Routing space exploration for scalable routing in the quantum Internet.

The entangled network structure of the quantum Internet formulates a high complexity routing space that is hard to explore. Scalable routing is a routing method that can determine an optimal routing at particular subnetwork conditions in the quantum Internet to perform a high-performance and low-complexity routing in the entangled structure. Here, we define a method for routing space exploration and scalable routing in the quantum Internet. We prove that scalable routing allows a compact and efficient routing in the entangled networks of the quantum Internet.

Introduction

Quantum information and quantum computations[1-19] will not only reformulate our view of the nature of computation and communication, but will also open new possibilities for realizing high-performance computer architectures and telecommunication networks[10-17,28-31,33-45,62-77]. Since our traditional data will no longer remain safe in the traditional Internet when quantum computers become available, there will be a need for a fundamentally different network structure: the quantum Internet[20-23,25-27,29,30].

In a quantum Internet scenario[20-31,43-53,55-61,78-80], a primary task is to distribute quantum entanglement[54,81-98] from a source quantum node to a target quantum node through a set of intermediate quantum nodes called quantum repeaters[32,99-112]. The entanglement distribution is realized in a step-by-step manner by the generation of short-distance entangled connections between quantum nodes. Next, the level of entanglement of the entangled connections is increased to generate longer-distance entangled connections. The entanglement level of an entangled connection determines the hop-distance (number of quantum nodes spanned by the particular entangled connection) between a source node and the target node of the given entangled connection. The level increment is realized by the so-called entanglement swapping (entanglement extension) procedure applied in the intermediate quantum repeaters. Specifically, the entanglement distribution is achieved within the framework of the so-called doubling architecture[28,43,44], where each increment of the level of entanglement doubles the hop-distance. Using the entanglem ent distribution procedure, the distant source node and the target node can share a long-distance entangled connection.

The entangled quantum network structure integrates several entangled paths between a distant source and destination quantum nodes. In a general Internet setting with several legal and transmit users, numerous entangled paths exist in parallel, so the quantum repeaters must process all paths simultaneously. The properties of the entangled paths, along with the internal and external attributes of quantum repeaters (quantum memory usage, auxiliary internal processes and communication between quantum repeaters), formulate an abstracted space, called the routing space, of the quantum Internet. Due to the complex mechanisms of the quantum Internet and to the large number of variables associated with modelling these processes, an efficient method for exploring the routing space of the quantum Internet is essential for high-performance and high-efficiency routing.

A fundamental problem in the quantum Internet is that while routing methods for finding the shortest path in a quantum network are available[28,43,44,46-49,53], a mathematical model for the working mechanism of the quantum repeaters in the quantum Internet is still missing. While the shortest paths can be determined only with respect to the cost function associated with the quantum links, these models omit the service capabilities and processes of quantum repeaters, which represent a bottleneck in experimental settings. As a corollary, a comprehensive and exhaustive model is required for the description of entanglement distribution and the construction of entangled paths.

Another issue is the lack of scalable routing in the quantum Internet. Scalable routing refers to a routing method that can determine the most appropriate routing mechanism of a particular subnetwork of the quantum Internet. Specifically, scalable routing can decide whether deterministic routing or adaptive routing would be optimal for a given subnetwork. In deterministic routing, the paths between a subset of quantum repeaters are fixed, while in adaptive routing, the paths are selected dynamically in an adaptive manner according to the actual status of the network. The main advantage of deterministic routing is a more compact and faster realization, since it requires no further path selection in a particular subnetwork. However, this is not generally applicable to the whole quantum Internet due to the dynamically changing conditions. However, performance improvement in the quantum Internet is possible if deterministic routing remains are applied in a particular set of subnetworks while quantum network adaptive routing is applied in the remaining parts. Thus, a scaled routing in the quantum Internet would have a more compact structure and be more efficient for path selection in the entangled network.

