1 Introduction
Quantum entanglement has a central role in the quantum Internet [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], quantum networking [11, 12, 13, 14, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], and longdistance quantum key distribution [1, 22, 30, 43]. Entanglement distribution is a crucial phase for the construction of the entangled core network structure of the quantum Internet. In a quantum Internet scenario, quantum entanglement is a preliminary condition of quantum networking protocols [30, 31, 32, 33, 34, 35, 36, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52]. Distant quantum nodes that share no quantum entanglement must communicate with their direct neighbors to distribute entanglement. To aim of entanglement distribution is to generate entanglement between a distant source node and a target node through a chain of intermediate quantum repeater nodes [55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67]. The intermediate quantum repeater nodes receive the entangled states, store them in their local quantum memories [43, 53, 54], and apply a unitary operation (called entanglement swapping [1, 2, 3, 22]) to extend the range of quantum entanglement. Storing quantum entanglement in the quantum nodes’ local quantum memories adds noise to the distribution process, since quantum memories are nonperfect devices [36, 37]
and the error probabilities evolve in time
[1, 59, 60, 61, 62, 63, 64]. As the error pattern of the evolution the quantum memories is predictable, the nodes can be characterized by a given storage success probability after a given time from the start of the storage.The fidelity of entanglement [38, 39, 40] is another critical parameter for entanglement distribution. In a quantum network with a chain of repeater nodes between a source and target nodes, for all pairs of entangled nodes (e.g., for nodes that share a common entanglement) a given lower bound in the fidelity of entanglement must be satisfied, otherwise the entanglement distribution fails [1, 41]. The stored entangled states have a given amount of fidelity that is determined by the transmission procedure, such as the noise of the quantum channel, etc. The evolution of a given entangled system’s fidelity parameter is timevarying in quantum memory, since it evolves through time, from the beginning of storage to the actual current time. Therefore, it is necessary to consider the predictability of the evolution of both the error patterns of the nodes’ local quantum memories, and the evolution of the stored quantum states’ fidelity of entanglement. In our model, using these parameters, we define an appropriate cost function for the realization of entanglement distribution.
Here we define the method of opportunistic entanglement distribution for the quantum Internet. The proposed scheme utilizes a cost function that accounts for the error patterns of local quantum memories and also the evolution of entanglement fidelities. The opportunistic model defines distribution sets in the entangled quantum network of the quantum Internet. A distribution set contains those quantum nodes for which our cost function picks up a local minimum in comparison to the cost of the other nodes in the given distributing set. The distribution set selects a lowestcost node from a given set of nodes to provide a maximal usability of stored entanglement. The cost function ensures that the nodes selected for entanglement distribution allow the lowest deviation in the entanglement fidelity from the start of storage, and that the behavior of the quantum memory error follows a predicted error model with respect to a given node pair. We also derive the computational complexity of the proposed method. The solution provides an easilyapplicable, lowcomplexity solution for highfidelity entanglement distribution in the quantum Internet.
The novel contributions of our manuscript are as follows:

We define the method of opportunistic entanglement distribution for the quantum Internet.

The proposed opportunistic model defines distribution sets that are aimed to select those quantum nodes for which the cost function picks up a local minimum. The cost function utilizes the error patterns of the local quantum memories and the predictability of the evolution of the entanglement fidelities.

Our method provides efficient entanglement distributing with respect to the actual statuses of the local quantum memories of the node pairs.

We derive the computational complexity of the model.
2 Preliminaries
2.1 System Model
The quantum Internet setting is modeled as follows [8]. Let refer to the nodes of an entangled quantum network , with a transmitter quantum node , a receiver quantum node , and quantum repeater nodes , . Let , , refer to a set of edges between the nodes of , where each identifies an level entangled connection, , between quantum nodes and of edge , respectively. The entanglement levels of the entangled connections in the entangled quantum network structure are defined as follows.
