& 0 & 0 & $\Delta_\mathrm{D}$ & 0 & $\frac{\pi}{\sqrt{2} |\Omega_\mathrm{Y}|}$ & $\Delta_\mathrm{D}$\
$ \propto H_1$ & -$\Omega_\mathrm{H} \in \mathbb{R}$ & 0 & $\sqrt{2} \Omega_\mathrm{H}$ & 0 & $\Delta_\mathrm{D}$ & 0 & $\frac{\pi}{2 \Omega_\mathrm{H}}$ & $\Delta_\mathrm{D}$\
SWAP$_{12}$ & 0 & 0 & 0 & $\Delta_\mathrm{S}$ & 0 & $\Delta_\mathrm{S}$ & $\frac{\pi}{\Delta_\mathrm{S}}$ & $\Delta_\mathrm{D}$\
C($-Z$)$_{12}$ & 0 & 0 & 0 & $\Delta_\mathrm{C}$ & 0 & 0 & $\frac{\pi 2 \sqrt{2}}{\Delta_\mathrm{C}}$ & $\Delta_\mathrm{D}$\

Definition of Logical Qubits and Computational Subspace
-------------------------------------------------------

Before finding the appropriate gates, we first need to define the logical qubits within the DFS. In principle, assuming an even number of atoms $N$, the dimension of the DFS is ${ {N \choose N/2} }$ and therefore, it is possible to encode $\log_2 { {N \choose N/2} }$ logical qubits. However, it is more useful to restrict the computational subspace to a smaller set of states in order to achieve universal quantum computation. As the DFS is spanned by all (tensor) products of singlet states over two atoms, it is natural to define the logical qubits as $$\begin{aligned}
    {|0\rangle}_{j}^{\mathrm{L}} \equiv & {|G_{j,j+1}\rangle} = {|g\rangle}_j \otimes {|g\rangle}_{j+1} \label{eq:logicalQB1} \\
    {|1\rangle}_{j}^{\mathrm{L}} \equiv & {|A_{j,j+1}\rangle} =  \left({|e\rangle}_j \otimes {|g\rangle}_{j+1}-{|g\rangle}_j \otimes {|e\rangle}_{j+1}\right)/\sqrt{2}.
    \label{eq:logicalQB2}\end{aligned}$$ 