Here, we define a method for routing space evaluation and for scalable routing in the quantum Internet. The routing space evaluation integrates the derivation of the external and internal characteristics of quantum repeaters and compacts them into a term called the service rate of quantum repeaters. The scalable routing method utilizes the results of routing space exploration to decompose the quantum Internet into subnetworks with deterministic and adaptive routing between the quantum repeaters of the subnetwork. By utilizing the fundaments of queueing theory[113-115], we define mathematical models for the service rate evaluation of quantum repeaters and entangled paths in the quantum Internet. The routing space exploration method utilizes the developed mathematical models to formulate the routing space that integrates the characteristics of the available paths, the service rates of quantum repeaters and the path service rates between the legal users of the network. Scalable routing results in more efficient routing overall, since it decreases the routing complexity and utilizes the available resources of the quantum Internet more conveniently than does unscaled routing.

The novel contributions of our manuscript are as follows: This paper is organized as follows. “System model and problem statement” section presents the system model and the problem statement. “Service rate of a quantum repeater” section defines the service rate evaluation model. “Service rate of a quantum repeater” section proves the evaluation of the service rate of an entangled path. “Routing space exploration and scalable routing” section provides the routing space exploration and scalable routing method. Finally, “Conclusions” section concludes the results. Supplemental information is included in the Appendix.

We define a mathematical model for the service rate evaluation of quantum repeaters and entangled quantum paths in the quantum Internet.

We propose a method for routing space exploration of the quantum Internet.

We conceal a method for scaled routing in the quantum Internet with deterministic and adaptive routing in the subnetworks.

The methods fuse the fundamentals of queueing theory and the theory of quantum networking and entangled networks.

System model and problem statement

System model

The quantum Internet setting is modeled as follows[46]. Let V refer to the nodes of an entangled quantum network N, with a transmitter quantum node , a receiver quantum node , and quantum repeater nodes , . Let , , refer to a set of edges between the nodes of V, 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.

Entanglement levels in the quantum Internet

In a quantum Internet setting, an entangled quantum network consists of single-hop and multi-hop entangled connections, such that the single-hop entangled nodes (The l-level entangled nodes x, y refer to quantum nodes x and y connected by an entangled connection .) are directly connected through an -level entanglement, while the multi-hop entangled nodes communicate through -level entanglement. Focusing on the doubling architecture[28-30] 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[28]). Therefore, the hop distance in N for the -level entangled connection between is denoted by[37,46]with intermediate quantum nodes between x and y. Therefore, refers to a direct entangled connection between two quantum nodes x and y without intermediate quantum repeaters, while identifies a multilevel entanglement.

Entanglement fidelity

Letbe the target Bell state subject to be created at the end of the entanglement distribution procedure between a particular source node A and receiver node B. The entanglement fidelity F at an actually created noisy quantum system between A and B iswhere F is a value between 0 and 1, for a perfect Bell state and for an imperfect state[28,30,37].

Routing space

Definition 1

The routing space of N is defined aswhere is an entangled path between source user and destination user , , where K is the total number of entangled paths in the quantum network N.

An entangled path is characterized aswhere is the service rate of defined aswhere is the service rate of source , is the service rate of the p-th quantum repeater in path , , q is the total number of quantum repeaters in [the service rate will be defined in (13)]; , is the service rate fluctuation of defined aswhere of a particular quantum node x will be defined in (48), is the number of available routes in the quantum Internet for the entanglement distribution from to , while is a set of the available routes for , aswhere is the k-th available route with and , with a shortest route Assuming a doubling architecture for the entanglement distribution, an entangled path consists of nodes aswhere is the hop-distance between A and B at an l-level entangled connection between A and B.

Cycle

Definition 2

A cycle C with cycle-time is set via an oscillator with frequency in the quantum nodes used for synchronization of a quantum network.

From Definition 2, sC cycles identify , where s is a nonzero real number.

Problem statement

The problem statement is given in Problems 1–3.

Problem 1

Determine the service rate of all quantum repeaters of the quantum network at a given set of incoming and outcoming entangled connections.

Problem 2

Evaluate the service rate of an entangled path in the quantum Internet between distant source quantum nodes A and B.

