# Surface-Codes-Based Quantum Communication Networks
## 5 Illustrative Numerical Results  
Although the channel loss dominates the performance of quantum repeaters, there will be quantum errors associated with each stage, which can be represented by using the quantum channel  
Figure 5: Illustrative four-node ring (**left**) and four-node mesh (**right**) quantum communication networks. Figure 6: Extending the distance between two nodes in a quantum communication network (QCN) by creating the logical qubit spanning two spatially separated surface code (SC) patches. model provided in Figure 7, where \(X\) and \(Z\) quantum errors occur with the same probability \(p\). The corresponding Kraus representation [1] is given by:  
\[\rho_{f}\ =\ \xi(\rho)\ =\ (1-2p)\rho+pX\,\rho X+pZ\,\rho Z. \tag{1}\]  
Let us consider the BB84 protocol by employing the approach introduced in previous section. The corresponding secret-key rate after \(N\) sections will be:  
\[SKR\ =\ \{[1-P(E)]T\}^{N}\text{max}\bigg{(}1-h_{2}\bigg{(}q_{N}^{(Z)} \bigg{)}-f_{e}h_{2}\bigg{(}q_{N}^{(X)}\bigg{)},0\bigg{)}, \tag{2}\]  
where \(f_{e}\) denotes the error correction inefficiency (\(f_{e}\geq 1\)), \(q_{N}^{(X)}\left[q_{N}^{(Z)}\right]\) denotes the quantum bit-error rate (QBER) in the \(X\)-basis (\(Z\)-basis) after \(N\) stages, \(T\) represents the single link transmissivity, and \(h_{2}(x)\) is the binary entropy function \(h_{2}(x)\ =\ -x\log_{2}(x)-(1-x)\log_{2}(1-x)\). The term \(h_{2}\bigg{(}q_{N}^{(Z)}\bigg{)}\) represents the amount of information [PERSON] was able to learn during the raw key transmission, which can be removed from the final key during the privacy amplification phase. The term \(f_{e}h_{2}\bigg{(}q_{N}^{(X)}\bigg{)}\) represents the amount of information revealed to [PERSON] during the information reconciliation stage. The dark counts, device imperfections, and errors introduced by [PERSON] are all contributed to the [PERSON] and included in transition probability \(p\). The QBER after \(N\) stages can be estimated by:  
\[q_{N}\ =\ \frac{1-s^{N}}{2},\ s\ =\ 1-2p. \tag{3}\]  
The probability of the syndrome decoding error is bounded by [1]:  
\[P(E)\leq\sum_{j\ =\ \lfloor(d-1)/2\rfloor+1}^{d^{2}}\left( \begin{array}{c}d ^{2}\\ j\end{array} \right)(1-s)^{j}s^{d-j},\ s\ =\ 1-2p, \tag{4}\]  
So, \([1-P(E)]T\) represents the success probability for the single stage. The total success probability can be estimated by \([[1-P(E)]T]^{N}\) and is illustrated in Figure 8 by setting the \(X\) (\(Z\)) qubit error probability to \(p=10^{-2}\) and transmissivity to \(T=1\), for different \(d\times d\) surface codes. Figure 7: Quantum channel model under study: (**a**) Pauli operator description and (**b**) density operator description. The numerical results for secret-key rate (SKR) for different transmissivities \(T\) (assuming that \(f_{e}=1\)) vs. the number of stages \(N\) are summarized in Figures 9 and 10. The channel transmittance in Figure 9 is set to \(T=0.95\), while in Figure 10 it is set to \(T=0.85\). The qubit error transition probability \(p\) is used as a parameter. In both figures, the \(7\times 7\) surface code is used. Given that the effective transmission distance of the fiber is given by [20]:  
\[L_{\rm eff}\ =\ \frac{1-e^{-\alpha L}}{\alpha}\approx 1/\alpha, \tag{5}\]  
Figure 8: Total success probability defined as \((1-P(E))^{N}\), where \(N\) is the number of stages, when the \(d\times d\) surface code is used, and syndrome decoding is applied. Figure 9: Normalized secret-key rate (SKR) vs. the number of stages \(N\) assuming that the \(7\times 7\) surface code is used, and single link channel transmittance is \(T=0.95\). For ultra-low loss fiber introduced in [21] with attenuation coefficient \(\alpha=0.1419\) dB/km, we obtain that \(L_{\text{eff}}=30.606\) km.