The total number of physical qubits is given by $n\times N_{\text{edges}}$, where $N_{\text{edges}}$ denotes the total number of lattice edges.

See [5, 12] for setups with a similar geometry. Another way to think about this is to consider the doubled picture, which corresponds to a ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ color code, see [36].

A single logical qubit will be our main focus in this work, however, the general theory we discuss is also applicable to different logical encodings: For example if the braiding is a phase $e^{2\pi i/d}$ the logical state will be a qudit, and if we have $k$ disjoint pairs of anyon triplets $((a1,a2,a3),(b1,b2,b3))$, the system will describe $k$ logical qudits (with a possibly different $d$ for each pair).

This is consistent with the choice of end-site $s_1$ for ribbon operators stretching from aboundary to the bulk made in [31].

Note that we can use this simplified expression since we are acting on ${\otimes }_l|\text{id}⟩$. For ribbon operators acting on a general state, see [30, 31, 44].

This follows from the modular $S$-matrix element $S_{a1,b2}$, whose general expression can be found in [31, 46].

This is analogous to the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ case discussed below (2.2).

We will require that the 2-cocycle representative $\alpha$ satisfies the properties of [31, Lemma 5.1], which we summarize in (a19).

The decomposition of a ${\mathbb{Z}}_4$ qudit as two qubits has previously been discussed in [47]. See also [48] for an alternative mapping of the $D_4$ operators onto a 3-qubit Hilbert space.

As explicit matrices in the basis $\{|00⟩,|01⟩,|10⟩,|11⟩\}$, we have $$\mathcal{X}=\left(\begin{array}{cccc}0& 0& 0& 1\\ 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\end{array}\right)=\left(\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right)\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\\ 0& 1& 0& 0\end{array}\right)=(\mathbb{I}\otimes X)CX.$$

This observation agrees with [12], in which the alternative presentation of $D(D_4)$ as $D^{\omega }({\mathbb{Z}}_2^3)$ was used. Certain Clifford-hierarchy stabilizer codes have previously been discussed in [25, 26].

$D_4$ corresponds to $N=1$, in which case the sums evaluate to: $$\underset{a=0}{\overset{1}{⨁}}[r^{2a}]=1\oplus [r^2],\underset{a=1}{\overset{1}{⨁}}[r^{2a-1}]=[r].$$

We include ${\mathrm{S}}_{\mathrm{p}}^{\mathrm{s}}$ (rather than only $({\mathrm{S}}_{\mathrm{p}}^{\mathrm{s}}{)}^{2N}$) for two reasons: The former will be necessary for the boundary stabilizers and will appear (with power of $-2$) as the result of thecommutator ${\mathrm{A}}_{\mathrm{v}}^{\mathrm{r}},{\mathrm{S}}_{\text{pNE}}^{\mathrm{r}}$.

We do not consider boundaries with non-trivial 2-cocycles in this work.

An alternative Hamiltonian with projectors onto the $[g]$ and $R$ sectors was proposed in [57].