Concurrent multipath quantum entanglement routing based on segment routing in quantum hybrid networks

In the future, quantum networks will be an important development direction. Quantum networks should be studied considering quantum's unique physical properties. In quantum networks with limited resources, decoherence will lead to qubits loss, which restricts the time that data qubits can be stored in memory before use. To realize the network function, it is very important to transmit data qubits as quickly as possible. In order to ensure timeliness and make use of entanglement resources as much as possible, this paper proposes a concurrent multipath entanglement routing scheme based on segment routing for quantum hybrid networks. In this scheme, centralized routing is used to calculate the “segments” of quantum relay path, and distributed routing is used to transmit control messages between the “segments”. The “concurrent” mechanism is fully adopted in classical messages passing, path nodes BSM (Bell State Measurement) execution and multipath shunt transmission. This scheme has obvious advantages in multipath concurrent transmission, traffic diversion, reducing transmission time and providing redundancy.


Introduction
A quantum network [1] can perform many functions and applications that classical networks cannot or cannot effectively perform, such as distributed quantum computing [2][3][4], quantum communication [5], quantum enhanced measurement network [6,7], and clock synchronization [8]. Data qubits must be communicated between remote sites in quantum networks for most of these applications.
Long-distance end-to-end communication of data qubits poses three key challenges: no-cloning theorem, decoherence, and transmission losses [9]. The no-cloning theorem states that any quantum data cannot be copied. Decoherence is the gradual loss of coherence due to the interaction between the system and the environment, resulting in quantum data loss. This limits the amount of time that data qubits can be stored in memory before they are required for use. In other words, this limits the lifetime of quantum memories [9]. Near-term quantum hardware offers limited memory lifetimes (at most seconds [10,11]), which means that entangled links cannot be stored for a long time [12]. Transmission or decoherence losses cannot be compensated by traditional amplification or retransmission techniques. Even though quantum error-correcting techniques for quantum repeaters [13][14][15] can compensate for both types of losses [16], it is currently difficult due to resources constraints.
Based on Briegel et al.'s entanglement swapping [17], quantum repeaters can transform short-distance entanglement pairs into long-distance entanglement pairs, as shown in Fig. 1a. The receiver and sender pre-share entanglement and transmit measurement outcomes through classical communication. Quantum teleportation [18] can achieve the purpose of transmitting data qubits, as shown in Fig. 1b. In this method, the data qubits are not directly transmitted along the entire path, which overcomes the transmission loss.
In addition to standard parameters like throughput and latency, fidelity is another important parameter in quantum networks [20]. Fidelity refers to the degree of keeping close to the desired state in the process of quantum data transmission and processing. Fidelity is a pure quantum metric with a value between 0 and 1, where 1 indicates that the state is in the desired state and below 0.5 indicates that the state is unavailable. Quantum applications will work as long as the fidelity is above an application-specific Fig. 1 Entanglement swapping and Quantum teleportation from Ref. [9,19]. a Entanglement swapping can create long-distance quantum entanglement; b Quantum teleportation consumes an entangled pair to transmit a data qubit threshold. Fidelity will decrease as a result of decoherence. Hence, shortening the waiting time of data qubits in memory before sending can delay decoherence reduction and thus reduce the loss of fidelity in this respect.
Many famous routing protocols for classical networks (like the Internet) have been widely developed and used, but many of them cannot be directly applied to quantum networks. As an example, in the Internet, connectionless datagrams are primarily used at the network layer. Due to the quantum no-cloning theorem, quantum data cannot be copied and retransmitted. Datagram is difficult to implement in quantum networks. For quantum networks, the virtual circuit may be more appropriate. That is, before communication, the connection (virtual circuit) is established, the path nodes are determined, and the entanglement resources needed for communication are ensured to provide the conditions for entanglement swapping.
As quantum networks expand, the number of quantum nodes increases, and the transmission process becomes more complex [21,22]. In some cases, some intermediate nodes will be very busy, while others will be relatively idle. As quantum nodes and data transmission increase, network congestion and bottlenecks are inevitable. The number of EPR (named after Einstein, Podolsky, and Rosen) pairs available on a data link or transmission path at any one time is finite. When the amount of data qubits to be transmitted is greater than the communication capacity of a single quantum path, if data qubits are transmitted on only one path, after the EPR pairs are exhausted by entanglement swapping, the remaining data qubits must wait for the path nodes to regenerate the EPR pairs before being transmitted. This increases the waiting time of data qubits in memory before sending, resulting in decoherence and reduced fidelity. In order to ensure timeliness (shortening the waiting time of data qubits in memory before sending can delay decoherence reduction), it is important to consider concurrent multipath transmission.
Hence, this paper proposes a concurrent multipath entanglement routing scheme for quantum hybrid networks. Routing metrics and certain constraints can produce multiple equal-cost paths, or multiple non-equal-cost paths (optimal, sub-optimal...). A routing protocol with such demands makes us think of the traffic engineering (TE) protocol used in classical networks, for example SR-TE (Segment Routing-Traffic Engineering) and RSVP-TE (Resource Reservation Protocol-Traffic Engineering). These multipath protocols based on TE combined with quantum network technology can be applied to multipath entanglement routing protocols. However, this paper does not focus on routing metrics and formulas. The aim of this paper is to describe the multipath quantum entanglement routing scheme based on segment routing (SR), as well as its execution flow, and demonstrate its benefits.
The key contributions of our research are as follows: (1) In this paper, a concurrent multipath Bell state entanglement routing scheme based on traffic diversion is proposed. It is the first time to combine the advanced technology of classical network traffic engineering "segment routing" with multipath quantum entanglement routing to form a concurrent multipath routing protocol that is suitable for quantum hybrid networks. The protocol adopts a hybrid routing mode of centralized and distributed routing. The centralized routing calculates the path "segments" (quantum relay routers), and the distributed routing is used to transmit control messages between the segments. The effect of "concurrency" is reflected in classical control messages passing, path nodes BSM execution and multipath shunt transmission of date qubits. (2) We design the Quantum Path Computation Reply (QPCRep) message format, which makes the quantum relay routers on the quantum path do not need to execute BSM in the path order, showing the "concurrency" advantage. (3) We design the execution flow of multipath concurrent routing protocol based on segment routing and illustrate the scheme using a specific example. (4) By adopting concurrency mechanism in many aspects and using entanglement resources in the network as much as possible, the protocol realizes concurrent multipath and improves the timeliness of data qubits transmission. Our analysis shows that this routing protocol has a significant effect in reducing transmission time of data qubits.
The rest of this paper is organized as follows: Section 2 provides background and related work on this topic. Section 3 describes several key technologies and execution flow of the concurrent multipath routing protocol based on segment routing. In Sect. 4, we compare and analyze our concurrent multipath routing protocol in terms of transmission timeliness and scalability, and we draw a conclusion. The article ends with an epilogue in Sect. 5.

