, 9, 10] consisting of the maps of the quantum nodes and the entangled connections of the overlay quantum network. The method requires no special devices for the practical implementation, therefore can be applied straightforwardly in quantum networking and in the quantum Internet. Our approach provides a solution for handling quantum memory failures through shortest nodedisjoint replacement paths, determined by efficiently adaptive decentralized routing in the quantum Internet.
The novel contributions of our manuscript are as follows:

We define a dynamic adaptive routing method for the management of quantum memory failures in the quantum Internet.

The proposed algorithm determines shortest nodedisjoint replacement paths in the entangled network structure of the quantum Internet, and minimizes the number of losable entangled contacts in a mainpath.

The shortest nodedisjoint replacement paths in the entangled quantum network are determined in a decentralized manner with high computational efficiency.
This paper is organized as follows. In Section 2, we characterize the problem and the impacts of a quantum memory failure scenario in an entangled quantum network. In Section 3, we discuss the determination of shortest nodedisjoint replacement paths and evaluate the complexity of the method. A performance evaluation is included in Section 4. Finally, Section 5 concludes the paper. Supplementary material is included in the Appendix.
2 System Model
Let us assume that an overlay entangled quantum network is given, where the set of entangled connections is determined in the basegraph [8] such that , where
is a threshold probability of an
level entangled connection .Without loss of generality, the level of an entangled connection is defined as follows. For an level entangled connection, the hop distance between quantum nodes and
(1) 
with intermediate nodes between the nodes and . The probability that an level entangled connection exists between nodes is , which depends on the actual network.
A repeater node is referred to as a highdegree node, , if , , where is the threshold degree of node set determined for the given entangled overlay quantum network .
In our network model, the highdegree nodes are omitted from the shortest main path . The explanation of this phenomenon is as follows. Since a highdegree node stores the highest number of entangled quantum systems in its local quantum memory, a quantum memory failure in will result in the loss of the highest number of entangled connections in the quantum network. A rational decision is therefore explained as follows. Use only the standard (nonhighdegree nodes) in , since if a quantum memory failure occurs in a standard node of , the highdegree nodes still can participate in a shortest replacement path . It is because a replacement path is used for only a short time while the reestablishment of the destroyed entangled contacts of in is in progress. As the entangled connections of are restored after the memory failure, the transmission continues on the main path .
Because the main path omits highdegree nodes, the average length (number of nodes) of the path increases with respect to the shortest path in , but the risk that a high number of entangled connections will vanish during a quantum memory failure is therefore reduced to a minimum. The length of the replacement path will be shorter than , for can contain the highdegree nodes.
Without loss of generality, in a quantum communication scenario, a quantum path with highest number of entangled connections usually means high communication efficiency and stability [1, 5, 6, 7]. It is also the situation for the shortest path determined by our method, however, these attributes are achieved through the application of an increased number of non highdegree nodes, instead of the use of some highdegree nodes. In the proposed model, a high number of the nonhighdegree nodes in the quantum network also yields a high number of entangled connections for the paths; therefore the shortest path will have a high efficiency, reliability and stability.
A quantum memory failure situation in an overlay quantum repeater network is illustrated Fig. 1. The actual shortest path contains no highdegree repeater nodes, for a memory failure would destroy a high number of entangled contacts in the network (Fig. 1(a)). A quantum memory failure in an intermediate (nonhighdegree) quantum repeater node requires the immediate establishment of a replacement shortest path in the basegraph. The new shortest path contains the highdegree repeater nodes and , for is used only while the reestablishment of the entangled contacts of is not completed (Fig. 1(b)).
2.1 Quantum Memory Failures
The proposed method utilizes different coefficients for the entangled connections of a shortest main path and shortest replacement path , which distinction defines the problem of finding nodedisjoint paths of a multilevel entangled quantum network. Particularly, this problem is analogous to a minsum (minimize the sum of costs of connection paths) problem [14, 15, 16, 17] in a multicost network, which is an NPcomplete problem [14, 16]. Note that similar to the determination of a shortest main path , the replacement path is also determined by our decentralized routing in a basegraph.