2.1.1 Entanglement Levels
In a quantum Internet setting, an entangled quantum network consists of singlehop and multihop entangled connections, such that the singlehop entangled nodes^{1}^{1}1The level entangled nodes refer to quantum nodes and connected by an entangled connection . are directly connected through an level entanglement, while the multihop entangled nodes communicate through level entanglement. Focusing on the doubling architecture [1, 2, 3] in the entanglement distribution procedure, the number of spanned nodes is doubled in each level of entanglement swapping (entanglement swapping is applied in an intermediate node to create a longer distance entanglement [1]). Therefore, the hop distance in for the level entangled connection between is denoted by [8]
(1) 
with intermediate quantum nodes between and . Therefore, refers to a direct entangled connection between two quantum nodes and without intermediate quantum repeaters, while identifies a multilevel entanglement.
An entangled quantum network is illustrated in Fig. 1. The quantum network integrates singlehop entangled nodes (depicted by gray nodes) and multihop entangled nodes (depicted by blue and orange nodes) connected by edges. The singlehop entangled nodes are directly connected through an level entangled connection, while the multihop entangled nodes are connected by and level entangled connection.
The fidelity of entanglement of an level entangled connection between depends on the physical attributes of the quantum network.
2.2 Terms and Definitions
2.2.1 Entanglement Fidelity
Let
(2) 
be the target Bell state subject to be created at the end of the entanglement distribution procedure between a particular source node and receiver node . The entanglement fidelity at an actually created noisy quantum system between and is
(3) 
where is a value between and , for a perfect Bell state and for an imperfect state. Without loss of generality, in an experimental quantum Internet setting, an aim is to reach over long distances [1, 3].
3 Method
Before giving the details of the algorithm, we briefly summarize the method of opportunistic entanglement distribution in a quantum Internet setting in Method 1.
3.1 Discussion
In Step 1, distributing entanglement between and the direct contacts of defines the distributing set , which can contain different levels of entangled contacts. In our opportunistic approach, the heterogonous entangled contacts leads to diverse hopdistances, specifically for an level entanglement , the , the hopdistance between a source node and target node from the distributing set is determined via (4) as [1, 2, 3, 8].
In Step 2, the next node from a set is selected by the same cost metric used for the selection of node in Step 1. The cost metric [15, 16] used in the opportunistic node selection procedure in Steps 12 ensures the selection of those nodes that can preserve the entangled quantum states with the highest fidelity in their local quantum memories. From a given distributing set , only one node is selected in each iteration step. The cost function will be clarified later in the algorithm.
Finally, Step 3 provides an iteration to reach from the source node to the target node.
Fig. 1 illustrates the method of opportunistic entanglement distribution in a quantum repeater network . A given distributing set of a node can contain several different levels of entangled contacts. From a distributing set , only one repeater node is selected in each level of iteration. Those quantum nodes that share level entanglement are referred to as singlehop entangled nodes, while the other entangled nodes are referred to as multihop entangled nodes.
4 Results
4.1 Parameterization
Let be a distributing set of a node , and let be the probability of the shared entanglement stored in the quantum memories of nodes , where identifies a repeater node from the distributing set , with a fidelity , where is a critical lower bound on the fidelity of entanglement.
Then, let be the probability that is satisfied between node and at least one repeater node from .
Using and , a cost function can be defined [15, 16] as
(7) 
Similarly, with as the probability that there exists entanglement between a node of and the target node with fidelity criterion , a cost function between a repeater node and a target node can be defined as
(8) 
where is the probability of fidelity entangled contact between a given and , while is the probability that a given repeater node is selected from for , defined [16] as
(9) 
where
(10) 
From equations (7) and (8), the cost function between a quantum node and a target node at a distributing set , such that holds for the fidelity of all entangled contacts from to , is therefore
(11) 
The probabilities depend on the actual state of the quantum memory (particularly, on the error probability of the quantum memory of node ) and on the fidelity of the stored entanglement. Parameters and are timevarying in our model, which is denoted by and , where refers to the storage time in quantum memory. Time refers to the start of the storage of a quantum system in quantum memory.
Let identify the error probability of quantum memories in nodes such that holds, which allows storing an fidelity entanglement in the nodes after time from start time . Therefore, at a critical upper bound on the quantum memories,
(12) 
holds, where characterizes the change of the error probability of the quantum memory of node at , as in
(13) 
where is the error probability of the quantum memory of node at , while is the change of from to .
