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Applied Sciences
, Volume 11 (22) – Nov 17, 2021

/lp/multidisciplinary-digital-publishing-institute/a-performance-consumption-balanced-scheme-of-multi-hop-quantum-mMKLvFMyHL

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applied sciences Article A Performance–Consumption Balanced Scheme of Multi-Hop Quantum Networks for Teleportation Jin Xu , Xiaoguang Chen *, Hanwei Xiao, Pingxun Wang and Mingzi Ma Department of Communication Science and Engineering, Fudan University, Shanghai 200433, China; 19210720064@fudan.edu.cn (J.X.); 19210720073@fudan.edu.cn (H.X.); wangpingxun0209@163.com (P.W.); 19210720078@fudan.edu.cn (M.M.) * Correspondence: xgchen@fudan.ac.cn or xiaoguangchen@fudan.edu.cn Abstract: Teleportation is an important protocol in quantum communication. Realizing teleportation between arbitrary nodes in multi-hop quantum networks is of great value. Most of the existing multi-hop quantum networks are based on Bell states or Greeberger–Horne–Zeilinger (GHZ) states. Bell state is more susceptible to noise than GHZ states after puriﬁcation, but generating a GHZ state consumes more basic states. In this paper, a new quantum multi-hop network scheme is proposed to improve the interference immunity of the network and avoid large consumption at the same time. Teleportation is realized in a network based on entanglement swapping, fusion, and puriﬁcation. To ensure the robustness of the system, we also design the puriﬁcation algorithm. The simulation results show the successful establishment of entanglement with high ﬁdelity. Cirq is used to verify the network on the Noisy Intermediate-Scale Quantum (NISQ) platform. The robustness of the fusion scheme is better than the Bell states scheme, especially with the increasing number of nodes. This paper provides a solution to balance the performance and consumption in a multi-hop quantum network. Citation: Xu, J.; Chen, X.; Xiao, H.; Keywords: multi-hop quantum networks; teleportation; fusion; entanglement puriﬁcation Wang, P.; Ma, M. A Performance– Consumption Balanced Scheme of Multi-Hop Quantum Networks for Teleportation. Appl. Sci. 2021, 11, 10869. https://doi.org/10.3390/ 1. Introduction app112210869 Teleportation is always used for transferring quantum information with the aid of maximally entangled states and classical channels. In 1993, Bennet proposed the ﬁrst Academic Editor: Roberto Zivieri teleportation protocol based on the Einstein–Podolsky–Rosen (EPR) state [1]. In 1997, Bouwmeester proved it in the practical experiment [2]. Since then, many other teleportation Received: 7 October 2021 protocols were proposed based on different entangled states, such as GHZ states, W states, Accepted: 15 November 2021 and cluster states [3–5]. Published: 17 November 2021 Teleportation can be easily realized between two adjacent nodes. However, in a multi- hop quantum network, most nodes are not directly connected. Realizing teleportation in a Publisher’s Note: MDPI stays neutral multi-hop quantum network is valuable. In 2005, Cheng proposed the Bridging protocol of with regard to jurisdictional claims in the network in this ﬁeld [6], and then, the ideas for teleportation network were given in published maps and institutional afﬁl- different entangled states [7,8]. However, they are all designed based on the teleportation iations. between directly connected nodes. Chen gave a classical solution to it [9]. Quantum states in all intermediate nodes are measured in the end to realize teleportation. Increasing the number of nodes between the source and the destination will bring difﬁculties with exponential growth to the calculation. Besides, the inﬂuence of noise is also ampliﬁed. Copyright: © 2021 by the authors. As an important technology in quantum relay, entanglement swapping has been Licensee MDPI, Basel, Switzerland. realized in the recent years [10–13]. The original entanglement swapping protocol took This article is an open access article advantage of the spontaneous parametric down-conversion (SPDC) source [14]. Today, distributed under the terms and scientists prefer to use hybrid protocols and photon interference to entangle spins or conditions of the Creative Commons atoms [15,16]. Scientists have proved that entanglement swapping can establish entangle- Attribution (CC BY) license (https:// ment between states in different entangled pairs. The price is the loss of ﬁdelity [17,18]. creativecommons.org/licenses/by/ This technology makes it possible for arbitrary nodes in the network to share entangled 4.0/). Appl. Sci. 2021, 11, 10869. https://doi.org/10.3390/app112210869 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 10869 2 of 15 states. When it comes to the noise in the environment, scientists think that puriﬁcation can solve this problem [19–21]. Because of a developed puriﬁcation [22–25], high ﬁdelity entanglement can be established in the multi-hop quantum network. Based on the entanglement swapping and puriﬁcation, we design a new scheme of a multi-hop quantum network. Taking advantage of entanglement swapping and puriﬁcation, nodes in the network can share entangled states with high ﬁdelity. To ensure the high robustness of the system, we design a puriﬁcation algorithm. To ensure the practicability of the system, we use bit-ﬂip, phase-ﬂip, and depolarizing noise models in the simulation. Although amplitude damping noise seems more practical in the simulation, we ﬁnd that it will lead to the disappearance of entanglement. Considering puriﬁcation always works on entangled states, thus, we did not use amplitude damping noise in this paper. Besides, we consider the decline in ﬁdelity of entanglement swapping based on the latest physical experiment. The number of entering nodes is decreased in the ﬁnal communication to protect the privacy. Thus, the computational complexity is reduced. In the end, we simulate the network on Cirq with bit-ﬂip and depolarizing noise in the true environment. Contributions of this paper can be stated as follows: • This paper proposes a new scheme of a multi-hop quantum network based on bipartite communication with fusion states. It improves the noise immunity of the network while avoiding the huge consumption. • A puriﬁcation algorithm is proposed to ensure the robustness of the system, providing a solution for future quantum communication. • The ﬁdelity for multi-particle entanglement puriﬁcation is calculated. The paper is outlined as follows. In Section 2, we review entangled states, teleportation, entanglement swapping, quantum noise, and ﬁdelity. In Section 3, we design the new multi-hop quantum network architecture based on bipartite communication. Besides, we give the puriﬁcation scheme for the Bell pair and GHZ states. In Section 4, we give the design for a puriﬁcation algorithm and simulation results. Finally, conclusions are given in Section 5. 2. Preliminary 2.1. Entangled State The Bell state is the most commonly used entangled state in quantum communication. The basic state in this paper is 1 0 j0i = ,j1i = . (1) 0 1 Then, the Bell states can be expressed as 1 1 p p F = (j00i +j11i), F = (j00ij11i) 2 2 (2) 1 1 Y = p (j01i +j10i), Y = p (j01ij10i). 2 2 If applying a unitary operation such as the Pauli matrix on Bell states, we obtain + + Y = (X 0 ) 0 + (X 1 ) 1 = X F j i j i j i j i F = (Zj0i) j0i + (Zj1i) j1i = Z F (3) Y = (Yj0i) j0i + (Yj1i) j1i = Y F , where 0 1 0 i 1 0 X = , Y = , Z = . 1 0 i 0 0 1 Appl. Sci. 2021, 11, 10869 3 of 15 Thus, we choose jF i to share in the network. Another entangled state commonly used in quantum communication is the GHZ state. Since the ﬁdelity of GHZ states decreases with the increasing number of qubits [25], we select 3-qubit GHZ states to use. 1 1 jY i = p (j000i +j111i),jY i = p (j001i +j110i) 1 2 2 2 1 1 p p jY i = (j010i +j101i),jY i = (j011i +j100i) 3 4 2 2 (4) 1 1 jY i = p (j000ij111i),jY i = p (j001ij110i) 5 6 2 2 1 1 jY i = p (j010ij101i),jY i = p (j011ij100i). 7 8 2 2 For the same reason, we choose jY i to share in the quantum network. 