2.1.1 Path Finding
Here, we summarize the path finding method of [8]. The basegraph of an entangled quantum network is determined as follows. Let be the set of nodes of the overlay quantum network. Then, let be the dimensional, sized finite squarelattice basegraph, with position assigned to an overlay quantum network node , where is a mapping function which achieves the mapping from onto [8].
Specifically, for two network nodes , the L1 metric in is , , and is defined as
(2) 
The basegraph contains all entangled contacts of all . The probability that and are connected through an level entanglement in is
(3) 
where
(4) 
is a normalizing term [8], which is taken over all entangled contacts of node in , while is a constant defined as
(5) 
where is the probability that nodes are connected through an level entanglement in the overlay quantum network .
For an level entanglement between and , in is evaluated as
(6) 
The routing in the dimensional basegraph is performed via a decentralized algorithm as follows. After we have determined the basegraph of the entangled overlay quantum network , we can apply the L1 metric to find the shortest paths. Since the probability that two arbitrary entangled nodes are connected through an level entanglement is (see (3
)), this probability distribution associated with the entangled connectivity in
allows us to achieve efficient decentralized routing via in the basegraph.Using the L1 distance function, a greedy routing (which always selects a neighbor node closest to the destination node in terms of distance function and does not select the same node twice) can be straightforwardly performed in to find the shortest path from any quantum node to any other quantum node, in
(7) 
steps on average, where is the size of the network of .
Note that the nodes know only their local connections (neighbor nodes) and the target position. It also allows us to avoid deadend nodes (where the routing would stop) by some constraints on the degrees of the nodes, which can be directly satisfied through the settings of the overlay quantum network.
2.1.2 Shortest Paths at a Quantum Memory Failure
Using the decentralized routing (see Section 2.1.1) in , the shortest path with respect to the scaled coefficient can be determined in at most steps, as follows.
Using (3), the term can be expressed as
(8) 
thus at a given , the distance function between is as
(9) 
Using the maps of the nodes and the scaled cost , a basegraph is constructed with transformed node positions as follows.
The aim is then to find at a given , via a target distance .
To determine the scaled positions, we first identify by in (3) as
(10) 
from which the target distance between is as
(11) 
The distance function in (11) is therefore results of the scaled positions in at a given reference position.
Specifically, to establish a given shortest main path between a source node and target node such that contains no highdegree repeater node , a given entangled connection , between nodes in the main path is weighted by the coefficient :
(12) 
where is defined as a normalization, precisely
(13) 
where
(14) 
where is the number of shortest paths (of the same minimal length) between nodes and traversing node , and where is the number of shortest paths (of the same minimal length) between nodes .
Let us assume that in an intermediate repeater node , a quantum memory failure occurs on the main path , which destroys all entangled contacts of that node. For seamless communication, an immediate shortest replacement path has to be established between nodes . The replacement shortest path, however, can contain the highdegree nodes , which have been removed from the main path . The replacement shortest path is aimed to serve as only a temporary path. It replaces the main path while the entangled contacts of have not been completely reestablished.
For a given connection of the replacement shortest path , the connection coefficient (12) is redefined as
(15) 
which provides a normalized metric for in .
Using coefficients (12) and (15), the optimization problem formulates a minimization, without loss of generality as
(16) 
where is the set of users, is the index of a given user , is a set of entangled (directed) connections, is the index of a given connection , is a variable that equals 1 if the entangled connection is used by the main path associated with user (0 otherwise), and is a variable that equals 1 if the entangled connection is used by the replacement path associated with user .
The optimization in (16) is therefore a minimization of the overall cost coefficient of the flows by means of main path and replacement path , subject to some constraints. These constraints are described next.