Quantum routing protocol
The role of quantum routing protocols is to determine the appropriate path before end-to-end entanglement is generated. Routing is more complicated in quantum network. To calculate the path, it needs to consider not only classical metrics (such as path length, cost, and throughput), but also end-to-end fidelity. The field of quantum network routing [12,19,[21][22][23][24][25][26][27][28][29][30][31][32] is undergoing rapid development. Routing in reference [23][24][25] considered single path. Reference [19] studies concurrent entanglement routing in multiple repeaters to find an optimal end-to-end path. Even if the protocol has a suboptimal path, it is used as a fail-back standby. Reference [26] describes three end-to-end multipath algorithms, but does not discuss non-overlapping paths. The literature [21] studied parallel multipath routing protocol with multi-qudit state. But the preparation of the multi-qudit channels (such as GHZ states, W states or cluster states) and the joint measurements of the multi-qudit state still remain challenging experimentally [21,30]. There exist constraints on efficiency and probability of successful generation distribution, or merging of multipartite entangled states [22]. Hence, not all the possible multipartite entangled states are proven to be successfully distributed over long distances through quantum repeaters [27]. The literature [12] proposed that multi-users transmitted data qubits by TDMA (Time division multiple access) through centralized scheduling. The literature [28] studied finding multiple entangled Bell pairs routes and then purifying them to obtain a single Bell pair route with higher fidelity.
The multipath protocol studied in the literature [31] is reflected in the repeater protocol, focusing on the analysis of the repeater entanglement generation rate, which belongs to the distributed routing mode. The literature [32] proposes a concurrent and efficient entanglement routing protocol for quantum wireless networks (QWNs) based on the design of network components, routing metrics and algorithms, and protocol processes. It can solve the problems of path collision, high energy consumption, and low utilization of quantum resource. In this paper, a concurrent multipath Bell state entanglement routing scheme based on SR-TE is proposed for the first time.