Problem 3

Define the routing space of the quantum Internet (available paths, service rates of quantum repeaters and service rates of the paths). Determine a scaled routing method with deterministic and adaptive routing in particular subnetworks of the quantum Internet.

The resolutions of Problems 1–3 are given in Theorems 1–3.

Service rate of a quantum repeater

By utilizing the fundamentals of queueing theory on priority queueing and quantum Shannon theory, we define the service rate of a quantum repeater as follows[113-115]. The system model utilizes a G/G/1 priority queueing model (also referred to as single-server queue with first-in-first-out serving in queueing theory)[113] for the service rate evaluation of a particular quantum repeater in the quantum Internet. In the proposed G/G/1 setting, the service rates (measured in Bell states per C) and the inverse incoming entanglement throughput values (measured in C per Bell states) are independent and identically distributed with a general distribution.

Theorem 1 derives the closed-form service rate of a quantum repeater in a G/G/1 setting.

Theorem 1

(Closed-form service rate of a quantum repeater in a G/G/1 setting) The service rate of an i-th quantum repeater can be expressed in a closed-form in a G/G/1 setting, where is an incoming entangled connection of , while is the outcoming entangled connection of .

Proof

The aim of the proof is to derive the service rate of in a closed-form. Let refer to an i-th quantum repeater node with a set of p input entangled connections,and an output entangled connection .

Then, let be the service rate (measured in Bell states per C cycle) of with incoming entangled connection and outcoming entangled connection ; let be the entanglement throughput (The entanglement throughput identifies the number of Bell states per cycle of a particular entanglement fidelity F.), measured in Bell states per cycle, of the output entangled connection of .

The optimization problem can be evaluated as a maximization,where is the service rate of source node A with outcoming entangled connection .

Then, by using the G/G/1 priority queueing model, the service rate for a quantum repeater with a given is defined in a closed-form aswherewhile is the number of incoming entangled states (measured in Bell states) in the input entangled connection of , refers to the situation if the input of is from a previous node such that , and where is a main path between A and B, and refers to the case if the input of is from a previous node such that is not part of the main path, .

The terms of are explained as follows.

The quantity is ratio that models the unavailability of the of output aswhere is the average entanglement throughput of output entangled connection of , is the average entanglement throughput of the input entangled connection , , , is the cardinality of , is the coefficient of variation[113-115] for the Z inverse of the sum of average entanglement throughput of all incoming entangled connections of connection in (measured in C per Bell states), aswherethuswhere is the average of Z,where if , while is the average of asThe term in (17) can be rewritten via (15) weighted by the ratio of (15) asThen, can be set as a constant[113-115] for all quantum repeaters, , for .

The term is the coefficient of variation[113-115] of cycles , where characterizes the cycles of the internal processes of (quantum memory usage, error correction, purification, etc), will be defined in (37). These cycles reduce the service rate through of .

The term (average Bell states per C) can be rewritten aswhere is the average output entanglement throughput (Bell states per C) of source node A, is the probability that a source A and a target B are connected an entangled path ,while is a routing function defined asthus the routing function in (24) therefore equals to 1, if quantum repeater is part of the path , and 0 otherwise.

The inverse of the service rate [see (13)] of (measured in C per Bell states) is defined asUsing (25), the objective function subject to a minimization (While the objective function in (12) subject to a maximization utilizes the service rate formula of (13) derived via the G/G/1 priority queueing model, the objective function in (26) utilizes the inverse of (13) and defines a minimization problem.) can be written asThe validation of the formula of (13) is as follows.

It can be verified[114,115], that [see (25)] can be decomposed aswhere is defined as a ratio of incoming and outcoming entanglement throughputswhile is the number of residual cycles (measured in C cycles) defined via (28) aswhere is given in (18).

The sum of in (29) can be rewritten via (15) aswhere is as given in (21).

As follows, (29) can be rewritten asThus, using (31), the term in (27) can be rewritten as given in (25), and as a corollary, the formula of (13) is validated.

