# Two-Excitation Routing via Linear Quantum Channels
## 2 Many-Body Dynamics in Non-Interacting Fermions on a Discrete Lattice  
Let us consider a discrete lattice model where each site can host one spinless fermion and whose dynamics is governed by the hopping Hamiltonian  
\[\hat{H}=\sum_{\langle ij\rangle}J_{ij}\left(\hat{c}_{i}^{\dagger}\hat{c}_{j}+h.c\right)\,, \tag{1}\]  
where \(\hat{c}_{i}^{\dagger}\) (\(\hat{c}_{i}\)) is the creation (annihilation) operator of a fermion on site \(i\) and \(J_{ij}\) is the kinetic term accounting for the hopping of a fermion between neighboring sites \(i\) and \(j\). This Hamiltonian conserves the total number of excitations (fermions) and can be block-diagonalised in each fixed particle-number sector. Moreover, because of the quadratic nature of the Hamiltonian, only the spectrum in the single-excitation subspace is needed in order to retrieve the full energy spectrum. This is a consequence of the non-interacting nature of the Hamiltonian witnessed by the absence of quartic terms accounting for particle-particle interactions [26]. We report here, for the sake of completeness, the main steps for the derivation of the many-body dynamics in terms of single-body dynamics for non-interacting fermions, which is standard procedure in the second-quantization formalism.  
The diagonalized form of the Hamiltonian in Equation (1) in the single-excitation sector reads  
\[\hat{H}=\sum_{k=1}^{N}E_{k}|E_{k}\rangle E_{k}|=E_{k}\hat{c}_{k}^{\dagger}\hat{ c}_{k} \tag{2}\]  
where \(\{E_{k},|E_{k}\rangle\}\) are the eigenvalues and eigenvectors of the \(N\)-dimensional adjacency matrix of the graph with entries \(J_{ij}\). Expressed in the position basis, \(|n\rangle\equiv\hat{c}_{n}^{\dagger}|\mathbf{0}\rangle=|00\dots 1n00\dots\rangle\), where \(|\mathbf{0}\rangle\) represents the fermionic vacuum state and \(|1_{n}\rangle\) denotes the presence of a fermion on site \(n\), the energy eigenstates in the single-excitation sector read \(|E_{k}\rangle=\sum_{n=1}^{N}a_{\mathbf{k}n}|n\rangle\), with \(a_{kn}=\langle n|E_{k}\rangle\). The single-particle transition amplitude of an excitation from site \(s\) to site \(r\) is given by  
\[f_{s}^{\prime}(t)=\langle r|e^{-i\hat{H}t}|s\rangle=\sum_{k=1}^{N}a_{r\in k}a_ {s,k}^{*}e^{-iE_{k}t}\,. \tag{3}\]  
Because of the non-interacting nature of the Hamiltonian in Equation (1), the energy eigenstates in the Hilbert space with \(m\) fermionic excitations are given by  
\[\left|E_{k_{1}k_{2}\dots k_{m}}\right\rangle=\sum_{n_{1}<n_{2}<\dots<n_{m}=1}^ {N}a_{n_{1}n_{2}\dots n_{m},k_{1}k_{2}\dots k_{m}}|n_{1}n_{2}\dots n_{m}\rangle\,, \tag{4}\]  
with eigenvalues \(E_{k_{1}k_{2}\dots k_{m}}=E_{k_{1}}+E_{k_{2}}+\dots+E_{k_{m}}\) and \(a_{n_{1}n_{2}\dots n_{m},k_{1}k_{2}\dots k_{m}}\) denoting the Slater determinant.  
The many-body transition amplitude of \(m\) excitations from sites \(\mathbf{s}=\left\{s_{1},s_{2},\ldots,s_{m}\right\}\) to sites \(\mathbf{r}=\left\{r_{1},r_{2},\ldots,r_{m}\right\}\) is readily obtained as a determinant of a matrix whose entries are the single-particle transition amplitudes in Equation (3),  
\[\begin{split}& f_{\mathbf{s}}^{\ast}(t)=\langle\mathbf{r}|e^{-i \hat{m}t}|\mathbf{s}\rangle=\sum\limits_{k_{1}<k_{2}<\cdots<k_{m}=1}^{N}e^{-i \left(\hat{t}_{k_{1}}+\hat{t}_{k_{2}}+\cdots+\hat{t}_{k_{m}}\right)t}\langle r_{ 1}r_{2}\ldots r_{m}|E_{k_{1}k_{2}-k_{m}}\rangle\langle E_{k_{1}k_{2}-k_{m}}|s_{ 1}s_{2}\ldots s_{m}\rangle\\ =&\begin{vmatrix}f_{1}^{\ast_{1}}(t)&f_{1}^{\ast_{1}}(t )&\cdots&f_{m}^{\ast_{m}}(t)\\ f_{2}^{\ast_{2}}(t)&\cdots&\cdots&f_{m}^{\ast_{m}}(t)\\ \vdots&\ddots&\vdots\\ f_{m}^{\ast_{m}}(t)&\cdots&f_{m}^{\ast_{m}}(t)\end{vmatrix}.\end{split} \tag{5}\]  
The expression given in Equation (5) holds for every fermionic quadratic model, whereas if the operators in Equation (1) represent bosons, then, instead of the determinant, the many-body transition amplitude is given by the permanent of the matrix [25; 26].