Let identify the quantum memory error probability and the stored entanglement fidelity in node at time . Then, the distance between and in identifies a , where is the maximal allowable distance in , which forms an upper bound to the distances for which holds, as
(14) 
and
(15) 
where
(16) 
where
(17) 
and
(18) 
The fidelities and refer to the fidelities of stored entanglement in a node at and , respectively, with relation for nodes
(19) 
where is the difference of entanglement fidelities and , while is an upper bound on the fidelity difference.
From equation (15), the following relation holds for :
(20) 
Fig. 2 illustrates the evolution of the error probabilities of the local quantum memories and the fidelities of the stored entangled states. Time refers to the start of storage in the quantum memory and time is the current time. The distance measures the difference of and of a node pair . From the time evolution of the error probability of the quantum memory, it follows that , and holds for the time evolution of the fidelity of the stored entanglement.
The normalized increase of distance between nodes therefore yields a quantity , , which characterizes the usability of a stored entanglement in from to , as
(21) 
As , if no change occurs in the initial distance in , thus
(22) 
while , if
(23) 
so the yielding relation between and is
(24) 
Equation (21) also characterizes the future behavior of the quantum memories in the nodes, i.e., the predictability of the error model of the quantum memories after a given time after beginning storage. The highest values of are therefore assigned to those memory units for which the error probabilities and the entanglement fidelities evolve by a given pattern.
For a given path with and as the source and target nodes, respectively, associated with a demand of user , , where is the number of users; thus, the overall usability coefficient for is
(25) 
To determine those quantum nodes for which both (21) and are high, a redefined cost function, , can be defined for a given as
(26) 
which assigns the lowest cost to those node pairs for which and are high.
4.2 Opportunistic Entanglement Distribution
The aim of the opportunistic entanglement distribution algorithm is to determine a shortest path with respect to our cost function. The shortest path contains those node pairs for which and are maximal, and therefore the resulting cost function (26) is the lowest in a quantum network . Thus, the algorithm finds the repeater nodes for entanglement distribution by maximizing the usability of stored entanglement.
Some preliminary notations for the algorithm are as follows. Let and be the source and target nodes, respectively, associated with a demand of user . The algorithm selects those nodes that provide the lowest cost with respect to for a given to distribute entanglement from to . Assume that for a set of nodes there exists a path to in a quantum network . Let refer to a set of nodes for which the shortest path to is not yet determined.
The algorithm of the minimalcost opportunistic entanglement distribution is detailed in Algorithm 1.
4.3 Computational Complexity
The computational complexity of the minimalcost opportunistic entanglement distribution is as follows.
Let be a quantum repeater network with quantum nodes. Applying a logarithmic search [17] to find a node with an actual minimal cost requires at most
(31) 
steps, via comparisons.
Since the number of nodes is , setting the final path cost for all quantum nodes requires at most
(32) 
steps.
5 Conclusions
Here we defined a method for entanglement distribution in the quantum Internet. Our method utilizes distributing sets for quantum nodes, which can preserve quantum entanglement with the highest fidelity in their local quantum memories. The algorithm is opportunistic, since in each iteration step a node is selected from a distributing set that can provide optimal conditions. The cost function includes the utilization of the evolution of the error model of the local quantum memories, and the fidelities of the stored entangled states. A usability parameter quantifies the predictability of the evolution of the error model, and of the evolution of the entanglement fidelity. The computational complexity of the method is moderate, which allows for direct application in experimental quantum Internet scenarios and in longdistance quantum communications.
Acknowledgements
This work was partially supported by the European Research Council through the Advanced Fellow Grant, in part by the Royal Society’s Wolfson Research Merit Award, in part by the Engineering and Physical Sciences Research Council under Grant EP/L018659/1, by the Hungarian Scientific Research Fund  OTKA K112125, and by the National Research Development and Innovation Office of Hungary (Project No. 20171.2.1NKP201700001), and in part by the BME Artificial Intelligence FIKP grant of EMMI (BME FIKPMI/SC).
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