Intrinsic Heralding and Optimal Decoders for Non-Abelian Topological Order

Abstract

Topological order (TO) provides a natural platform for storing and manipulating quantum information. However, its stability to noise has only been systematically understood for Abelian TOs. In this work, we exploit the non-deterministic fusion of non-Abelian anyons to inform active error correction and design decoders where the fusion products, instead of flag qubits, herald the noise. This intrinsic heralding enhances thresholds over those of Abelian counterparts when noise is dominated by a single non-Abelian anyon type. Furthermore, we use Bayesian inference to obtain a statistical mechanics model for fixed-point non-Abelian TOs with perfect measurements under any noise model, which yields the optimal threshold conditioned on measuring anyon syndromes. We numerically illustrate these results for $D_4 \cong \mathbb Z_4 \rtimes \mathbb Z_2$ TO. In particular, for non-Abelian charge noise and perfect syndrome measurement, we find a conditioned optimal threshold $p_c=0.218(1)$, whereas an intrinsically heralded minimal-weight perfect-matching (MWPM) decoder already gives $p_c=0.20842(2)$, outperforming standard MWPM with $p_c = 0.15860(1)$. Our work highlights how non-Abelian properties can enhance stability, rather than reduce it, and discusses potential generalizations for achieving fault tolerance.

Figures (6)

• Figure 1: Intrinsic heralding from non-Abelian anyons. A constant-depth error string not only creates a pair of non-Abelian anyons at its endpoints, but also leaves behind a superposition over possible fusion outcomes along its path. Intermediate anyon syndromes can be extracted by collapsing this superposition, providing information about the original error path. This additional information can improve error correction, particularly when the noise model is biased toward the non-Abelian anyon of interest. Such intrinsic heralding arises from the non-deterministic fusion of non-Abelian anyons, without the need for flag qubits.
• Figure 2: Error correction thresholds for $D_4$ TO with charge noise. The phase diagram of the stat-mech models to which we map the error correction problems of the $D_4$ TO in the presence of non-Abelian charge noise, with example snapshots. The black phase boundary corresponds to the random-bond Ising model on the triangular lattice, associated with the unheralded matching decoder which only sees the non-Abelian charge anyon (red dot in snapshots); the black square and star mark the thresholds of the MWPM and maximum-likelihood decoders (Eq. \ref{['TC']}), respectively. The unheralded matching decoder considers only $P(E) \propto t^{|E|}$ with $t = e^{-2\beta}$. The yellow symbol is the intrinsically heralded MWPM decoder, where the error string is forced to pass through the Abelian charge fusion products (blue and green dots in snapshots), with threshold $p_c=0.20842(2)$. The orange phase boundary corresponds to the threshold using the full set of syndromes, where each snapshot is weighted by $P(\bm s|E) P(E)$ (Eq. \ref{['Bayes']}). Here, $P(\bm s |E)$ (Eq. \ref{['Probability']}) includes a factor of $\frac{1}{2}$ for each vertex of the error string not passing through a non-Abelian anyon, and a factor of $2$ for each Abelian parity constraint, as illustrated by the last two error strings above the phase diagram. The optimal decoder is marked by the orange star along the Nishimori line (dashed), $t = \frac{p}{1-p}$, with $p_c = 0.218(1)$. Details on the stat-mech models, the calculation of $P(\bm{s}|E)$, and the numerical simulations can be found in the Supplemental Material SM.
• Figure 3: Stability of heralding. Error correction threshold of non-Abelian charges in the $D_4$ TO as a function of the intermediate Abelian charge pair-creation rate $p_I$. The yellow solid line shows thresholds from a naïve application of the intrinsically heralded MWPM decoder for non-Abelian charges, while the red solid line reflects thresholds after incorporating an algorithm that identifies isolated Abelian charge pairs. The black dashed line indicates the unheralded MWPM threshold. The advantage from intrinsic heralding persists up to $p_I \approx 0.5\%$ when the decoder is applied naïvely, and is further improved by the algorithm.
• Figure 4: Ising strings emerging along $E^{{\color{red} R }}_X$ errors and ungauging. The conjugate action of $\prod_{i\in E^{{\color{red} R }}_X}X_i$ on star operators $A^c_s$ with $c={\color{DarkGreen} G}, {\color{blue} B}$ (as obtained in Eq. \ref{['neighbor_dressing']}) dresses $A^c_s$ with a product of two $Z$'s. The ungauging then maps these two $\tilde{Z}$'s lying along the error string $E_X^{{\color{red} R }}$.
• Figure 5: $O(4)$ loop model from absence of syndrome fluxes.