Centralized routing and distributed routing
A routing protocol can be implemented in a centralized [12,23,24,26] or distributed [33] fashion. Because the paper studies the concurrent multipath quantum entanglement routing protocol, the quantum entanglement routing calculation must have a global view. Additionally, due to the unstable and fragile nature of entanglement resources, routing calculation will be updated more frequently. Because centralized mode can focus high-performance computing engines on a small number of devices, this paper adopts the centralized mode to compute quantum entanglement routing. However, the transmission amount of BSM outcomes and various control messages between quantum relay routers (See Sect. 3.1) is not large. The transmission of these classical bits is the task of classical networks. The classical routers can use the distributed routing mode to complete the information transfer between nodes competently. Therefore, the routing implementation of our protocol is a hybrid architecture of centralized/distributed mode. This mode can better disperse complexity, guarantee delay and bandwidth, and achieve disjoint multipath.

Segment routing (SR)
Segment routing [34][35][36] technology initiates routes by source nodes. The service path is encoded into an ordered list of segments by a specific algorithm, stored using a label stack, and encapsulated into the packet header for explicit identification. Intermediate nodes are only responsible for fast forwarding of data. Segment routing can realize traffic load balancing and reduce the resource overhead of flow table with the help of SDN (Software Defined Network) centralized control and appropriate routing algorithms [37]. In classical networks, the "segment" of segment routing is often the key node in the routing path, not all nodes. This can control the number of tags and reduce the overhead. In this paper, the concept of segmentation is applied to quantum networks. It proposes for the first time that "segment" nodes are quantum relay routers with both classical and quantum data planes. See Sect. 3 for details.

Explicit path and dynamic path
The candidate paths calculated by centralized routing can be divided into explicit paths and dynamic paths by analogy with traffic engineering (TE) in classical networks.
The explicit mode is equivalent to offline computing. The multipaths between the sender and the receiver are precomputed and stored (through configuration or message triggering) as common information on all nodes. The dynamic mode is equivalent to online computing. In the dynamic mode, according to the current topology and link state information, the quantum routing algorithm calculates multipaths to the receiver for the sender. The calculation results will be dynamically updated as the quantum network topology or link status change. Due to the variability of entanglement links, we adopt a dynamic mode in this paper.

Proactive and reactive entanglement strategy
There exit two different approaches for entanglement generation, namely proactive and reactive [27,31]. Proactive strategy aims at early distribution of entanglement resources, with a new generation process starting as soon as the entanglement resource is depleted. Conversely, reactive strategy aims to distribute entanglement on-the-fly, starting a new generation process when needed. In this paper, multipath shunt is used to transmit data qubits, which is often the case of large quantum flow, so our network chooses the proactive entanglement strategy.

Scheme design
In this section, several key technologies in the scheme will be elaborated. This section also explains the specific design of concurrent multipath entanglement routing protocol based on segment routing and illustrates the scheme execution flow using a specific example.

