the computational subspace given by: $$\mathcal{P}_\mathrm{CS} H_{\mathrm{las}}\mathcal{P}_\mathrm{CS}
    = \sum_{j \ \mathrm{odd}} 
        \left(\frac{\Omega_{j,j+1}^{-}}{\sqrt{2}} {|1\rangle}^\mathrm{L}_j{\langle 0|} + \mathrm{h.c.} \right)
        + \Delta_{j,j+1}^+ {|1\rangle}^\mathrm{L}_j {\langle 1|},$$ where we used the following notation $x_{i,j}^\pm = \frac{1}{2} \left(x_i \pm x_{j}\right)$ to abbreviate the combination of parameters.

![(a) Hilbert space of two TLS in the basis of non-DFS and DFS states with energies in the interaction picture with respect to $H_\mathrm{qb}$. The blue arrows denote the transitions necessary for the Pauli-X gate. The triplet states ${|E\rangle} = {|ee\rangle}$ and ${|S\rangle} = {|eg+ge\rangle}/\sqrt{2}$ are not inside the DFS. (b) DFS of 4 TLS consists of 6 states that split into the $2^2$-dimensional computational subspace and two states in the auxiliary subspace. The blue \[green\] arrows denote the transitions necessary for the Pauli-X \[and Controlled-(-Z)\] gate.[]{data-label="fig2"}](figure2v1.pdf){width="98.00000%"}

Single-Qubit Gates
------------------

The goal is to find the $\{\Omega_n,\Delta_n\}$ such that they define both the phase and Pauli-$X$ (and $Y$) gates over the computational subspace.

#### **Two atoms:** 

This is the simplest situation because the size of the computational space is the same as the one of the DFS. In this case (see also [@beige00b]), a phase shift gate on the logical qubit ($\alpha {|0\rangle}^\mathrm{L} + \beta {|1\rangle}^\mathrm{L}\rightarrow \alpha {|0\rangle}^\mathrm{L} + \beta 