# On Global Quantum Communication Networking
## 2 Proposed Cluster States-Based Quantum Communications Networks  
To enable the next generation of quantum communication networking, we envision a scenario in which disconnected terrestrial cluster states-based QCNs are coupled through the LEO satellite (cluster state) quantum network, thus providing global coverage. The proposed quantum network will be highly robust against turbulence encountered by FSO links, as the envisioned quantum satellite network will communicate to ground nodes only through the LEO satellite-to-ground links, exhibiting a vertical downlink profile through vacuum followed by a turbulence layer with strength that is altitude-dependent. The cluster states belong to the class of the graph states, which also include Bell states, Greenberger-Horne-Zeilinger (GHZ) states, W-states, and various entangled states used in quantum error correction [22]. When the cluster \(C\) is defined as a connected subset on a \(d\)-dimensional lattice, it obeys the set of eigenvalue equations \(S_{a}\big{|}\phi\big{\rangle}_{C}=\big{|}\phi\big{\rangle}_{C}\), \(S_{a}=X_{a}\underset{b\in N(a)}{\otimes}Z_{b}\), where \(S_{a}\) are _stabilizer operators_ with \(N(a)\) denoting the neighborhood of \(a\in C\). To create a 2-D cluster state, the approach proposed by [PERSON] et al. [23] is applicable; it employs linear states, generated by spontaneous parametric down conversion (SPDC), local unitaries, and type I fusion to create the desired 2-D cluster state. The type I fusion is illustrated in Figure 1, based on [23]. The vertical photon is reflected by the polarization beam splitter (PBS), while the horizontal photon is transmitted through the PBS. Given the probabilistic nature of the PBS, with the photons present at both the left and right input ports, there are four possible outcomes, each occurring with probability 0.25. Two outcomes correspond to the desired fusion operators, and the success probability of the fusion is 0.5. When a single photon is detected by the detector, a successful fusion is declared. The procedure to create the T-shape cluster state is described in Figure 2. To create the box-cluster state, we start with a four-qubit linear cluster state, re-label the qubits 2 and 3, and apply the Hadamard gates to qubits 2 and 3, which effectively establish the bond between qubits 1 and 4. Namely, relabeling the qubits is equivalent to the SWAP gate action. To create the box-on-chain cluster state, we start with a longer linear chain of qubits and apply the same approach as in a box-state creation. Two T-shape cluster states can be fused together to get the \(H\)-shape cluster state, etc. Figure 1: Illustrating the type I fusion process. PBS: polarization beam splitter. Once the 2-D cluster state of nodes is created, we can use properly selected \(Y\) and \(Z\) measurements to create the EPR pair between any two arbitrary nodes in the quantum network. As a reminder, the role of the \(Z\) measurement is to remove the particular node (qubit) from the cluster, whereas the role of \(Y\) measurement is to remove a given node and link neighboring nodes. As an illustration, the 2-D cluster state with nine nodes is shown in Figure 3. Let us assume that we are interested in establishing EPR pairs between nodes 3 and 7 as well as nodes 1 and 9. We first perform \(Y\) measurements in the following order: \(Y_{8}\), \(Y_{5}\), and \(Y_{6}\) to get the intermediate stage. We then perform \(Z\)-measurement on node 2 and Y measurement on node 4 to get the two desired EPR pairs. Given that the 2-D cluster state is universal, it is possible to use the same network architecture for both QCN and distributed quantum computing. We also imagine the scenario in which each node is equipped with multiple qubits, wherein several layers of 2-D cluster states are active at the same time, which will allow us to simultaneously perform QCN and distributed quantum computing. Moreover, when several 2-D cluster states are run in parallel on the same set of network nodes, we will be able to reconfigure the QCN as needed.