"Segment" in quantum hybrid network
At present, classical Internet infrastructure is relatively common and popular. A more realistic and reasonable design is that quantum networks will use existing network infrastructure to exchange classical messages for the purposes of running quantum protocols as well as the control and management of the network itself [9]. The schematic diagram of quantum network proposed in this paper is shown in Fig. 2. In Fig. 2a, on the basis of classical Internet, quantum terminals are added, quantum relay routers with quantum relay function are added (or upgrade some classical routers) in order to form a hybrid network. The quantum relay router has both classical and quantum data planes. In addition to performing routing protocol and network management functions in classical Internet, it also adds quantum storage, quantum transceivers, quantum entanglement and related measurement control functions to help generate end-to-end quantum entanglement. Several scenarios shown in Fig. 2a need to be explained here: (1) There is a direct physical link between nodes 2 and 3. There is also a quantum device on the link interface to establish a single-hop quantum entangled link (the short red dashed line between nodes 2 and 3). (2) There is no direct physical link between nodes 1 and 2, but nodes 5 can connect to each other. Nodes 1 and 2 share EPR pairs In order to form a quantum hybrid network, on the basis of classical Internet, quantum terminals are added, and quantum relay routers are added (or upgrade some classical routers). Quantum relay routers have both classical and quantum data planes; b is a quantum virtual network abstracted from (a). Since entanglement swapping can be used to form a quantum augmented link, node 4 is taken as an example to show its quantum augmented links through node 5 (only physical link conversion, not measurement) to form an entangled link. (3) There is no direct physical link between nodes 3 and 4, and the nearest physical path needs to be diverted through node 7, but this path has no quantum device at the interface of nodes 3 and 4. Since there is a single-hop quantum entangled link between nodes 3 and 6 and between nodes 6 and 4, the quantum augmented link [27] between nodes 3 and 4 can be formed through entanglement swapping (blue dotted line). A quantum virtual network is shown in Fig. 2b abstracted from Fig. 2a, and the quantum augmented link is related only to node 4. Using entanglement swapping, node 4 in this topology can establish an entanglement path (single-hop quantum link or quantum augmented link) with any remaining quantum relay routers.
In order to study the routing of quantum entanglement, we need to look at the topology of Fig. 2b, and for the transmission of BSM outcomes, we need to look at the topology (black solid line) of Fig. 2a. Therefore, in such a quantum hybrid network design, the quantum entanglement routing path between the quantum source and the destination is not necessarily consistent with the transmission path of the measurement outcomes or other control messages.
The dual role of quantum relay router makes it more complicated. It is the key node in hybrid network. Therefore, we consider the segment of two adjacent quantum relay routers with a single-hop quantum link (long dashed line) as a "segment" of segment routing.

Hybrid architecture of centralized/distributed routing
As described in Sect. 2.2, the proposed scheme adopts a centralized/distributed routing hybrid architecture. Centralized routing is responsible for computation of quantum entanglement routing. This task is performed by the quantum path computing unit (QPCE), a network element that provides path computing services to the quantum path computing client (QPCC). When the quantum terminal wants to send data qubits, it submits the sending request to the quantum relay router (also known as source routing node) directly connected with it. The source routing node is a quantum relay router that can access the quantum terminal. It has more powerful capabilities than ordinary quantum relay routers, such as quantum memory with larger capacity, WDM (Wavelength Division Multiplexing) equipment (this paper takes photons as quantum and optical fiber network as media network for example). Each source routing node has a classical channel connected with QPCE to transmit various control messages (such as performing entanglement and BSM) and messages related to quantum link state (such as the number of EPR pairs, fidelity and other quantum routing metrics). The source routing node that receives the sending request from the quantum terminal can act as a QPCC, which requests a QPCE to calculate paths using the client/server (request/response) model. The QPCE, like any other router in the network, learns quantum link status and topology from routing protocols (such as OSPF/ISIS/BGP combined with quantum-metrics), and stores them in the database, then computes quantum paths using routing algorithms. As soon as the QPCE receives the request, it calculates paths based on quantum routing algorithm. QPCE allocates the total amount of data qubits to be sent according to each quantum path's capacity and priority, calculates the appropriate path (or multipaths), and feeds back the result to the QPCC in the reply message. The nodes on the abstract quantum paths in Fig. 2b are all quantum relay routers, which are regarded as "segment" nodes. After receiving the reply message, QPCC adopts the distributed routing method of the classical network (the network in Fig. 2a) to inform the quantum relay routers on each path to establish entanglement links. Then, end-to-end entanglement swapping is implemented to generate concurrent entanglement quantum paths. In the following step, data qubits are transferred using quantum teleportation. BSM outcomes and other control messages are routed in the same way as distributed routing in classical networks. To summarize, the hybrid architecture of centralized/distributed routing proposed in this paper is to use centralized routing to derive quantum paths with only quantum relay routers, while other classical messages are transmitted by distributed routing.