2.2. Teleportation Teleportation is a safe communication method when facing eavesdropper. It sends only measurement results in a classical channel, instead of sending the quantum state directly. The receiver can rebuild the quantum state based on the measurement results. Even if the classical channel is eavesdropped, qubit still cannot be rebuilt because of the lacking entangled state. Teleportation based on the Bell pair is shown in Figure 1. Nodes A and B share a Bell pair jF i. Node A wants to send an arbitrary state jyi = aj0i + bj1i to node B, where 2 2 jaj +jbj = 1. The control state and the entangled state in node A are measured by the circuit shown in Figure 1. State y can be reconstructed in node B by a quantum gate j i based on the entangled state. Quantum gate matching to the measurement results is shown in Appendix A. Figure 1. (a) Entangled states in teleportation model and (b) circuit in node A based on Bell pair. Classical channel is used to transfer measurement results, quantum channel is for entanglement distribution. Here, H represents the Hadamard gate. jyi is the control qubit of the CNOT gate. Teleportation based on GHZ states is shown in Figure 2. Node A holds two entangled states and an arbitrary state jyi. Figure 2. (a) Teleportation model and (b) circuit based on GHZ states. GHZ states jY i , i = 1, 2, 3 are shared between nodes A and B. Node A holds three qubits operated by CNOT and the H gate in the circuit, and sends measurement results to node B to rebuild jYi based on jY i . 3 Appl. Sci. 2021, 11, 10869 4 of 15 2.3. Entanglement Swapping In a multi-hop quantum network, entanglement swapping can establish the entan- glement between nodes without being directly connected. So, it is also an important technology to realize a quantum relay [13]. Node B has shared entanglement both with nodes A and C, as shown in Figure 3. Entanglement swapping can help to establish entanglement between node A and C. It is described in detail in Appendix B. Figure 3. (a) Entanglement swapping based on Bell pairs and (b) circuit in node B. F is shared AB between nodes A and B. F is shared between nodes B and C. Once two states in node B are BC measured, entanglement between nodes A and C is established. 2.4. Fidelity Because of the channel noise, the quantum state we ﬁnally get may be different from the theoretical value. Fidelity is used to measure this deviation, deﬁned in [26]: F = hyjrjyi, (5) where jyi is the wanted quantum state, r is the density matrix of the quantum state received ﬁnally. The density matrix is r = p y y , (6) j ih j å i i i where p is the probability of state y . Fidelity takes values from 0 to 1. j i i i 2.5. Noisy Channel The main noisy channels in this paper are the bit-ﬂip channel, phase-ﬂip channel and depolarizing channel in [26]. The bit-ﬂip channel ﬂips a qubit from j1i to j0i with probability p, 0 < p < 1. Its operation elements are: E = 1 p I, E = pX. (7) 0 1 For the phase-ﬂip channel, the operation elements are: E = 1 p I, E = pZ. (8) 0 1 For the depolarizing channel, with error probability p, the state can be described as #(r) = (1 p)r + (XrX + YrY + ZrZ), (9) where X, Y, and Z are Pauli operations. There are also other noise models, such as Gaussian noise, but they are hard to realize on the NISQ platform. So, we choose the three most commonly used quantum noise models for the channel environment simulation. Since puriﬁcation results under bit-ﬂip and phase- Appl. Sci. 2021, 11, 10869 5 of 15 ﬂip noise are similar, we only provide simulations under bit-ﬂip and depolarizing noise in Section 4. 