The mapping to the two-dimensional RBIM holds only under the assumption of perfect syndrome measurements. If one incorporates imperfect measurements as part of the physical process, then the underlying model becomes the three-dimensional random-plaquette model [55, 56].

More than for $D(S_3)$ TO [48], which counts six anyons, or double Fibonacci (dFib) [75], which has only four

The fact that this TO can be obtained by a single round of measurements is believed to be impossible [77] for smaller TOs such as $D(S_3)$ and dFib TO.

More concretely, it is obtained by gauging a ${\mathbb{Z}}_2\times {\mathbb{Z}}_2\times {\mathbb{Z}}_2$ symmetry protected topological phase on a triangular lattice [78].

Calling certain anyons “charges” and others “fluxes” depends on how one describes it as a gauge theory, which is a matter of choice. Here, we follow the convention of this state as a twisted ${\mathbb{Z}}_2^3$ gauge theory as in Refs. [73, 78, 79, 80, 81].

While the arXiv version of this manuscript includes colored text and formulas that can facilitate the reading, we here include whenever necessary either a subscript or superscript to clarify the color of the corresponding sublattice: red ($R$ or $r$), green ($G$ or $g$), and blue ($B$ or $b$).

The orientation of the string is set by the fact that the initial triangle $t_i$ lives in either a blue or green plaquette.

For completeness, $A_s^B\to X_v^B{\prod }_{⟨v,v^{\prime }⟩}{(CB)}_{v^{\prime },t}$, where we are controlling the action of the three-body $B_t^R$ term on the state of the three neighboring green vertices on the honeycomb lattice—clearly, if $B_t=1$, one can forget about this controlled term.

If we had instead put it in the state, then Eq. (5) would have an additional finite-depth “CCZ” circuit along the bonds of the honeycomb lattice, and, e.g., Eq. (3) would simplify to just mapping $X\to X$.

For loop models with intersections, one has to take care with how to define components; for $N=1$, this issue does not arise.

This implies the trivialization of the linear-depth circuit necessary for the Wilson operators ${\mathcal{X}}_{t_i}^{t_f}$ and ${\mathcal{X}}_{t_i}^{t_f}$ to produce a pair of non-Abelian fluxes in the undeformed $D_4$ state, since they now become Abelian anyons.

In the present work, we do not focus on the details of these beyond-threshold phases, but see Ref. [111] for subtleties that arise.

We note this physical mechanism has also recently been explored from a Hamiltonian perspective in Ref. [114], where it was found to lead to an infinitely robust spin liquid.

Specht’s theorem says that two Hermitian matrices $A$ and $B$ are unitarily equivalent if and only if $\mathrm{tr}(A^n)=\mathrm{tr}(B^n)$ for all $n\in \mathbb{N}$.

Only contractible loops contribute, as otherwise the eigenvalue of the logical $\mathcal{Z}$ operator is toggled when considering the initial state $|D_4⟩$—this condition is relaxed when one instead considers, e.g., the superposition $|D_4⟩+\mathcal{X}|D_4⟩$.

The reason we find a loop model instead of a trivalent graph is that $m_R$ does not appear in the fusion $m_R\times m_R$.

Notice that this high-temperature expansion includes both contractible and noncontractible loop configurations.

Intriguingly, the first equality can be interpreted as a mixed model where blue sites have “face-cubic” spins and green sites have “corner-cubic” spins—it is known that such spins lead to local spin models for $O(N)$ honeycomb loop models [89, 141], but usually this covers only the range of tension $t\le (1/n)$; our mixed model gives a manifestly positive stat-mech model for $t\le (1/\sqrt{n})$.

Interestingly, there is a phase transition involving crystalline symmetry breaking for very large string tension [141, 143], which is far outside the physical regime under consideration here.

Notice that the quantum channel resulting from applying the channel ${\mathcal{E}}^X$ with the error strength $p_R$ $m$ times is a quantum channel with the same form but error strength $p_R^{(m)}=\frac{1}{2}-(\frac{1}{2}-p_R)(1-2p_R{)}^m\in [0,\frac{1}{2}]$.

More precisely, it holds for any “flux-free” configuration where ${\prod }_{⟨i,j⟩\in ⎔}{\eta }_{ij}=1$ for each hexagon, which means we can write ${\eta }_{ij}={\tau }_i{\tau }_j$; by a change of variables ${\sigma }_i\to {\sigma }_i{\tau }_i$, this leads to the same partition function as ${\eta }_{ij}=1$.

If $p_c^{(\infty )}<\frac{1}{2}$, it implies we have entered a phase with algebraic correlations.

Notice the difference with the analysis of the toric code [6, 56], where the paramagnetic phase of the RBIM corresponds to breakdown of the quantum memory.

In particular, the initial $|D_4⟩$ and the state obtained by applying the logical $\mathcal{X}$ string operator [Eq. (2)] around the torus.

We define $\mathrm{\Delta }{F}_{\mathcal{C}}=-\ln [Z_{-}(\eta )/Z(\eta )]$, where $\mathcal{Z}(\eta )\propto P(\eta )$ and ${\mathcal{Z}}_{-}(\eta )$ is the partition function with a line $\mathcal{C}$ of flipped antiferromagnetic bonds.

Notice that $X$ and $Z$ pure wave-function deformations acting on the same lattice do not commute. One can show that the resulting stat-mech model for a given ordering of the deformations corresponds to coupled self-avoiding O(2) and O(1) loop models.

Here, the factor of 2 relates to the degeneracy of domain wall configurations leading to the same loop configuration.

While this is a nontrivial assumption, notice that it correctly reproduces the known BKT critical point of the O(2) loop model.

Our convention for the normalization of the fields connects to that in Ref. [162] by sending $\phi \to 2\phi$. In this convention, the canonical commutator becomes $[\phi (x),{\partial }_y\theta (y)]=i\pi \delta (x-y)$, and the local Hamiltonian density $\mathcal{H}=[1/(2\pi )](K({\partial }_x\theta {)}^2+(1/K)({\partial }_x\phi {)}^2)$.

We thank Sagar Vijay for an insightful discussion on this point.