Paths with non-overlapping links
The following study is based on these situations: At some point, a source routing node (QPCC) needs to send more data qubits than a single entanglement path can transmit at once (The number of entangled pairs on the path at that moment is less than the number of data qubits that need to be transmitted). Concurrent multipath entanglement routing can divert traffic, thereby shortening the waiting time of data qubits in the memory of the source routing node and reducing decoherence. A point-to-point link Both cases belong to the paths with non-overlapping links defined in this paper. That is, physical paths cannot overlap, but nodes can intersect uniquely corresponds to one physical interface on each of two neighboring nodes. Since they are concurrent paths, to reduce the complexity and pressure of quantum nodes, these multipaths must be the paths with non-overlapping links between source routing node and destination routing node, as shown in Fig. 3. The intermediate route nodes on paths 1 and 2 in Fig. 3a do not intersect. In Fig. 3b, intermediate routing nodes intersect. The load of intersecting nodes will increase. If the intersection node fails, the quantum entanglement path passing through the node may be affected. Whether it will be affected and how much it will be affected depends on the performance of dynamic link-state routing algorithm. Both cases represented in Fig. 3 belong to the paths with non-overlapping links defined in this paper. That is, physical paths cannot overlap, but nodes can intersect.
In practice, the real source and destination of data qubits are the quantum terminals, that is, the starting point and ending point of the entanglement path are the quantum terminals at both ends. Therefore, when building a concurrent multipath network, it is necessary that the quantum terminal has multiple optical ports and fibers connected to the source routing node, namely multiple physical lines. The number of physical lines is proportional to the quantum traffic to be sent by quantum terminal, and is limited by the number of outgoing ports/fibers of the source routing node. As shown in Fig. 4a, since the source routing node has at most two physical lines connected to the other quantum relay routers, that is, the maximum number of multipaths with link non-overlapping is 2, the source quantum terminal can set up at most two ports/fibers. More generally, if there is only one physical line between the quantum terminal and the source routing node, the link non-overlap requirement cannot be met. WDM can be used to realize multipath [38] between quantum terminal and source routing node, as shown in Fig. 4b. How to distinguish WDM multipaths on the same physical line? Different optical wavelengths correspond to different logical ports of a same physical port (identified by logical port numbers).
Why not also use WDM among intermediate quantum relay routers? The reason is that intermediate quantum relay routers do not connect to quantum terminals. To reduce complexity and cost, they do not require the same capabilities as the source routing nodes described in Sect. 3.2, nor do they need to be configured with WDM splitters. For the convenience of drawing, the following scheme in this paper adopts the method of Fig. 4b and omits the WDM diagram.

Path ID description
Inspired by the segmented Routing Traffic Engineering (SRTE), we describe a quantum path in terms of explicitness ( "explicitness" means that the external controller calculates the source routing path and programs it to the head routing node [39]). However, we do not use the segment list label stack method, because the quantum relay router is not required to do the Bell state measurement (BSM) in the path order. However, we do not use segment list label stack because the quantum relay routers on the quantum path do not need to be notified in the order of the path to perform the BSM. We design the Quantum Path Computation Reply (QPCRep) message to place quantum path information and data qubit transfer quantity of the path calculated by QPCE, as shown in Fig. 5. After receiving the QPCRep, the source routing node learns the path-node relationship, then carries out BSM operation. At the same time, the source routing node copies and distributes the QPCRep messages to inform other quantum relay routers (including destination routing node) on the path to carry out BSM and implement entanglement swapping, so as to establish remote entanglement  The source routing node reads the QPCRep and learns the path-node relationship, then carries out BSM operation. At the same time, the source routing node copies and distributes the QPCRep to inform other quantum relay routers (including destination routing node) on the path. c After receiving the QPCRep, the other quantum relay routers carry out BSM and implement entanglement swapping. d Remote entanglement between the source and destination quantum terminals is established between the source and destination quantum terminal, as shown in Fig. 6. It should be noted that: (1) The source routing node copies the QPCRep message according to the number of quantum relay routers (including destination routing node) on the quantum path. Then QPCRep messages are concurrently distributed (like multicast) to quantum relay routers. The purpose of this is to make each quantum relay router on the path receive the QPCRep as "simultaneously" as possible. Of course, the "simultaneity" here emphasizes a parallel way of improving efficiency. The distribution of the QPCRep messages is the task of classical networks. QPCRep messages are routed individually (for example in packet switched datagram mode), so the time and path to each quantum relay router are not necessarily the same. (2) Once a quantum relay router receives the QPCRep message, it can immediately conduct BSM by checking the quantum path neighbors (Fig. 6c) and finally realize the remote entanglement between source and destination quantum terminal (Fig. 6d).