3. The Scheme of Network Architecture and Puriﬁcation 3.1. Network Architecture Based on Bipartite Communication For a network based on Bell states, a Bell pair is pre-shared between adjacent nodes. Assume that A is the source node while E is the destination node, and they are not directly connected. Entanglement swapping can establish entanglement between them. However, physical experiment has proved that each swapping will bring about 12.77 4.17% loss of entanglement ﬁdelity [17,18]. The number of times a qubit taking part in entangle- ment swapping should be as small as possible. Node A can select the node between the source and destination for entanglement swapping through transferring commands in the classical channel. Compared with the Bell state, a multi-particle state such as the GHZ state has a better entanglement property [27,28]. Therefore, the GHZ state is more widely used in telepor- tation. However, once we measure one particle of GHZ states, the whole entanglement will be destroyed. To solve this problem in bipartite communication, fusion is used to form GHZ states based on Bell pairs [25]. The principle and circuit of fusion are shown in Figure 4. Figure 4. (a) Fusion principle and (b) circuit. Assume that nodes A and B share a pair of Bell states b1 and b2, and node A holds the other Bell pair a1 and a2. Circuit in (b) can be applied in two entangled particles a2 and b1. One only needs measure b1 in the Z basis and apply the Pauli gate correction on a2 based on the measurement result. Then, a1, a2 and b2 are changed into GHZ states. The fusion scheme in the 5-node network is shown in Figure 5. The ﬁrst round of entanglement swapping in the network occurs in nodes B and D, while the second round occurs in node C. Thus, entanglement is established between nodes A and E. Figure 5. Entanglement swapping and fusion in a quantum network. Entanglement swapping can help to establish entanglement between nodes A and E based on Bell states. Fusion in node A can help to form GHZ states. Appl. Sci. 2021, 11, 10869 6 of 15 3.2. Puriﬁcation Circuit for Different States The accurate realization of teleportation needs high-ﬁdelity entangled states. Puriﬁca- tion is a common method to improve the ﬁdelity. 3.2.1. Puriﬁcation of Bell Pairs The standard entanglement puriﬁcation protocol (EPP) is the way to reduce the inﬂuence of bit-ﬂip noise. A Bell pair is measured to improve the ﬁdelity of another pair. To solve the phase-ﬂip noise, quantum privacy ampliﬁcation (QPA) was proposed in [29], shown in Figure 6. Figure 6. Puriﬁcation circuit for a Bell pair. Assume two Bell pairs are distributed to the source and the destination node. U and U are rotation operators to solve phase-ﬂip noise. Fidelity of one A B entangled pair can be improved by measuring the other entangled pair. U and U are rotation operators for q about x-axes, noted as R (q) [30]. A B x q q jq X/2 R (q) = e = cos I j sin X, (10) 2 2 where q is p/2 for U , and p/2 for U . Assume that the two Bell pairs are both jF i A B with the ﬁdelity of 1 at ﬁrst. After quantum distribution, U acts on the quantum state in the source node, while U acts on the state in the destination node. For bit-flip noise, assume that the error probability of each qubit is p, where 0 < p < 0.5. 2 2 + Let q = 1 (1 p) p . The two nodes share jF i with the probability 1 q. The Bell pairs pre-shared by two nodes with probability are shown in Table 1. 2 2 Table 1. Shared Bell pairs and the matching probability. Here, q = 1 (1 p) p . One Shared Pair Probability Two Shared Pairs Probability 2 + + 2 j00i +j11i (1 p) jF i jF i (1 q) + + 10 + 01 p(1 p) F Y (1 q)q j i j i j i j i + + 01 + 10 (1 p) p Y F q(1 q) j i j i j i j i 2 + + 2 j11i +j00i p jY i jY i q + + + F is the control pair, the other is target pair. Prepare two Bell pairs F j i j i j i c c at ﬁrst. Because of the noise, four possible scenarios will thus occur for the two Bell pairs before puriﬁcation. We can obtain the ﬁnal results as C NOT Measure + + + + + F ! F F ! F c c c C NOT Measure + + + + Y ! F Y ! discard c c (11) C NOT Measure + + + + F ! Y Y ! discard c c C NOT Measure + + + + + Y ! Y F ! Y . c c c If the measurement results of the target pair are equal, we save the control pair and judge puriﬁcation to be successful. Otherwise, we discard the control pair with using the Appl. Sci. 2021, 11, 10869 7 of 15 measurement and prepare new pairs until the puriﬁcation succeeds. Finally, we obtain (1q) jF i with probability . 2 2 (1q) +q Fidelity of entanglement states before puriﬁcation is described as F , initial 2 + + 2 + + + + F = (1 q)( F jF ) + q F jY Y jF = 1 q. (12) initial Fidelity after puriﬁcation is F , puri f y 2 2 (1 q) q 2 + + + + + + + + F = F jF F jF + F jY Y jF puri f y 2 2 2 2 (1 q) + q (1 q) + q (13) (1 q) = . 