Next, let study the case if there are multiple possible output paths are available for a given incoming entangled connection. Let be a source neighbor node of , associated with an incoming entangled connection of , and let us assume that has outcoming entangled connections, where is the set of r output entangled connections of , .

Using (13), the average service rate for the output entangled connection of a particular from set can be evaluated aswhere is the first moment of defined aswhere is evaluated via (27), and is the probability that an incoming entangled state from of is distributed through the output of , evaluated aswhere is defined in (21); term is the sum of additional internal and external C cycles related to , aswhere is an external term associated to the transmission process between nodes and , while identifies the cycles of usage of the internal quantum memory of , aswhere is the number of entangled states received from and stored in the quantum memory of , is the number of entangled states readout from the quantum memory of and distributed through connection .

Then, the coefficient of variation[113-115] from [see (33)] can be evaluated aswhere is the second moment of , asThen, using (37), the term can be evaluated via (25).

The proof is concluded here. □

Figure 1 depicts the proposed system model for the service rate evaluation of a quantum repeater.

Figure 1

A network situation in a quantum Internet setting with an i-th quantum repeater with service rate in a main path between a source quantum node A and receiver quantum node B. The previous neighbour of is and the next neighbour of is . The node has an incoming entangled connection from the main path with incoming Bell states, and with other incoming entangled connections from other nodes, where is the number of incoming entangled connections and an outcoming entangled connection . The entangled connections have different l levels of entanglement (depicted by different colours: the entangled connections of the main path are denoted by red arrows). A C cycle in the quantum network is set via an oscillator in the quantum nodes (depicted by a dashed grey line), , at a particular oscillator frequency .

Service rate of entangled paths

Theorem 2 derives the closed-form service rate of an entangled path in a G/G/1 setting, at a doubling architecture.

Theorem 2

(Closed-form service rate of an entangled path) The service rate of an entangled path between distant quantum nodes A and B at a doubling architecture is , where q is the total number of quantum repeaters in , , x and y are level entangled source and target quantum nodes connected by , while is the service rate decrement in the entanglement distribution caused by the entanglement swapping operation.

Let and be two quantum repeaters connected by an level entangled connection , with service rates and determined via Theorem 1.

Then, the service rate between and is aswhere term refers to service rate degradation, defined aswhere the simplified notations of and are used for and , respectively.

Let assume that N is utilized via the doubling architecture, with hop-distance for an -level entangled connection between quantum nodes x and y. Then, for a source A and destination B, the entanglement distribution process and the generation of the entangled path is characterized via the service rate (The formula of (41) assumes a path in a doubling architecture with a source node A, destination node B, and with q intermediate quantum repeaters. The formula defines the inverse of the sum of service rate inverses—service rate inverse is given in (25)—taken for the total nodes of the path, amended by a residual quantity (42) that identifies a service rate decrement caused by entanglement swapping in the nodes.), aswhere q is the total number of quantum repeaters of , is the total number of intermediate quantum repeater pairs on the path excluding the boundary nodes, is the service rate decrement (measured in C cycles) in the entanglement distribution caused by the entanglement swapping operationswhere is the number of entanglement swapping operations required for an -level entangled connection between distant A and B,while is the cycles required for the entanglement swapping an i-th swapping quantum repeater . □

Path service rate algorithm

Using (41), a cumulative service rate can be evaluated for all source A and destination B in the network N aswhere is the probability of an entangled path between A and B, and .

The steps are detailed in Algorithm 1. The algorithm utilizes the proposed system parameterization for a given path between source node A and target node B, with q intermediate quantum repeaters. The algorithm evaluates via (22), the coefficient via (21), determines via (34), evaluates via (33), via (38), and via (37), along with the determination of the cycle reduction via the usage of the internal quantum memory by (28), and via (29). Finally, the service rates of node with incoming entangled connection and outcoming entangled connection are determined using (13) for all nodes of the path, and outputs of the entangled path via (41).

Input: Parameterization of path between source node A and target node B, with q intermediate quantum repeaters.

Output: Service rate of .

Step 1. Set the C cycle via oscillator used in the quantum nodes of the quantum network N.