Concurrent multipath entangled routing process instance
To illustrate the implementation process of concurrent multipath entanglement routing, the quantum hybrid network shown in Fig. 7 is used as an example. Figure 7a shows a quantum hybrid network. The quantum relay routers in the figure are capable of classical and quantum communication, and the red dots and dotted line between them represent single-hop quantum entanglement link. As mentioned in Fig. 7 Schematic diagram of concurrent multipath quantum entanglement routing process. a Source quantum terminal A wants to send 5 data qubits to destination quantum terminal B; b Step 1-3; c, d Step 4 Sect. 2.5, in this paper, the method of generating single-hop entanglement between quantum relay routers is the proactive strategy (a new generation process starting as soon as the entanglement resource is depleted). Therefore, Fig. 7a shows the singlehop quantum entanglement links between various quantum devices at a certain time. Suppose that source quantum terminal A want to send 5 data qubits (white dots in the figure) to destination quantum terminal B, and the implementation process of concurrent multipath entanglement routing is as follows.
Step 1 When a source quantum terminal wants to send data qubits, it will send a Quantum Path Computation Request (QPCReq) message to its directly connected source routing node. The source routing node acts as a QPCC and requests the QPCE to calculate the path to the destination quantum terminal. The source routing node forwards a QPCReq message to the QPCE. The message provides all the necessary information to calculate the path: source quantum terminal address, destination quantum terminal address, number of data qubits to be sent, constraints.
Let's take Fig. 7b as an example. Source quantum terminal A wants to send 5 data qubits to destination quantum terminal B. It sends a QPCReq message to its directly connected source routing node 1, as shown in Fig. 7b➀. Node 1 acts as QPCC, forwarding the QPCReq message to QPCE, as shown in Fig. 7b➁. The following necessary information is contained in the QPCReq: the address of quantum terminal B, the address of node 1, the number of data qubits to be sent (5 data qubits), and the constraints: (1) Meet the quantum metrics constraints, such as the fidelity threshold higher than the application requirements, for basic QKD the threshold fidelity is about 0.8, as well as the time upper limit for keeping data qubits in memory (e.g., a few milliseconds [9]), etc.; (2) When a single path cannot transmit all data qubits at one time, multiple paths with non-overlapping links shall be arranged to meet the transfer quantity. If all the multipaths that meet the constraints cannot complete the transfer quantity, the source can only obtain the new path information after the entangled resource is regenerated and the link state information is updated.
Step 2 QPCE uses quantum routing algorithm to calculate quantum paths with the support of quantum link state database information. If an optimal path cannot meet the transfer quantity, multiple paths meeting the constraints will be generated.
As shown in Fig. 7b➂, the path calculation is carried out on the QPCE. Suppose, at some point, with the support of quantum link state database information such as topology and quantum link status, QPCE calls the routing algorithm to calculate three non-overlapping quantum paths that meet the constraints. The optimal quantum path is node 1-10-11, which can transmit 3 data qubits. Since this path cannot transmit all 5 data qubits at once at this moment, other concurrent paths need to be calculated. Paths 1-7-8-11 and 1-12-13-11 are two suboptimal paths, each of which can transmit 1 data qubit. The three concurrent paths can complete the transfer of 5 data qubits.
Step 3 A Quantum Path Computation Reply (QPCRep) message is transmitted by QPCE to QPCC to convey the result of quantum path computation.
As shown in Fig. 7b➃, QPCE sends a QPCRep message to Node 1, and tells Node 1 the information about the three paths.
Step 4 After receiving the QPCRep, the source routing node carries out BSM operation and, at the same time, copies and distributes the QPCRep messages to inform other quantum relay routers (including destination quantum routing node) on the path to carry out BSM and implement entanglement swapping, so as to establish remote entanglement between the source and destination quantum terminal. After that, quantum teleportation is used to complete data qubits transmission. Take Fig. 7c as an example. After receiving the QPCRep message, source node 1 checks it with its own address (number 1 here). The source node 1 learns that the quantum links neighbors of the three paths are terminal A and node 1, terminal A and node 7, terminal A and node 12, respectively. After checking the transfer quantity of the corresponding path, source node 1 performs 3-qubit BSM, 1-qubit BSM, and 1-qubit BSM, respectively, to implement entanglement swapping. At the same time, source node 1 checks the QPCRep message to know the address of the quantum relay routers, copies and sends the QPCRep message to them: node 10, 11, 7, 8, 12, 13 (if there are duplicate nodes, the QPCRep message will only be sent to them once). The time that QPCRep messages are individually routed by routers (including classical routers and classical routing devices of quantum relay routers) to each quantum relay routing node is not necessarily in path order. For example, when node 11 receives the QPCRep message, node 8 may not. Once a quantum relay router receives the QPCRep message, it checks the quantum links neighbors and performs BSM of the corresponding transfer quantity. Finally, three remote quantum entanglements from quantum terminal A to B are established. Subsequently, a total of 5 data qubits (3 + 1 + 1) were transmitted through quantum teleportation, as shown in Fig. 7d.