2 2 (1 q) + q For phase-ﬂip noise, the calculation and result are consistent thanks to the rotation operators. 3.2.2. Puriﬁcation of GHZ States A puriﬁcation circuit for GHZ states is designed in Figure 7. Figure 7. Puriﬁcation circuit for GHZ states. Assume that two groups of 3-qubit GHZ states are distributed to the source and the destination node. No rotation operator exists here that can help to solve phase-ﬂip noise, but the design is still useful for bit-ﬂip noise. The circuit helps improve the ﬁdelity of the control group by measuring the target group. For bit-ﬂip noise, assume bit-ﬂip noise occurs with probability p, then q = 1 (1 3 3 p) p . The states shared before puriﬁcation and the matching probability are shown in Table 2. jY i is the control group in CNOT gate, the other is the target group. Table 2. Shared GHZ states and the matching probability (bit-ﬂip noise). Shared States Probability 3 3 jY i (1 p) + p 2 2 Y p(1 p) + p (1 p) j i 2 2 jY i p(1 p) + p (1 p) 2 2 jY i p(1 p) + p (1 p) jY i jY i (1 q) 1 1 jY i jY i (1 q)q/3, i = 2, 3, 4 1 i Y Y q(1 q)/3, i = 2, 3, 4 j i j i i 1 jY i Y q /9, i, j = 2, 3, 4 i j If the measurement results of the target group are equal, we save the control group and judge puriﬁcation to be successful. Otherwise, we discard the control group and Appl. Sci. 2021, 11, 10869 8 of 15 prepare new groups until puriﬁcation succeeds. After puriﬁcation, we obtain jY i with (1q) probability . 2 2 (1q) +q /3 The ﬁdelity after puriﬁcation is F puri f y 2 2 (1 q) hY jY ihY jY i q hY jY ihY jY i 1 1 1 1 1 2 2 1 F = + puri f y 2 2 2 2 (1 q) + q /3 3(1 q) + q 2 2 q hY jY ihY jY i q hY jY ihY jY i 1 3 3 1 1 4 4 1 + + (14) 2 2 2 2 3(1 q) + q 3(1 q) + q (1 q) = . 2 2 (1 q) + q /3 For phase-ﬂip noise, a similar rotation operator does not exist. However, we can take advantage of quantum channel recognition by a quantum neural network [31]. If the noise is phase-ﬂip, we can use the H gate to transform the noise into bit-ﬂip before the transmission. n n For n-particle GHZ states, if p is the bit-ﬂip probability, then q = 1 (1 p) p . The ﬁdelity before puriﬁcation is F = 1 q. (15) initial The ﬁdelity after puriﬁcation is (1 q) F = . (16) puri f y 2 2 n1 (1 q) + q /(2 1) 4. Design of Puriﬁcation Algorithm and Simulation Results 4.1. Puriﬁcation Algorithm Puriﬁcation is realized in a quantum network under the pumping manner as reported in [22] and shown in Figure 8. Figure 8. Entanglement pumping for puriﬁcation. F is the ﬁdelity of the entangled states before 00 0 00 puriﬁcation, satisfying F > F > F. Puriﬁcation increases the ﬁdelity of control states to F . Target states are always with ﬁdelity F. The ﬁdelity as a function of the error probability is shown in Figure 9. Puriﬁcation time (PT) means the number of successful puriﬁcations. When PT = 0, Bell states are more resilient against noise than GHZ states. With the help of puriﬁcation, we can see a huge improvement in ﬁdelity. Under the existence of puriﬁcation, lines of GHZ states and Bell states have an intersection point under the same PT. When p is on the left side of the intersection point, the ﬁdelity of the GHZ state is higher than that of the Bell state. With the increasing number of PT, the intersection point moves to the right side. Appl. Sci. 2021, 11, 10869 9 of 15 Figure 9. Fidelity as a function of the error probability with different PT under bit-ﬂip noise. We use different colors to indicate different puriﬁcation time, solid line to represent ﬁdelity of Bell states, and dotted line to represent ﬁdelity of GHZ states. The same color represents the same PT. PT = 0 with blue line means that puriﬁcation is not applied on entangled states. With the increasing number of PT, ﬁdelity of the entangled states will be higher. We observe that entanglement swapping can bring a 12.77% decline in ﬁdelity. Noise in the quantum channel only affects the entanglement distribution in the network. Puriﬁ- cation in the initial entanglement