1 Introduction
Quantum Internet is a communication network with quantum nodes and quantum links [1, 2, 3, 4, 5, 6, 7, 31, 42, 43, 44, 45, 46] that allows to the parties to perform efficient quantum communications [34, 35, 36]. The aim of quantum Internet [5, 31] and quantum repeater networks [1, 34, 35, 36, 37, 38, 39, 40] is to distribute quantum entanglement between distant nodes through a chain of intermediate quantum repeater nodes [21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. In the quantum Internet, the quantum nodes share entangled connections that formulate entangled links [1, 2, 3, 4, 5, 6]. The quantum nodes store the quantum states in their local quantum memory for path selection and path recovery purposes [1, 2, 3, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. Since setting several attributes must be optimized in parallel in an arbitrary quantum network, the optimization problem formulates a multiobjective procedure. Formally, multiobjective optimization covers the minimization of quantum memory usage time (storage time), the maximization of entanglement throughput (number of transmitted entangled states per second of a particular fidelity) of entangled links [1, 2, 3, 4, 5, 6], and the reduction of the number of entangled links between a source and a target quantum node [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. However, the problem does not end here since a quantum repeater network can be approached on the quantum transmission (quantum layer) level and on the auxiliary classical communication (classical layer) level that is required for the dynamic functioning of the quantum layer. Therefore, the problem is not just a multiobjective optimization problem in the quantum layer but also a multilayer optimization issue that covers the development of both the quantum and classical layers of a quantum repeater network.
Here, we define a multilayer optimization method for quantum repeater networks. This covers both the quantum layer and the classical layer of a quantum repeater network.
By utilizing the tools of quantum Shannon theory [1, 2, 3, 4, 5, 6, 7], the optimization of the quantum layer includes minimizing the usage of quantum memories in the nodes to reduce the storage time of entangled states, the maximization of entanglement throughput of the entangled links, and also these conditions have to be satisfied for the shortest path between a given source node and target quantum node (i.e., a multiobjective optimization of the quantum layer).
The aim of classicallayer optimization is to curtail the cost of auxiliary classical communications, which is required for such optimization. The cost of the classical communication covers all communication costs required to achieve the optimal quantum network state including the classical communication steps for overall quantum storage time minimization, entanglement throughput maximization, and the selection of a shortest path.
The multilayer optimization employs advanced methods to solve the multiobjective optimization of the quantum layer. We define the structures of the quantum memory utilization graph and the entanglement throughput tree for the multiobjective optimization of the quantum layer of a quantum repeater network. The quantum memory utilization graph models the quantum memory usage for entanglement storage. The entanglement throughput tree shows the entanglement throughput of entangled links with respect to the number of transmittable entangled states at a particular fidelity. Using these advanced constructions, we also define a method for the optimal assignment of entangled states in the repeater nodes. The input of the quantum layer optimization procedure is the quantum memory utilization graph, while the output of the method is a set of entanglement throughput trees. The output identifies the optimal states of the quantum network with respect to the multiobjective optimization function.
Classicallayer optimization focuses on the minimization of the total cost of classical communications by utilizing swarm intelligence[13, 14, 15, 16, 17]. This also defines a multiobjective problem since the cost has to be reduced with respect to the classical communication cost required for the minimization of quantum memory usage, the classical cost of entanglement throughput maximization of entangled links, and for the selection of the shortest path in the quantum layer. Classicallayer optimization uses some fundamentals of bacteria foraging models [11, 12, 13, 17, 18, 19, 20] and probabilistic multiobjective uncertainty characterization [8, 9, 10, 11, 12, 13].
The optimization framework requires no changes in the physical layer, so the framework is directly implementable by the current physical devices [1, 2, 3, 4, 5, 6, 34, 35, 36] and quantum networking elements [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. The method is useful in quantum networking environments with diverse physical attributes (different quantum memory characteristics, quantum error correction, physical quantum nodes attributes, and transmission capabilities of noisy quantum links).
The novel contributions of our paper are as follows:

We conceive a complex optimization framework for quantum networks. It integrates the development of the quantum and classical layers of quantum repeater networks.

Quantumlayer optimization utilizes the attributes of physicallayer quantum transmissions, quantum memory usage, and entanglement distribution via the framework of quantum Shannon theory.