Particularly, for the main path , a flow conservation rule [14, 16] leads to precisely
(17) 
where
(18) 
where is a node associated with the demand of user , where is an egress connection incident associated with node , where is an ingress connection incident associated with node , and where
(19) 
Using (18) and (19), can be evaluated as
(20) 
Similarly, for the replacement path , the following constraint is defined via (18) and (19):
(21) 
For a given entangled connection , the following relation states that the requested number of entangled states of a particular fidelity on the main path and replacement path is bounded as:
(22) 
where refers to the number of dimensional maximally entangled states per second of a particular fidelity available through the entangled connection (entanglement throughput), and where is the demand of associated with the entangled connection with respect to the number of dimensional maximally entangled states per second of a particular fidelity .
Because and are nodedisjoint paths in our model, the following constraint holds for a given and :
(23) 
3 Shortest Replacement Paths in the Entangled Network
Here we discuss a heuristic to solve (
16) via the determination of a set of nodedisjoint replacement paths in the basegraph of an entangled quantum repeater network . The algorithm focuses on a given demand of a user , with a source node and target node .Let identify the cost function of the algorithm such that if a given connection belongs to the main path , then (12), whereas if belongs to a replacement path , then (15). Thus,
(24) 
Utilizing the notations of the frameworks KPA [16, 17] and KPI [14, 15], some preliminaries for our scheme are as follows. Let set refer to the previously discovered nodedisjoint paths, where refers to the main path, i.e., . Specifically, for each nodedisjoint path , , a cost matrix is defined, which is a matrix of connection costs , where is an auxiliary cost of entangled connection of the th path. The matrix is used to calculate the concurring path cost such that the cost of the concurring entangled connections is increasing by a given value.
Particularly, for a given set of already discovered paths, a th path identifies a current (candidate) path to be discovered. A given entangled connection of a path is identified as a prohibited entangled connection with respect to if is incident to any transit quantum node of path [14, 16].
A given is referred to as a concurring entangled connection with respect to a given path if is incident to any common transit quantum node of also used by any other of the paths from the set of the previously discovered paths. Without loss of generality, the initial matrix provides the initial path cost for a given path to increase the cost of each concurring entangled connection of , where is an initial cost of entangled connection and where is an entangled connection on path .
Let be the matrix of replacement path coefficients, with for all entangled connections of a current path associated with a user . For a given matrix , let refer to the total cost of a path .
For a current path index , let refer to a matrix of coefficients of entangled connection , where is an auxiliary cost of .
The steps of the method are summarized in Algorithm 1. The algorithm determines nodedisjoint paths for a demand of a user. The main path is identified first and followed by the replacement paths. For a given th path, the cost of any prohibited entangled connection is increased by the cost of all previously discovered paths of demand for which the given entangled connection is prohibited [14, 16]. Traversing the prohibited entangled connections with respect to a given th path therefore results in an increased coefficient. The cost of concurring entangled connections increases if a given th path has common entangled connections with the paths.
3.1 Discussion
A brief description of the algorithm follows.
In Step 1, some initialization steps are made and an actual shortest main path is determined for the next calculations.
In Step 2, some steps for the next (th) nodedisjoint path (candidate path) are performed. In particular, the coefficients of the prohibited entangled connections that are traversed by the actual main path are increased by a given quantity. This step aims to avoid a situation in which the establishment of the next nodedisjoint path of a given user fails. Some calculations are performed for the th path using the already determined set of nodedisjoint paths . The cost of any prohibited entangled connection is increased by the total cost of a given path .
In Step 3, the th disjoint (shortest) path is determined by the decentralized algorithm in the basegraph , which contains the scaled of the nodes of . The basegraph is evaluated from , and it contains the scaled maps of the nodes of the entangled overlay quantum network and the scaled coefficients of the entangled connection, using .
In , a given contact between two nodes is characterized by the scaled coefficient , where is the cost of an entangled connection in . Particularly, is constructed such that the distribution of the scaled coefficients follows an inverse power distribution, and the decentralized routing scheme can determine the shortest path in in at most (7) steps [8].
Step 4 deals with the situation when the th path is not nodedisjoint with the paths of . If more than one already discovered path from traverses a given entangled connection, then the cost of the concurring entangled connection is increased by the total cost of path . Step 4 is completed by the incrementing and checking of a concurrence counter.