distribution is not so effective because of subsequent entanglement swapping. Thus, when designing the scheme, we focus on the puriﬁcation after each entanglement swapping. In the end, we add an extra puriﬁcation for fusion to obtain a higher ﬁdelity. So, we set m = 1, and m = 1. m is the PT for entanglement 1 2 1 swapping, m is the PT for fusion. The puriﬁcation algorithm of a multi-hop quantum network is shown in Algorithm 1. Algorithm 1 Puriﬁcation Algorithm 1: procedure P URIFIC ATION(m , m , n) . Swapping round n 1 2 2: while m > 0 do 3: while n > 0 do 4: ﬁnish entanglement distribution between adjacent nodes 5: ﬁnish i round entanglement swapping 6: if puriﬁcation condition for swapping is true then 7: ﬁnish swapping puriﬁcation for once 8: m m 1 1 1 9: end if 10: if m == 0 then 11: n n 1 12: end if 13: end while 14: if puriﬁcation condition for fusion is true then 15: ﬁnish fusion puriﬁcation for once 16: m m 1 2 2 17: end if 18: end while 19: return ﬁdelity 20: end procedure Since we design the network for teleportation, only the entanglement distribution contains transferring quantum states in the channel. To avoid collision of quantum states, the quantum channel is set to simplex. If node B wants to share the entangled pairs with Appl. Sci. 2021, 11, 10869 10 of 15 node A, it can ask node A for an entangled state through the classical channel. Thus, duplex communication is not necessary in the quantum channel. 4.2. Simulation Results The simulation result of the 5-node network in Figure 5 is shown in Figure 10. We should point out that the simulation is inﬂuenced by both channel noise and ﬁdelity loss in entanglement swapping. Entanglement swapping does not transfer qubit in the channel, but it surly brings a huge ﬁdelity loss. Thus, we set the ﬁdelity loss at a ﬁxed value of 12.77% in entanglement swapping. Because of the ﬁxed ﬁdelity loss, the system ﬁdelity cannot reach 1 even without noise in the channel. Figure 10. Fidelity as a function of bit-ﬂip and depolarizing noise with puriﬁcation. Puriﬁcation is applied after every entanglement swapping with time of m = 1. After the entanglement is established in the source and distribution node, extra puriﬁcation is applied with time of m = 1. In Figure 10, the blue line represents the 5-node network based on Bell states without puriﬁcation. After puriﬁcation, the orange line represents the Bell states scheme only using Bell states, the purple line represents the GHZ states scheme only using GHZ states, and the yellow line represents the fusion scheme we have proposed. The puriﬁcation algorithm can improve the ﬁdelity of the ﬁnal entangled states. When p is small, the fusion scheme and GHZ states scheme are far better than the Bell states scheme under bit-ﬂip noise. However, under depolarizing noise, the advantage of the GHZ states scheme is not obvious, which is even worse than the Bell states scheme most of the time. We can treat depolarizing noise as a mixed noise of bit-ﬂip and phase-ﬂip. In puriﬁcation of Bell states, there is no effective way to solve the noise with bit-ﬂip and phase-ﬂip together. In addition, in puriﬁcation of GHZ states, there is no effective way to solve the presence of phase-ﬂip noise. Under bit-ﬂip noise, there is no need to consider the existence of phase-ﬂip noise. Thus, the puriﬁcation performance under depolarizing noise is most of the time worse than that under bit-ﬂip noise. However, the fusion scheme we have proposed is surely better than the Bell states scheme. The performance of the fusion scheme is close to that of the GHZ states scheme under bit-ﬂip noise. Under depolarizing noise, the fusion scheme is the best. In the puriﬁcation scheme we have introduced, each successful puriﬁcation needs to measure one entangled group. Entanglement swapping also needs to measure lots of entan- gled states. Puriﬁcation does not always succeed, which will waste more entanglement. To measure the resource consumption in different schemes, we calculate the minimum basic states required in Algorithm 1. We generate one Bell pair with two basic states