Classicallayer optimization focuses on minimizing any auxiliary communications related to the quantum layer and optimization.

Multilayer optimization is applicable by current physical devices and quantum networking elements providing a solution for the optimization of arbitrary quantum networking scenarios with diverse physical attributes and environments.
This paper is organized as follows. In Section 2, the system model is proposed. In Section 3, the optimization procedure of the quantum layer of quantum repeater networks is defined. Section 4 studies the optimization of the classical layer. Section 5 provides a performance evaluation. Finally, Section 6 concludes the paper. Supplemental material is included in the Appendix.
2 System Model
Our model assumes that the quantum repeater network consists of a source and target node with intermediate repeater nodes and a quantum switcher node. A quantum switcher node operates as follows. Node is a quantum repeater node capable of switching between the entangled connections stored in its local quantum memory and a permit of applying entanglement swapping on the selected connections. While an th quantum repeater node establishes only the entangled connections with the neighbor quantum repeater nodes, a switcher node is equipped with an extended knowledge about the quantum network to select between the entangled links. A general repeater node is not allowed to perform any link selection, since it is assumed in the model that a quantum repeater node has only local knowledge about the network. A switcher node based on its network knowledge can also send entanglement swapping commands to the quantum repeater nodes to define new paths in the network.
Let be a quantum network, , where is a set of nodes, is a set of entangled links. Without loss of generality, the level of an entangled link is defined as follows. For an level entangled link, the hop distance between quantum nodes and is [1, 41]
(1) 
with intermediate nodes between the nodes and
. The probability that an
level entangled link exists between nodes is , which depends on the actual network.The quantum switcher is modeled as a quantum node with the following attributes and permissions:

knowledge about the physical attributes of distant quantum repeater nodes and the entangled connections of (e.g., entanglement fidelity, quantum memory status, link noise, etc),

internal quantum memory for the storage of entangled states,

quantum functionality:

permission to set new entangled connections between its local quantum system and a selected quantum node of the quantum network,

permission to switch between the stored entangled states to construct new paths,


classical functionality:

permission to command distant quantum nodes of via classical links (to construct new entangled connections in the network, to perform entanglement swapping between the selected nodes, other).