Step 2. Parameterize the quantum network via , , for a source node A and target node B, and via , and for an i-th quantum repeater, .

Step 3. Determine via (22), and via (21).

Step 4. Evaluate via (34).

Step 5. Determine via (33), via (38), and via (37).

Step 6. Evaluate the cycle reduction via the usage of the internal quantum memory by (28), and via (29).

Step 7. Compute the service rate of node with incoming entangled connection and outcoming entangled connection via (13).

Step 8. Output of the entangled path via (41).

Routing space exploration and scalable routing

Theorem 3 derives the optimal routing for the subnetworks, using the closed-form service rate formulas of Theorems 1–2. The derivations utilize a G/G/1 setting and a doubling architecture.

Theorem 3

(Scalable routing in the quantum Internet) An scaled routing function for the quantum Internet can be determined as , where and are the probabilities of deterministic routing and adaptive routing in the network N, , while and is the cardinality, is a set of subnetworks in which deterministic routing can be applied, , where is the number of subsets, and is a set of subnetworks in which adaptive routing can be applied, , where is the number of subsets.

The proof includes a predictive method for routing space exploration in the quantum Internet. The method utilizes the properties of the quantum nodes, such as transmission between the nodes, and also integrates an updating mechanism motivated by machine learning approaches[116-118] to find the highest service rate path in the quantum network. The scalable routing function is derived via Algorithm 2.

Let A and B be the source and target quantum nodes. Then, let refer to the service rate of as defined in (13). Let be a neighbor node connected by an entangled connection with , and let be a next neighbor of , with service rates and , respectively.

Let be the maximal weighted service rate from the k-th quantum repeater to the destination B evaluated in the j-th node , aswhere is the set of next (i.e., toward destination) quantum nodes that share entangled connection with , , while is the weighted service rate from to B, defined aswhere is the number of quantum repeaters of path from to B, defined aswhere is refers to the total number of nodes of path , while is the entanglement throughput reduction associated with , defined aswhere is a delay between and , (); in (48) the simplified notations of and are used for and , respectively.

By propagating backward (see Fig. 2) the value of [see (45)] to node , node can determine the estimation as (A backpropagation method is also used in Q-learning based routing methods[117, 118].)Using the side information available, an algorithm can be defined to determine the routing space of the quantum Internet. The details are given in Algorithm 2.

Figure 2

Routing space exploration in the quantum Internet. This method evaluates the service rates of paths between a source quantum repeater and target node B; is the entanglement throughput reduction associated with the nodes and coefficients , and for a given . An i-th quantum repeater has a next neighbour set (depicted by a cloud) with a set of neighbouring quantum repeaters. From each , a particular quantum repeater is selected such that the weighted service rate from to B is maximized (the quantum repeaters of the initial path are depicted by green nodes). The service rate information is backpropagated from to as side information, and is backpropagated from to to update estimation (classical links are depicted by grey dashed lines).

The proof concludes here. □

Figure 2 shows the procedure for determining the service rates of the paths. The initial path between an i-th quantum repeater and the target node B is .

Routing space exploration algorithm

Algorithm 2 utilizes local and nonlocal information for the determination of the service rate of a particular path. The algorithm uses the proposed service rates formulas of quantum nodes of N, and evaluates the related coefficients. The algorithm determines the maximized weighted service rate between and B in an iterative manner. The algorithm also utilizes an learning rate coefficient in the parameter updating mechanism. The algorithm evaluates , , [see (8)] and [see (9)] of path , for all paths , . In the internal steps, it evaluates via (7), along with the number of available routes for , determines via (8), and via in (9). Finally, the algorithm outputs the routing space via (4).

Output: Routing space of N.

Step 1. Let and be quantum repeaters with service rates and evaluated via (13) and let be an initial path from to B with quantum repeaters. Set the initial value in for the weighted service rate (46) from to to B, as

,

where is the initial value of the weighted service rate in , as

,

and is determined via (13), while is via (48) for all quantum repeaters.

Step 2. In , evaluate via the maximization of (45) for all neighbors of ,.