based on a H gate and CNOT gate. Although we obtain a GHZ group in the fusion scheme by using two Bell pairs, we use 3 basic states to get the entangled group in the GHZ states scheme. Appl. Sci. 2021, 11, 10869 11 of 15 Considering that measuring only one qubit in the GHZ group leads to the disappearance of the whole entanglement, entanglement swapping in the GHZ states scheme needs to measure two qubits in one group at a time to keep the system entangled. The number of states needed in the network with different node numbers is shown in Table 3. The fusion scheme improves the network performance in comparison to the Bell states scheme and consumes less qubits than the GHZ states scheme. Table 3. Basic states consumed in different schemes, writing as required/measured qubits. Node Number Bell States Scheme Fusion Scheme GHZ States Scheme 3 16/14 20/17 36/33 5 64/62 68/65 216/213 9 256/254 260/257 1296/1293 Simulation is performed for networks spanning different numbers of nodes and is shown in Figure 11. With increasing number of nodes, the ﬁdelity decreases. The performance of the fusion network is signiﬁcantly better than that of the Bell states scheme. Figure 11. The ﬁdelity in simulation of networks with different numbers of nodes in the presence of bit-ﬂip noise, where q = 0.15. In the fusion scheme, the calculation complexity is O(1). After entanglement swapping and puriﬁcation, high-ﬁdelity entanglement has been established between source and destination node. One can directly use the entangled group to ﬁnish teleportation at last. The price is sacriﬁcing a lot of entanglement resources in puriﬁcation and entanglement swapping. To verify the scheme in the true environment, we ﬁnish the simulation on the NISQ platform Cirq. Here, we choose bit-ﬂip noise and depolarizing noise. Since successful puriﬁcation is a probabilistic event, not all the simulation results are of reference value. Thus, the consumption is huge. With limited computing power, we set the simulation time for the 3-node network to 5000 and the simulation time for the 5-node network to 1000. The circuit of the 5-node network on Cirq is designed based on the puriﬁcation algorithm and shown in Figure 12. Simulation results of the 3-node and 5-node network on Cirq are shown in Figure 13. Appl. Sci. 2021, 11, 10869 12 of 15 Figure 12. The circuit of the 5-node network on Cirq. A, B, B2, C, C2, D, D2, and E are the initial entangled states shared by adjacent nodes. The fusion resources are p1 and p2. Target states in puriﬁcation are pui_j, where i = 1, 2, 3 and j = 0, 1, 2, 3. The output of the 5-node circuit comes from states in E, p1, and p2. To avoid the huge circuit, we prepare the target states with ﬁdelity 1 in pui_j. They should have come from entanglement swapping with less ﬁdelity, but the calculation capacity of the lab does not allow us realizing it on the circuit. Figure 13. The ﬁdelity of the 3-node and 5-node networks under bit-ﬂip noise and depolarizing noise. In each sub-ﬁgure, the circuits are the network based on Bell state, the network based on fusion state, and two networks with puriﬁcation. The simulation results of the realistic platform can be found to be far worse than the theoretical expectation. When the noise parameter is 0, the ﬁdelity of the network without Appl. Sci. 2021, 11, 10869 13 of 15 puriﬁcation stays around 0.25 on Cirq no matter how many nodes there are. Entanglement swapping can bring ﬁdelity loss not only because of the noise in the channel. The loss value in entanglement swapping we use comes from the work in [18], which is hard to realize on Cirq. However, we can still observe that the ﬁdelity gain comes from puriﬁcation. The network based on fusion state shows stronger robustness under depolarizing noise. Since entanglement swapping brings serious loss of ﬁdelity, the inﬂuence of bit-ﬂip noise is hard to observe. After puriﬁcation, the network based on fusion state has higher ﬁdelity than that of the Bell state network under both bit-ﬂip and depolarizing noise. 