Time advantage analysis
In this section, we will analyze and demonstrate the time saving advantage of concurrent multipath of this protocol. As mentioned above, concurrency manifests in many ways: concurrent transmission of control messages, concurrent execution of BSMs, multipath concurrent transmission of quantum information.

Time advantage analysis of concurrent transmission of control messages
This analysis is illustrated by our design of QPCRep as an example. The concurrent execution of the BSMs is related to concurrent transmission of control messages, so it is not analyzed separately. Here, we only need to analyze the QPCRep message passing on one of the multipaths, because the rest are the same. Here is a comparison of QPCRep messages transmitted in sequential path order (as shown in Fig. 8a) and in parallel (as shown in Fig. 8b). It is assumed that t1 is the propagation time of QPCRep message between quantum relay routers, t2 is the execution time of quantum relay routers (including source quantum routing node) to calculate classical routing algorithm, and t3 is the execution time of BSM (quantum bandwidth is not considered here). The number of quantum relay routers on the path is r (including source/destination quantum routing nodes).
Then, the total time S1 of QPCRep messages transmitted in sequential path order to establish end-to-end remote quantum entanglement link is represented by formula (1). The total time S2 of QPCRep messages transmitted in parallel to establish end-to-end remote quantum entanglement link is expressed by formula (2). S1 (r − 1)t 1 + (r − 1) max(t 2 , t 3 ) + t 3 (1) In the formulas, the max(t 2 , t 3 ) indicates that the routing algorithm and BSM can be executed parallelly on a same quantum relay router, and the last t3 means that the routing algorithm is not performed but the BSM is performed on the destination quantum routing node. It can be seen from Fig. 8b that except for the source/destination quantum routing nodes, the max(t 2 , t 3 ) s of the intermediate quantum relay routers are covered by (r − 1)t 1 . From the definition of r, we know that r is an integer ≥ 2, so S2 is less than S1, and the larger r is, the less S2 is. That is, the parallel transmission of QPCRep messages has the advantage of time saving, and this advantage becomes more prominent with the increase in the number of quantum relay routers in the path.

Time advantage analysis of multipath concurrent transmission of data qubits
To demonstrate the advantages of concurrent multipath entanglement routing protocol over single-path routing protocol, a comparative analysis will be conducted below.
For simplicity, we simplify the network topology as a multipath entanglement topology between source and destination routing nodes, as shown in Fig. 9. At the same time, the assumption conditions are as follows: I. The quantum entanglement links Fig. 9 Simplified multipath entanglement topology between source routing node and source quantum terminal are sufficient, and the situation of the destination is the same, so the condition for implementing concurrent multipath is available. II. The number (bandwidth) of quantum entangled links on each quantum path is consistent. III. Use unit time to describe the time dimension. The time required to establish end-to-end entanglement on every path from source to destination (regardless of how many intermediate nodes exist) is the same, and is denoted as 1 Unit Time T1. If quantum entanglement links of a path are exhausted, EPR pairs need to be prepared again to regenerate single-hop quantum entanglement links, and the reconstruction time is recorded as 1 Unit Time T2. Suppose that the quantum link bandwidth of each path is k, the number of data qubits to be sent is D, and the number of quantum paths from the source to the destination is p. Then T1 can be represented by formula (3), and T2 can be represented by formula (4). Where represents the round up operation.
According to the above principles, the data qubits transmission time of single-path protocol and multipath protocol is shown in Fig. 10. In Fig. 10a, the quantum link bandwidth of each path is 1, Fig. 10b is 2, and Fig. 10c is 5. The abscissa is the number of data qubits to be sent, and the ordinate is the Unit Time consumed. Due to the concurrent effect of multipath protocol, the establishment time of the end-toend quantum entanglement link and the regeneration time of the single-hop quantum entanglement link in the three cases are smaller than those in single-path. The more concurrent paths, the greater the time savings. When the quantum link bandwidth of each path is smaller, the time saving effect of multipath is more obvious, as shown in Fig. 10a. As the quantum link bandwidth of each path increases, this advantage decreases, as shown in Fig. 10c. This is because when the total transfer demand is unchanged, increased quantum link bandwidths of each path increase the transferred traffic of a single path, obscuring the advantages of multipath. As the total transmission demand increases, the multipath advantage will once more become apparent, as shown in Fig. 10d. So, we can conclude that concurrent multipath entanglement routing protocol outperform single-path protocol in terms of time saving.