Step 3. Propagate back to , and update (50) via estimation (49) to as

,

where and is the learning rate, .

Step 4. Repeat step 3 to for all neighbors of , .

Step 5. Output the maximized weighted service rate between and B as

,

where is determined via (52).

Step 6. Repeat the procedure until source node A to output , , [see (8)] and [see (9)] of path .

Step 7. Repeat the steps for all paths , . Determine via (7), the number of available routes for , via (8), and via in (9).

Step 8. Output routing space via (4).

Routing scaling algorithm

Using the results of Algorithm 2 for determining the service rates of paths, the neighbouring quantum repeaters can be selected from each quantum repeater to establish scalable routing. The routing scaling algorithm uses the path service rate information to find the entangled path with the highest weighted service rate , and uses deterministic routing if the service rate degradation coefficient [see (40)] between a particular source and destination quantum repeater in is below a critical threshold value ; otherwise, it uses adaptive routing between the quantum nodes.

Algorithm 3 provides the steps of the routing scaling. The algorithm outputs a highest service path with a scaled routing function for quantum network N. The algorithm utilizes the parameterized routing space of N outputted via Algorithm 2. As a main contribution of the algorithm, it evaluates between and via (39) as , where and are determined via (13), while is as in (40). Then, it makes a decision, using the relation of . If the relation is true, then it sets a deterministic routing between and . Otherwise, the algorithm sets a adaptive routing between and . After some internal steps and calculations, the algorithm determines for all source and destinations, and it establishes the selection of the appropriate routing method for all quantum repeater pairs of path . Finally, the algorithm determines the appropriate routing mechanism for all sub-networks of the particular quantum Internet scenario.

Input: Routing space of N.

Output: routing scaling for the quantum network N.

Step 1. Set a critical upper bound on the service rate fluctuation in N. Assume that is constant for all swapping quantum repeaters (i.e., has no impact on routing).

Step 2. Determine and in nodes and via Algorithm 2.

Step 3. In , select a neighbour node that maximizes and set it as . In , select a neighbour node that maximizes and set it as .

Step 4. Update the initial path to path with the highest service rate (53) between and B.

Step 5. Compute between and via (39) as , where and are determined via (13), while is as in (40).

Step 6. If , then use deterministic routing between and .

Step 7. If , then use adaptive routing between and .

Step 8. Repeat steps 5–7 for all quantum repeater pairs of path .

Step 9. Apply step 4 to find for all source and destinations, and repeat step 8 for all quantum repeater pairs of path .

Step 10. Output sets , and , where and are the number of subsets of quantum nodes for which and routing functions can be applied in N. Output the scaled routing function for the quantum network N as

,

where and are the probabilities of deterministic and adaptive routing in the network, , while and is the cardinality.

Deterministic and adaptive routing

In a deterministic routing between and , the shortest path is fixed such that is always selected as the neighbouring node of from the set of possible neighbours in (A shortest path is selected with respect to a particular cost function, in our setting the cost function of the path selection is the inverse of ). deterministic routing is theoretically more compact and faster than adaptive routing . Practically, this also means that in (22) is predetermined[114, 115], regardless of topology and cost function. A straightforward selection for iswhere is a current node processed by , while function selects the next neighbour node of .

In adaptive routing between and , the shortest path is not fixed. The next neighbour is adaptively selected from set according to the current network situation. adaptive routing requires more resources and computational power than does . Practically, this also means that in (22) is not predetermined and depends on the actual topology and cost function of . Algorithm 2 is a straightforward selection for in our setting.

Figure 3 depicts scaled routing in a quantum Internet setting. The network consists of K transmit users, , and K receiver users . As the highest service rate path is determined between all transmit and receiver users, the quantum repeaters nodes of the quantum network are partitioned into subnetworks with deterministic routing and with adaptive routing between the nodes.

Figure 3

Scaled routing in the quantum Internet. The network is decomposed into subnetworks with adaptive routing (depicted by yellow clouds) and deterministic routing (depicted by grey-blue clouds) between the nodes of the subnetwork. Each subset consists of quantum repeaters and entangled connections with heterogeneous entanglement levels.