The network model is illustrated in Fig. 1. The example network in Fig. 1(a) consists of six quantum repeater nodes , , and a quantum switcher that switches between the entangled connections using its local quantum memory. The switcher also can perform entanglement swapping in the network. The switcher node has knowledge about the physical attributes (e.g., entanglement fidelity, quantum memory status, etc) of quantum repeater nodes to make a decision on a path. The knowledge about the repeater nodes can be transmitted over a classical link to the quantum switcher (classical links are not depicted). As it is depicted in Fig. 1(b), the switcher node has a permission to set new entangled connections via its local quantum state with a selected quantum node. The switcher node decided to set a new entangled connection between its local quantum system and repeater node . A standard quantum repeater is not allowed to perform these operations (except with the direct neighbors in the entanglement distribution phase) without a dedicated command from the switcher.
Note, path in Fig. 1(a) provides a shortest path at a particular network situation at an initial network time . Since the quantum network evolves in time (quality of the entangled links, the status of the nodes, internal quantum memories, etc), at a given time , by utilizing the functions of the switcher node, the switcher node determined a new shortest path, , as depicted in Fig. 1(b).
2.1 Quantum Memory Scheduling
In this section, we define a structure for scheduling quantum memory usage of the quantum nodes called the quantum memory utilization graph . This is a directed graph mapped from the network model, with several abstracted nodes and links.
Proposition 1
The quantum memory utilization graph is a directed graph with abstract nodes and links to schedule the quantum memory usage mapped from the quantum repeater network.
The graph of quantum memory utilization is constructed as follows. Assuming quantum nodes (excluding the switcher node) in the network, the graph contains abstracted transmitter nodes and abstracted receiver nodes with directed connections. Note if the quantum switcher is modeled as the th node and , where is the number of quantum switchers in , the graph also can be constructed via transmitter and receiver quantum nodes. A given with an arbitrary number of quantum switcher nodes defines a particular , therefore the graph is a combination of all other possible switcher modes. The graph contains directed edges between quantum node pairs and of a particular switcher mode , , where is the total number of switcher modes, depicted via nodes labeled as and .
Let us assume that has a single switcher node , and it has two states, and . In both modes and , repeater node serves as a transmitter node for node .
In switcher mode , the shared entangled connection defines the following relations. Repeater node serves as a transmitter node for nodes and . Node serves as a transmitter node for nodes and . Node serves as a transmitter node for nodes and . Node serves as a transmitter node for node .
In switcher mode , the shared entangled connection defines the following relation. Repeater node serves as a transmitter node for node .
2.2 Entanglement Throughput Tree
In this section, we define the structure of the entanglement throughput tree, which aims to extract information from the quantum memory utilization graph. The entanglement throughput tree is also the output format of the multiobjective optimization procedure of the quantum layer.
Lemma 1
The entanglement throughput tree is a structure modeling the multiobjective optimization problem of quantum repeater networks. The tree structure is derived from the quantum memory utilization graph.
Proof. As a given source node is selected, the next nodes are added to the path with a given probability.
Let us assume that is determined. Let us index the nodes by an identifier tag, , and let be the set of unvisited neighbor nodes of a node .
Let refer to the entanglement throughput of a given level entangled link between nodes measured in the number of dimensional entangled states per second at a particular fidelity [1, 3, 4].
Further, let be a neighbor node of with entangled connection and with entanglement throughput .
A cost function, , between nodes is defined as
(2) 
where is the cost of entangled link defined as
(3) 
while is the cost of quantum storage in node .
Let be the entanglement utility coefficient of entangled link between nodes and , initialized as . This amount is equivalent to the utility of the entangled link that it has taken to arrive at the current node from .
At a given , the initial entanglement utility [4] of link is updated to as
(4) 
Using these cost functions, the probability that from node a node is selected is as follows:
(5) 
where and are weighting coefficients [4].
Using (2), (3), and (5) for all node pairs, the method to build a random entanglement throughput tree using a graph is as follows [8, 9].
Let refer to the set of already reached destination nodes, and let be the set of initial nodes, the set of feasible neighboring nodes to node . Let be the set of destination nodes.
The method is given in Procedure 1.
The proof is concluded here.
The structure of a entanglement throughput tree is illustrated in Fig. 3.
2.3 Entanglement Assignment Cycle
In this section, we propose a solution for an optimal assignment (scheduling) of stored entanglement called the entanglement assignment cycle, . The goal of is to achieve a minimal overall storage time at a given entanglement throughput tree.
Lemma 2
An entanglement assignment cycle can be determined by a weighted graph coloring method.
Proof. To determine the minimal overall storage time for a entanglement throughput tree, the conflict graph of that is constructed first. In the graph, each vertex corresponds to a directed link of (an entangled connection). An edge exists between two vertices of , if only the vertices (entangled connections) have a conflict. A conflict occurs if two (stored) entangled connects are associated to the same physical link. The problem is therefore to associate each link of a storage schedule (optimal assignment of stored entanglement), which includes the list of time slots when a given link can transmit the stored entangled states such that total number of time units is minimized. Therefore, our goal is to determine what entangled connects of the given should be scheduled in which time unit, such that the total storage time is minimal in .
Let be an indicator variable, defined as
(6) 
For a periodic scheduling,
(7) 
where is a period and is a constant.
