Throughout this work, fault-tolerant encodings refer to those that have an error-correcting threshold in the presence of local errors and faulty measurements.

Therefore the $K$ matrix is $K=(\genfrac{}{}{0ex}{}{2}{0}\genfrac{}{}{0ex}{}{0}{-4})$.

Moreover, unlike plane waves, the code states of an $N$-mode GKP code are not simple distributions over $L^2(\mathbb{R}{)}^{\otimes N}$. This is despite the fact that they are sums of delta functions, each of which is well-defined distribution on $L^2(\mathbb{R})$. For a single-mode GKP code, this can be seen by evaluating the code state on a wave function with an amplitude that goes as $1/|x|$ for large $x$. This does not evaluate to a finite value, due to the divergence of the sum ${\sum }_{k}1/k$. One way to give precise mathematical meaning to GKP code states is as tempered distributions. These are distributions on the space of Schwartz functions, i.e., smooth functions that decay faster than any polynomial for large $|x|$.

Note that the string operators are only ambiguous up to a phase, which does not effect the excitations that they create.

Since the anyon theory is Abelian, the braiding relations $B(a,b)$ of $a$ and $b$ can be determined from the exchange statistics by $B(a,b)=\theta (ab)/\theta (a)\theta (b)$.

This can be computed explicitly by making use of the translation invariance and applying a Fourier transform.

For concreteness, we think of the ball as being defined with respect to the Euclidean metric on ${\mathbb{R}}^D$.

Formally, we can identify that the logical algebra—the commutant of the stabilizer algebra—decomposes into independent subfactors, $$\mathcal{C}(\mathcal{S})\simeq \mathcal{S}\otimes \mathcal{L}\text{with}\mathcal{L}=\underset{i=1}{\overset{N_d}{⨂}}{\mathcal{M}}_{d_i}\underset{j=1}{\overset{N_r}{⨂}}\mathbb{L}(L^2(\mathrm{U}(1)))\underset{k=1}{\overset{N_c}{⨂}}\mathbb{L}(L^2(\mathbb{R})),$$(33)where $\mathcal{L}$ is the algebra that acts transitively on the codespace, ${\mathcal{M}}_{d_i}$ denotes the algebra of $d_i\times d_i$ complex matrices, acting irreducibly on ${\mathbb{C}}^{d_i}$, and $\mathbb{L}(\mathcal{H})$ is the algebra of (bounded) linear operators on $\mathcal{H}$. We say that a code defined by $\mathcal{S}$ encodes $N_d$ qudits with dimensions $\{d_i{\}}_i$, $N_r$ rotors, and $N_c$ CVs.

Note that although the bosons are labeled by $\mathbb{R}\times \mathbb{Z}$ their fusion is not described by that group. Two bosons can fuse to an excitation that is not a boson.

That is, the stabilizers of the $\mathbb{R}$ model that commute exactly with each element in ${\mathcal{B}}_{2\pi }$.

Here, we take $e\in \gamma$ to mean that the cocycle $\gamma$ evaluated on $e$ is nonzero.

Since condensing $\mathcal{B}$ can be interpreted as a two-step process where $\mathcal{B}$ is condensed first, the final logical operators are a subset of the logical operators from the first condensation.

Concretely, given a constant $s\in \mathbb{R}$ we can rescale $(\phi ,c)\mapsto (\phi s,c/s)$ without changing the data of the topological excitations. In the lattice model this symmetry transformation can be implemented via an on-site squeezing operation.

There are choices of $n_2,n^{\prime }$ such that $(2\pi n_2/c_2,c_2)\in ⟨(2\pi ,n_1)⟩$ or vice versa. In that case, $\mathcal{B}$ would only need a single generator and hence correspond to a single condensation.

We use the direct sum symbol loosely here, as the index set is not discrete and not compact. For our purposes, however, it makes sense as elements in the algebra that correspond to bounded operators in $\mathbb{L}(\mathcal{H})$ can be expressed as integrals over displacement operators with coefficients that obey a physicality condition, i.e., they fall off sufficiently quickly for large $(\phi ,a)$. Note that we have not defined a norm on the above algebra in this work.

In fact, it is the symplectic form on the vector space spanned by the displacement operators modulo phases, in the basis defined by $W_e$.

Bilinearity can be checked straight forwardly, $$\begin{gathered} b[(\phi ,c)+({\phi }^{\prime },c^{\prime }),(\vartheta ,b)]=\frac{1}{2}[(\phi +{\phi }^{\prime })b+\vartheta (c+c^{\prime })]=\frac{1}{2}(\phi b+\vartheta c)+\frac{1}{2}({\phi }^{\prime }b+\vartheta c^{\prime }) \\ =b[(\phi ,c),(\vartheta ,b)]+b[({\phi }^{\prime },c^{\prime }),(\vartheta ,b)], \end{gathered}$$which holds similarly in the second argument.

We do not make any assumptions on $\mathbf{c}$ here. This allows unbounded operators and unphysical density matrices in $\mathcal{S}$. However, we can still make formal calculations in that algebra given the structure as described above.

We can write each cocycle class as $\alpha [\gamma ]+\beta [{\gamma }^{\prime }]$ where $[\gamma ]\ne [{\gamma }^{\prime }]$ and $\alpha ,\beta \in {\mathbb{R}}^2$.

Since $H$ is a noncompact group, and can even be continuous, the direct sum symbol should be interpreted loosely. We use it to refer to a formal vector space whose elements can be expressed as group integrals over operators in $V_{(\phi ,c)}^e$, see the expansion in Eq. (a19).

Since the labels $q$ and $\alpha$ enter linearly into the definition of the coset representatives the group operation is preserved.