# Two-Excitation Routing via Linear Quantum Channels
## 4 Routing in Spin Systems  
In the previous Sections, we have shown how, in the weak-coupling regime, routing of two-excitations from a sender to a receiver block can be achieved both in a switchable and a permanent coupling configuration in quadratic Hamiltonians. In this Section, we will consider the case when the network is made up of spin-\(\frac{1}{2}\) particles interacting via an \begin{table}
\begin{tabular}{l c c} \hline \hline \(k\) & \(J_{r}\) & \(w_{i}\) \\ \hline
1 & \(\frac{\sqrt{3}-1}{2}\) & 1,11 \\
2 & \(1\) & 1,2,4,5,7,8,10,11 \\
3 & \(\sqrt{2}\) & 1,3,5,7,9,11 \\
4 & \(\sqrt{3}\) & 1,5,7,11 \\
5 & \(\frac{\sqrt{3}+1}{2}\) & 1,11 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values of the intraspin couplings for the receiver blocks and available wire’s connection sites for the receiver block for a wire’s length \(n_{w}=11\). Figure 4: (**left**) Single-excitation energy levels in the permanently coupled routing scheme. The sender’s energy level can be tuned to be in resonance with a different pair of wire’s (and receiver’s) energy levels by tuning \(J_{s}\). (**right**) Excitation transfer in the permanent coupling configuration of Section 3.2 with \(n_{w}=11\), \(J_{0}=0.01\) and coupling scheme as in Figure 3. The different curves correspond to the transfer fidelity of the two excitation to different receiver block by tuning \(J_{s}\) to \(J_{r_{i}}\). The colors of the curves correspond to the energy levels in the left panel. \(XX\)-type Heisenberg Hamiltonian. We will consider the switchable routing configuration depicted in Figure 1 with Hamiltonian  
\[\hat{H}=\sum_{\langle ij\rangle}\frac{l_{ij}}{2}\left(\hat{\sigma}_{i}^{x}\hat{ \sigma}_{j}^{x}+\hat{\sigma}_{i}^{y}\hat{\sigma}_{j}^{y}\right)\,, \tag{18}\]  
where \(\langle\rangle\) denotes the summation running over nearest-neighbor sites. Notice that Equation (18) differs from Equation (12) because of the 2D nature of the network. By introducing the ladder operators \(\hat{\sigma}^{\pm}=\frac{\hat{\sigma}^{\pm}\pm\hat{\sigma}^{y}}{2}\), the Hamiltonian of the system can be obtained from Equations (6)-(10) by substitution of \(\hat{\epsilon}\rightarrow\hat{\sigma}^{-}\) and \(\hat{\epsilon}^{\dagger}\rightarrow\hat{\sigma}^{+}\). As already stated in Section 3, were the network one-dimensional, i.e., the receiver block coupled to the last spin of the wire, then the Jordan-Wigner transformation would map Equation (18) to a quadratic spinless fermion Hamiltonian. Such a case would constitute a special instance of the analysis in Section 3.1 and several works on two-qubit quantum state transfer can be found in the literature. However, in the general case, where the receiver block is coupled to an arbitrary site of the wire, the Jordan-Wigner mapping does not apply as the system looses its one-dimensional nature. However, for a configuration such as the one depicted in Figure 1, there is only one spin belonging to the wire that has three nearest-neighboring spins; therefore, the one-dimensional nature of the model is only locally broken with the lowest possible coordination number. It is therefore interesting to investigate if the routing properties of the spinless non-interacting one-dimensional model of Section 3 still persist also when the network is made of spins when a rigorous mapping to fermions is not possible. Now, in order to evaluate the transition probability of two-excitations, we need to diagonalise the Hamiltonian in Equation (18) in the two-excitations sectors, being the reduction to one-particle transition amplitudes not possible. The dimension of the Hamiltonian in the two-excitation Hilbert space is the binomial factor \(\dim\!\left[\hat{H}^{(2)}\right]=\binom{N}{2}\) and we diagonalise the Hamiltonian numerically for \(N=306\) using the QuSpin package [29]. From Figure 5 we see that, as for the free-fermion network in Section 3.1, the transition probability of the two excitations from the sender block to the receiver block is negligible whenever the latter is coupled to every third spin of the linear chain.