For an entangled connection , let us define as the set of entangled connects that are scheduled in the same time unit , but the physical link can transmit only or . As follows, for any ,
(8) 
As it can be concluded, this problem is analogous to the coloring of the conflict graph .
Then, let us assume that each entangled connection (entangled link) has a weight , which is defined as
(9) 
where is the fidelity of entangled connection and is the largest fidelity available. As follows, for a lowerfidelity entangled connection, the transmission of a given amount of information requires more time units.
As the weights are determined, the problem is analogous to a weighted graph coloring of the conflict graph , which is the assignment of at least distinct colors to each entangled connection such that no two entangled connections sharing the same color interfere with each other on the physical link. For this purpose, a simple distributed weighted coloring algorithm [10] can be straightforwardly applied.
The method for the weighted coloring of conflict graph is summarized in Procedure 2.
After an entanglement assignment cycle has been determined by the weighted graph coloring method, in the next step the
time intervals between each time unit of a given cycle is computed. For this purpose, a linear programming method
[8] can be applied.3 QuantumLayer Optimization
In this section, we define the multiobjective optimization for the quantum layer. The multiobjective function covers a parallel optimization of quantum memory usage and entanglement throughput for a shortest path.
Theorem 1
At a given , the quantum layer of a quantum repeater network can be optimized by a procedure that achieves a parallel minimization of the quantum memory usage time , the maximization of entanglement throughput , and the minimization of the number of entangled links between two arbitrary nodes.
Proof. The aim of the procedure is to find an optimal entanglement throughput tree for which the overall storage time is minimal, that is,
(10) 
the entanglement throughput for all links is maximal,
(11) 
and the number of entangled links of a path is minimal, thus
(12) 
where is the shortest path.
The procedure is based on the fact that for a given with conflict graph , from the knowledge of weighted coloring of conflict graph , time intervals between each time unit of a given cycle, the required objective values of (10), (11), and (12) can be determined.
Let be the set of optimal entanglement throughput trees. Then the aim of the procedure is to determine .
The problem can be rewritten via a solution set with decision variables [8]
(13) 
where is the number of all links in a given quantum memory utilization graph , and is defined as
(14) 
Let be a solution from solution set . Let refer to solution dominating solution , which is
(15) 
and for these relations, there is at least strict inequality [8]. If dominates all other possible solutions, then is a set of nondominated solutions.
For each set , let refer to the set that contains the best nondominated solutions that have been found at a particular iteration. If changes, then the entanglement utilities of the links are updated. If is stationary, then the elements of set are used to update the entanglement utilities. The former serves as an improvement in the exploration process whereas the latter aims to yield more information via the best solutions [8].
At a given graph of quantum memory utilization, the procedure for the optimization of the quantum layer is defined as follows. The output of is an optimal set of entanglement throughput trees that realize the conditions of the multiobjective optimization function. The steps of the quantum layer optimization are summarized in Algorithm 1.
Note if there are no nondominated solutions, then the values of the weighting coefficients , , and in Step 5 of Algorithm 1 can be selected according to the actual tradeoff requirements.
The proof is therefore concluded here.
4 ClassicalLayer Optimization
In this section we characterize the classical layer optimization procedure.
Theorem 2
The cost function of classical communications can be minimized by a procedure .
Proof. Let
(16) 
be the cost function of classical communication of the multilayer optimization procedure, where is a
dimensional real vector of an
th system state of the quantum network, , , are the number of nodes that require the determination of optimal , and , , , refer to the number of classical steps required to find , and for a particular node of network , while , , and are the costs of classical link used for the determination of , and for a given .Then, let introduce indices for as
(17) 
where is the index of a desired optimal system state, is the index of an optimal system state reproduction step, is the index of a nonoptimal system state event [11].
Let assume that there is a set of substates in the network, then the total network state is evaluated as
(18) 
Let be the cost function of classical communication at a given . Then, if
(19) 
then the system state evolves from to as
(20) 
where is the number of random system states, while quantifies a unit cost of system change [11, 12].
Therefore, for a set of substates and current vector , the total cost of classical communication is yielded as
(21) 
which can be rewritten [11] as
(22) 
where is the distributionentity of a current system state, is the information transmission rate of , is the distributionentity of a system state, is the information transmission rate of , while is the th element of a current network state vector , and is the th element of .
Using (see (22)), an environmentdependent cost function is defined as
(23) 
where is a tuning parameter [11].
From (23), the cost function at a given is defined as
(24) 
Using the proposed basic model, we define an optimization procedure of the classicallayer to achieve a minimized cost function. The method is summarized in Algorithm 2.
These results conclude the proof.
4.1 LargeConstrained Optimization
The optimization efficiency can be further improved for a large constrained network scenario. This step can be replayed by a different approach, which allows an optimization of the classical layer for an arbitrary constrained setting. The solution is based on the idea of system state merging.
Lemma 3
The optimization of the classicallayer can be extended to a largeconstrained optimization by state merging.
Proof. An extended model of the classicallayer optimization is as follows.
Let , , be the number of nodes that require the determination of optimal , and . Then, let be the objective function [13], as
(25) 
where
(26) 
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