# Two-Excitation Routing via Linear Quantum Channels
## 5 Discussion  
In this paper, we have investigated the routing of two fermionic excitations across a quantum network. In the proposed protocol we were able to show that two fermions, initially located on a sender block composed of two sites, can be efficiently routed to a receiver block of two sites, provided that both the former and the latter are weakly coupled to a one-dimensional quantum wire, modeled by a fermionic nearest-neighbor hopping Hamiltonian. We have proposed two different protocols: in the first one, we have assumed switchable couplings and derived the connection points of the wire which yields high-quality routing; in the second one we have assumed permanent couplings and envisaged in the tunability of the sender's intrasite coupling a mean to route the two excitations to the desired location. In each considered configuration we obtained a perturbatively-perfect fidelity, i.e., \(F=1-O(J_{0})\), where \(J_{0}\) is the weak coupling of the sender and receiver block to the wire, with a transfer time scaling as \(O(10J_{0}^{-1})\). We also compared the fermionic network with a spin-\(\frac{1}{2}\) network interacting via an \(XX\)-Heisenberg Hamiltonian. Due to the 2D nature of the network, the analysis had to rely on numerical evaluation and we found that, apart from the scenario where the receiver blocks are located towards the end of the wire, efficient routing is not achievable with qubits. However, our work hints towards the possibility to utilise very long quantum wires for the proposed 2-qubit routing protocol as we observed an enhancement of the routing fidelity by increasing the length of the wire. In such a scenario, our protocol may be utilised also for two-qubit entanglement routing, similarly to Refs. [17; 18] where the tranfer is achieved between the edges of the chain.  
For a realistic implementation of our protocol one should however consider possible experimental imperfections in the couplings, the time-dependence of the sender/receiver couplings to the chain, and decoherence due to interaction with the surrounding environment. While an extensive analysis of the performance of our protocol under these conditions has not been presented here, and may be left to future investigation, it is reasonable to assume that similar analyses done for the one-particle transfer scenario may apply also for two-particle routing as our results are derived from single-particle transition amplitudes. In this regard, disorder in couplings [21; 30; 31; 32; 33], time-dependent couplings [34; 35], and decoherence [36; 37] have been extensively addressed and several strategies to counter the detrimental effect on the transfer quality have been devised which may find application also in our two-particle routing protocol.  
Conceptualization, T.J.G.A. and W.J.C.; Data curation, T.J.G.A. and W.J.C.; Software, T.J.G.A. and W.J.C.; Writing--original draft, T.J.G.A. and W.J.C. All authors have read and agreed to the published version of the manuscript.  
This research has been carried out using computational facilities procured through the European Regional Development Fund, Project No. ERDF-080 A supercomputing laboratory for the University of Malta.  
The data presented in this study are available on request from the corresponding author.  
We thank [PERSON] and [PERSON] for technical support with the computational facilities.  
The authors declare no conflict of interest.