Performance evaluation

Service rate

Assume that R is a quantum repeater in N with a standard quality optical fiber with a link loss . In Fig. 4, the service rate of quantum repeater R is depicted in function of the number of incoming Bell states at a particular fidelity F through , and in function of , where is evaluated via (25). The service rate is measured as the number of outcoming Bell states per sC cycles, where s is selected such that , while the quantity is measured in sC cycles. The values of are set to the range of [0, 250], while is scaled between sC cycles (Fig. 4a) and between sC cycles (Fig. 4b), at node efficiency (ratio of outcoming and incoming number of Bell states) .

Figure 4

The service rate of a quantum repeater R with an optical fiber with a standard link loss . The service rate is depicted in function of the number of incoming Bell states at a particular fidelity F and , where is evaluated via (25), node efficiency (ratio of outcoming and incoming number of Bell states) . a The values of are set to the range of [0, 250], while is scaled between sC cycles. b The values of are set to the range of [0, 250], while is scaled between cycles.

Computational complexity

Let be the total number of quantum repeaters in N, with an average number of incoming entangled connections per quantum repeater R, and with an average number of outcoming entangled connections. Then, by utilizing the complexity of a service rate determination[114, 115], the computational complexity of the routing scaling algorithm isAssuming that in the nodes, the computational complexity isFor a detailed proof, see Section A.1 of the Appendix.

Comparison

The computational complexity of the algorithm is compared with the computational complexity of the PG (performance queueing) G/G/1 routing method[115].

For analytical purposes, we assume a realistic network setting with , and for simplicity we set , with , and .

Figure 5a depicts the resulting complexity of our algorithm, the complexity of the PG routing method is depicted in Fig. 5b.

Figure 5

a The computational complexity (: number of operations) of the proposed algorithm at a realistic network setting, , with , in function of and , , . b The computational complexity of the PG method at a realistic network setting, , at a particular , , .

Conclusions

Here, we defined a method for routing space evaluation and scalable routing in the quantum Internet. The derived methods utilize the framework of queueing theory along with the characteristics and physical attributes of the quantum Internet. We proved the service rate formulas of quantum repeaters and entangled paths. We defined a method for routing space evaluation to explore the service rates of quantum repeaters and entangled paths of the quantum Internet. Using the results of the routing space exploration, we defined scalable routing for the quantum Internet. The scaled routing function determines the most appropriate routing mechanism for the subnetworks to realize high efficiency and routing in the quantum Internet.

Ethics statement

This work did not involve any active collection of human data.

Supplementary information

Algorithm 1 Service rate of a path

Input: Parameterization of path PA→B between source node A and target node B, with q intermediate quantum repeaters. Output: Service rate SPA→B of PA→B. Step 1. Set the C cycle via oscillator O_C used in the quantum nodes of the quantum network N. Step 2. Parameterize the quantum network via μB_FA, χin2A, PrlA,B for a source node A and target node B, and via μB_Fl_k, and CMR_il_k for an R_i i-th quantum repeater, i=1,…,q. Step 3. Determine μB_Fl_i→l_k via (22), and ω_kR_j via (21). Step 4. Evaluate PrlR_il_j,R_il_k via (34). Step 5. Determine θR_jl_k via (33), νR_jl_k via (38), and χ^2MR_il_j,l_k via (37). Step 6. Evaluate the cycle reduction via the usage of the internal quantum memory ςR_il_j,l_k by (28), and ψ_kR_i via (29). Step 7. Compute the SR_il_j,l_k service rate of node R_i with incoming entangled connection l_j and outcoming entangled connection l_k via (13). Step 8. Output SPA→B of the entangled path PA→B via (41).

