Author Contributions
Conceptualization, J.X. and H.X.; methodology, J.X.; software, J.X.; validation, 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.
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. is the control qubit of the CNOT gate.
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. is the control qubit of the CNOT gate.
Figure 2.
(a) Teleportation model and (b) circuit based on GHZ states. GHZ states 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 based on .
Figure 2.
(a) Teleportation model and (b) circuit based on GHZ states. GHZ states 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 based on .
Figure 3.
(a) Entanglement swapping based on Bell pairs and (b) circuit in node B. is shared between nodes A and B. is shared between nodes B and C. Once two states in node B are measured, entanglement between nodes A and C is established.
Figure 3.
(a) Entanglement swapping based on Bell pairs and (b) circuit in node B. is shared between nodes A and B. is shared between nodes B and C. Once two states in node B are measured, entanglement between nodes A and C is established.
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.
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.
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.
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.
Figure 6.
Purification circuit for a Bell pair. Assume two Bell pairs are distributed to the source and the destination node. and are rotation operators to solve phase-flip noise. Fidelity of one entangled pair can be improved by measuring the other entangled pair.
Figure 6.
Purification circuit for a Bell pair. Assume two Bell pairs are distributed to the source and the destination node. and are rotation operators to solve phase-flip noise. Fidelity of one entangled pair can be improved by measuring the other entangled pair.
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.
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.
Figure 8.
Entanglement pumping for purification. F is the fidelity of the entangled states before purification, satisfying . Purification increases the fidelity of control states to . Target states are always with fidelity F.
Figure 8.
Entanglement pumping for purification. F is the fidelity of the entangled states before purification, satisfying . Purification increases the fidelity of control states to . Target states are always with fidelity F.
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.
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.
Figure 10.
Fidelity as a function of bit-flip and depolarizing noise with purification. Purification is applied after every entanglement swapping with time of . After the entanglement is established in the source and distribution node, extra purification is applied with time of .
Figure 10.
Fidelity as a function of bit-flip and depolarizing noise with purification. Purification is applied after every entanglement swapping with time of . After the entanglement is established in the source and distribution node, extra purification is applied with time of .
Figure 11.
The fidelity in simulation of networks with different numbers of nodes in the presence of bit-flip noise, where .
Figure 11.
The fidelity in simulation of networks with different numbers of nodes in the presence of bit-flip noise, where .
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 , where and . 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 . 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 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 , where and . 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 . 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.
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.
Table 1.
Shared Bell pairs and the matching probability. Here, .
Table 1.
Shared Bell pairs and the matching probability. Here, .
One Shared Pair | Probability | Two Shared Pairs | Probability |
---|
| | | |
| | | |
| | | |
| | | |
Table 2.
Shared GHZ states and the matching probability (bit-flip noise).
Table 2.
Shared GHZ states and the matching probability (bit-flip noise).
Shared States | Probability |
---|
| |
| |
| |
| |
| |
| |
| |
| |
Table 3.
Basic states consumed in different schemes, writing as required/measured qubits.
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 |