## 4 Multi-Hop Bidirectional Quantum Communication
### N-Hop Bidirectional Quantum Communication  
Specifically, in the \(n\)-hop case, the entanglement channel between the node \(N_{1}\) and the nodes \(N_{n+1}\) is shown in Figure 5. Assume that the node \(N_{1}\) intends to transmit an arbitrary qubit \(|\chi\rangle_{a}=\alpha|0\rangle_{a}+\beta|1\rangle_{a}\) to the node \(N_{n+1}\), while the node \(N_{n+1}\) intends to transmit \(|\chi\rangle_{b}=\gamma|0\rangle_{b}+\delta|1\rangle_{b}\) to the node \(N_{1}\) at the same time, where \(|\alpha|^{2}+\left|\beta\right|^{2}=1\) and \(\left|\gamma\right|^{2}+|\delta|^{2}=1\).  
To achieve bidirectional \(n\)-hop qubit transmission, each node on the quantum path performs quantum measurements simultaneously. After that, the node \(N_{1}\) and all intermediate nodes send their measurement results together with the types of the composite GHZ-GHZ states shared between themselves and the next-hop nodes to the node \(N_{n+1}\) through classical channels independently. Similarly, the node \(N_{n+1}\) and all intermediate nodes send their measurement results together with the types of the GHZ-GHZ states to the node \(N_{1}\) independently. Based on the received information, the node \(N_{1}\) and the node \(N_{n+1}\) perform appropriate unitary operations locally to recover the original qubit.  
Figure 4: Quantum circuit for two-hop bidirectional quantum communication.  
According to the analysis in Section 3, the unitary operations of \(N_{1}\) and \(N_{n+1}\) can be determined in a hop-by-hop manner. We take the qubit transmission from the node \(N_{1}\) to the node \(N_{n+1}\) as an example. Given that node \(N_{1}\) intends to transmit \(|\chi\rangle_{a}\) to the node \(N_{n+1}\), after \(N_{1}\)'s CNOT, Hadamard operations, and quantum projective measurement, the quantum state of \(N_{2}\)'s particle would be \(T_{1}|\chi\rangle_{a}\), where \(T_{i}(i=1,2,\cdots,n)\) is a unitary matrix. After \(N_{2}\) performs CNOT, Hadamard operations, and quantum projective measurement, the quantum state of \(N_{3}\)'s particle would be \(T_{2}(T_{1}|\chi\rangle_{a})\). Similarly, after all the intermediate nodes perform CNOT, Hadamard operations, and quantum projective measurements, the quantum state of the node \(N_{n+1}\)'s particle would be \(T_{n}\cdots T_{2}T_{1}|\chi\rangle_{a}\).  
To recover the original qubit \(|\chi\rangle_{a}\) locally, the node \(N_{n+1}\) needs to perform the unitary operation \(U\) on his particle \(N_{n+1}^{1}\), given by  
\[U=\prod_{i=1}^{n}T_{i} \tag{9}\]  
where \(T_{i}(i=1,2,\cdots,n)\) is determined according to Table 2, based on the measurement results of the node \(N_{i}\) and the composite GHZ-GHZ state previously shared between the node \(N_{i}\) and \(N_{i+1}\).  
Similarly, for the other direction of qubit transmission, the node \(N_{1}\) can perform the unitary operation \(V\) on its particle \(N_{1}^{3}\) to recover the original qubit \(|\chi\rangle_{b}\),  
\[V=\prod_{j=n}^{1}C_{j+1} \tag{10}\]  
where \(C_{j+1}(j=1,2,\cdots,n)\) is determined according to Table 2, based on the measurement results of the node \(N_{j+1}\) and the composite GHZ-GHZ state previously shared between the node \(N_{j}\) and \(N_{j+1}\).