Extension
The concurrent multipath scheme proposed in this paper aims to make use of entanglement resources of quantum network as much as possible and split the same source and destination based on traffic engineering, so as to save time and improve fidelity. However, from the perspective of the literature [28,40], multipath can be used to purify into an end-to-end entanglement with better fidelity. Therefore, the multipath scheme provided in this paper can also be used for the purpose of purification.
In addition, limited quantum storage limits the number of entanglement links that can be stored at the same time [12,41], so we expect the quantum relay routers on each path to be disjoint. Such disjoint multipath concurrent transmission can increase the quantum network bandwidth, avoid network congestion and bottleneck, and improve redundancy protection.
The scheme discussed in this paper is based on a pair of source/destination quantum terminals, and it is feasible to extend the scheme to multiple pairs of source/destination quantum terminals.

Conclusions
This paper is the first time to combine the advanced technology of classical network traffic engineering "segment routing" with multipath quantum entanglement routing to form a multipath concurrent routing protocol based on segment routing, which can be better applied to quantum hybrid networks. The protocol adopts a hybrid routing mode of centralized and distributed routing. The centralized routing calculates the path "segments" (quantum relay routers), and the distributed routing is used to transmit control messages between the segments. The effect of "concurrency" is reflected in classical control messages passing, path nodes BSM execution and multipath shunt transmission of date qubits. We also design the QPCRep message format and execution flow of the protocol. Finally, we analyze and demonstrate the time saving advantage of concurrent multipath of this protocol.
Inspired by other recent studies, future work can study the multipath entanglement routing protocol for purification [28,40], or it can be extended to more rich and complex network configurations (regarding a QR chain with simultaneous BSMs, parallel channels and time-multiplexing, to the case of N given optical fiber links with different lengths [42]).
Author contributions LZ contributed to conceptualization and methodology; LZ contributed to writing-original draft preparation; QL contributed to resources; LZ and QL contributed to writing-review and editing; QL contributed to supervision; LZ contributed to funding acquisition. All authors have read and agreed to the published version of the manuscript.

Data availability
The used and analyzed data during the present study are available from the corresponding author on reasonable request.

Conflict of interest
The authors declare no conflict of interest. Figure 11 shows a schematic diagram of a quantum entanglement multipath system. There are several quantum entanglement paths from the quantum source node A to the quantum destination node B. The path in the middle is the p-th path. No matter how many quantum relay nodes are passed on the p-th path, the quantum path can always be changed into node A-X-B model at last due to entanglement swapping. Suppose the number of quantum paths from the source to the destination is n. According to the principle of quantum teleportation [18], on the P-th (p 1,2… n) path, the system composed of Bell pairs A ( p) ,A ( p) and B ( p) ,B ( p) is expressed by formula (5). Fig. 11 Schematic diagram of quantum entanglement multipath system. There are several quantum entanglement paths from the quantum source node A to the quantum destination node B. No matter how many quantum relay nodes are passed on the p-th path, the quantum path can always be changed into node A-X-B model at last due to entanglement swapping. Then, nodes A and X share a Bell pair of A ( p) ,A ( p) , nodes B and X share a Bell pair of B ( p) , B ( p) . By performing BSM on node X, a remote end-to-end quantum entanglement path can be established between node A and node B. The other n-1 paths and so on

Quantum entanglement multipath system
Quantum relay node X performs BSM on A ( p) and B ( p) , and the expression of the Bell bases is (6).
For example, if |ψ − is chosen, the state of particles A ( p) and B ( p) is |ψ + A ( p) B ( p) after BSM, and entanglement is realized. An entangled quantum channel is successfully established between nodes A and B. Formula (5)- (7) are the derivation of the p-th path assuming that the quantum link bandwidth is 1 (only 1 data qubit can be transmitted). If each path quantum link bandwidth is k (k 1,2… m), then the system composed of n paths is expressed by formula (8).