Step 5 handles the case when a th path is nodedisjoint with the paths of . In this situation, is incremented: ; if , the iteration stops and returns the nodedisjoint shortest replacement paths for a given main path ; otherwise, the algorithm jumps back to Step 1 with the actual value of .
In Step 6, after set is determined, the total cost of the replacement paths of demand of is precisely as follows:
(25) 
where is an entangled connection on path . The total cost for the set of the nodedisjoint paths is therefore
(26) 
The basegraphs and of a given overlay quantum repeater network are illustrated in Fig. 2.
3.2 Computational Complexity
In particular, each of the shortest paths is determined by a decentralized routing algorithm in a dimensional size basegraph, therefore the overall complexity the proposed method is bounded from above by
(27) 
for a given maximum allowable number of concurring entangled connections.
3.3 Practical Implementations
An experimental implementation of the proposed method in a stationary quantum node can be achieved by standard photonics devices, quantum memories, optical cavities and other fundamental physical devices [11, 12, 13, 28, 38, 39, 40, 41, 42, 43, 44] required for practical quantum network communications [46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60]. The quantum transmission and the auxiliary classical communications between the nodes can be realized via standard links (e.g., optical fibers, wireless quantum channels, freespace optical channels, etc) via the integration of the fundamental quantum transmission protocols of quantum networks [1].
4 Performance Evaluation
In this section, we compare the performance of our scheme with the KPA, KPI, and the successively shortest linkdisjoint paths (KSP) [15] algorithms.
The KPA and KPI algorithms are based on Dijkstra’s shortest path [49] algorithm. As follows, at a particular , the computational complexity of these schemes is [14, 15, 17].
The worstcase complexity of the KSP algorithm can be evaluated in function of the number of disjoint paths, as [15].
Fig. 3. illustrates the performance comparisons, and refers to the number of operations. The performance of our scheme is depicted Fig. 3(a). In Fig. 2(b) the performance of the KPA, KPI algorithms is depicted. In Fig. 3(c) the performance of the KSP algorithm is shown.
For the analyzed parameter ranges, in our algorithm is maximized as , while for the compared methods the resulting quantities are , and , respectively.
As a conclusion, in comparison with the KPA, KPI and KSP methods, our solution provides a moderate complexity solution to determine the connectiondisjoint paths in the quantum network, for an arbitrary number of and .
5 Conclusions
We defined an adaptive routing scheme for the handling of quantum memory failures in the entangled quantum networks of the quantum Internet. The main path contains no highdegree repeater nodes, for in a case of quantum memory failure, these quantum nodes result in the highest number of lost entangled contacts in the network. The method finds the shortest nodedisjoint replacement paths between source and target quantum nodes. The replacement paths serve as temporary paths until all destroyed entangled contacts are completely reestablished between the repeater nodes. All pathsearching phases are performed on a basegraph of the overlay quantum network to provide an efficient computational solution. The scheme can be directly applied in practical quantum networks, including longdistance quantum communications.
Acknowledgements
This work was partially supported by the National Research Development and Innovation Office of Hungary (Project No. 20171.2.1NKP201700001), by the Hungarian Scientific Research Fund  OTKA K112125 and in part by the BME Artificial Intelligence FIKP grant of EMMI (BME FIKPMI/SC).
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Appendix A Appendix
a.1 Abbreviations
 NP
 KPA

KPenalty
 KPI

KPenalty with Initial cost matrix
 KSP

KSuccessively shortest link disjoint Paths
a.2 Notations
The notations of the manuscript are summarized in Table A.1.
Notation  Description 
L1  Manhattan distance (L1 metric). 
Level of entanglement.  
Fidelity of entanglement.  
An level entangled connection. For an entangled connection, the hopdistance is .  
Hopdistance of an level entangled connection between nodes and .  
level (direct) entanglement, .  
level entanglement, .  
level entanglement, .  
An edge between quantum nodes and , refers to an level entangled connection.  
Probability of existence of an entangled connection , .  
Overlay quantum network, , where is the set of nodes, is the set of edges.  
Set of nodes of .  
Set of edges of .  
An size, 
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