5. Conclusions In this paper, we propose a novel multi-hop quantum network based on fusion to balance the network performance and consumption. Entanglement swapping can establish entanglement between nodes in multi-hop networks, while fusion can transform entanglement from Bell states to GHZ states. Puriﬁcation can ensure high ﬁdelity of the entanglement in the network. A puriﬁcation algorithm is given to ensure the robustness of the communication system. In the algorithm, we set m = 1 and m = 1, which means applying a one-time puriﬁcation after each entanglement swapping and a one- time puriﬁcation after fusion. Simulation results indicate that our new scheme has better robustness against the noise than the Bell states scheme. It also consumes less entanglement than the GHZ states scheme. The calculation complexity is kept at a low level. Moreover, we give the circuit of a 5-node network and simulate it under bit-ﬂip and depolarizing noise on the NISQ platform Cirq to verify our network. Finally, simulation results conﬁrm the superiority of the proposed scheme in balancing network performance and consumption. Some problems still need to be solved. First, the inﬂuence of distance between nodes in the quantum network is not considered. Second, the performance of entanglement swap- ping can also be improved by developing a new technology. In the end, since puriﬁcation needs huge amounts of entanglement and often fails, the number of nodes in the simulation is not enough because of the limited computer capacity. Author Contributions: Conceptualization, J.X. and H.X.; methodology, J.X.; software, J.X.; valida- tion, J.X., M.M. and P.W.; formal analysis, J.X.; investigation, J.X.; resources, J.X.; data curation, J.X.; writing—original draft preparation, J.X.; writing—review and editing, J.X.; visualization, J.X.; supervision, J.X.; project administration, X.C.; funding acquisition, X.C. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conﬂicts of Interest: The authors declare no conﬂicts of interest. Appendix A Assume that node A wants to send a quantum state jyi to node B, the state of the system is jGi. The Bell pair shared between two nodes is jF i. jGi =jyi F = (aj0i + bj1i) p (j0 0 i +j1 1 i) A B A B (A1) =p [a(j00 0 i+ j 01 1 i) + b(j10 0 i +j11 1 i)]. A B A B A B A B 2 Appl. Sci. 2021, 11, 10869 14 of 15 jyi is the control state, while the state with subscript A is the target state. Node A operates CNOT gate, and then applies the H gate to jyi. jGi is transformed to jG i. jG i = [j00 i(aj0 i + bj1 i) +j01 i(aj1 i + bj0 i) 1 A B B A B B (A2) +j10 i(aj0 i bj1 i) +j11 i(aj1 i bj0 i)]. A B B A B B Two qubits in node A are measured. Measurement results and the qubit state owned by node B are in Table A1. Quantum gate in node B can rebuild jyi with entangled state. Relevant data based on GHZ states are also given in Table A1. Table A1. Teleportation based on Bell pairs and GHZ states. States in A State in B Operation in B j00 i aj0 i + bj1 i / A B B j01 i aj1 i + bj0 i X gate A B B j10 i aj0 i bj1 i Z gate A B B j11 i aj1 i bj0 i ZX gate A B B j00 0 i aj0 i + bj1 i / A A B B j01 1 i aj1 i + bj0 i X gate A A B B j10 0 i aj0 i bj1 i Z gate A A B B j11 1 i aj1 i bj0 i ZX gate A A B B Appendix B Both nodes A and C share a Bell pair jF i with node B. The system can be described as jLi . 1 1 jLi =p (j0 0 i +j1 1 i) p (j0 0 i +j1 1 i) A B A B B C B C 2 2 1 (A3) = (j0 0 ij00i +j0 1 ij01i +j1 0 ij10i A c B A C B A C B +j1 1 ij11i ). A C B Node B operates CNOT gate on the states it has held, and the H gate on the entangled state. The entangled state is the control state, while the other is target state. Now, the system becomes jL i . 0 0 + 1 1 0 1 + 1 0 j i j i j i j i A C A C A C A C jL i = p j00i + p j01i 1 B B 2 2 2 2 (A4) j0 0 ij1 1 i j0 1 ij1 0 i A C A C A C A C + p j10i + p j11i . 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Applied Sciences – Multidisciplinary Digital Publishing Institute

**Published: ** Nov 17, 2021

**Keywords: **multi-hop quantum networks; teleportation; fusion; entanglement purification

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