Algorithm 2 Routing space exploration

Input: Service rates of quantum nodes of N. Output: Routing space S_RN of N. Step 1. Let R_i and R_j be quantum repeaters with service rates SR_j and SR_k evaluated via (13) and let P^0R_i→B be an initial path from R_i to B with V_R_i→B=P^0R_i→B-1 quantum repeaters. Set the Ri0W_R_j→B initial value in R_i for the W_R_j→B weighted service rate (46) from to R_i to B, as Ri0W_R_j→B=WR_j→B0, where WR_j→B0 is the initial value of the weighted service rate in R_i, as WR_j→B0=∑p=1V_R_j→BS^0R_p+γ^0R_p, and S^0R_p is determined via (13), while γ^0R_p is via (48) for all quantum repeaters. Step 2. In R_j, evaluate R_jW_R_k→B via the maximization of (45) for all neighbors of R_j,∀R_n∈S_NR_j. Step 3. Propagate back R_jW_R_k→B to R_i, and update Ri0W_R_j→B (50) via estimation ER_iW_R_j→B (49) to R_iW_R_j→B as R_iW_R_j→B=Ri0W_R_j→B+ℓER_iW_R_j→B-Ri0W_R_j→B, where ER_iW_R_j→B=R_jW_R_k→B+SR_j+γR_j, and ℓ is the learning rate, ℓ∈0,1. Step 4. Repeat step 3 to for all neighbors of R_i, ∀R_n∈S_NR_i. Step 5. Output the W_R_i→B maximized weighted service rate between R_i and B as W_R_i→B=maxR_n∈S_NR_iR_nW_R_j→B, where R_nW_R_j→B is determined via (52). Step 6. Repeat the procedure until source node A to output S_A→B, Ω, SΩ [see (8)] and R_∗ [see (9)] of path PA→B. Step 7. Repeat the steps for all paths PA_i→B_i, i=1,…,K. Determine γ_i via (7), the number Ω_i of available routes for PA_i→B_i, SΩ_i via (8), and R∗i via in (9). Step 8. Output routing space S_RN via (4).

(50) (51) (52) (53)

Algorithm 3 Scalable routing in the quantum Internet

Input: Routing space S_RN of N. Output: R_SN routing scaling for the quantum network N. Step 1. Set a critical upper bound ∂S^∗≥0 on the service rate fluctuation in N. Assume that ΥU_swap is constant for all swapping quantum repeaters (i.e., ΥU_swap has no impact on routing). Step 2. Determine R_jW_R_k→B and R_iW_R_j→B in nodes R_j and R_i via Algorithm 2. Step 3. In R_j, select a neighbour node R_n that maximizes R_jW_R_k→B and set it as R_n=R_k. In R_i, select a neighbour node R_n that maximizes R_iW_R_j→B and set it as R_n=R_j. Step 4. Update the initial path P^0R_i→B to path P^∗R_i→B with the highest service rate (53) W_R_i→B between R_i and B. Step 5. Compute S_l=1R_i,R_j between R_i and R_j via (39) as S_l=1R_i,R_j=SR_i+SR_j+ξR_i,R_j, where SR_i and SR_j are determined via (13), while ξR_i,R_j is as in (40). Step 6. If ξR_i,R_j<∂S^∗, then use R_d deterministic routing between R_i and R_j. Step 7. If ξR_i,R_j≥∂S^∗, then use R_a adaptive routing between R_i and R_j. Step 8. Repeat steps 5–7 for all R_i,R_j quantum repeater pairs of path P^∗R_i→B. Step 9. Apply step 4 to find P^∗R_S→R_D for all R_S source and R_D destinations, and repeat step 8 for all R_i,R_j quantum repeater pairs of path P^∗R_S→R_D. Step 10. Output sets S_R_d=S_R_d,1⋃⋯⋃S_R_d,D, and S_R_a=S_R_a,1⋃⋯⋃S_R_a,A, where D and A are the number of subsets of quantum nodes for which R_d and R_a routing functions can be applied in N. Output the R_SN scaled routing function for the quantum network N as R_SN=p_dR_d+p_aR_a, where p_d=S_R_d/V_R and p_a=S_R_a/V_R are the probabilities of deterministic and adaptive routing in the network, p_a+p_d=1, while V_R=S_R_a+S_R_d and · is the cardinality.

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