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 $D_4\cong {\mathbb{Z}}_4⋊{\mathbb{Z}}_2$ 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.

References in corpus (9)

• Non-Abelian Anyons and Topological Quantum Computation
• Fault-tolerant quantum computation with high threshold in two dimensions
• The 3D +-J Ising model at the ferromagnetic transition line
• Strong-disorder paramagnetic-ferromagnetic fixed point in the square-lattice +- J Ising model
• Improved HDRG decoders for qudit and non-Abelian quantum error correction
• On locations and properties of the multicritical point of Gaussian and +/-J Ising spin glasses
• Critical behavior of the random-anisotropy model in the strong-anisotropy limit
• Properties of the multicritical point of +/- J Ising spin glasses on the square lattice
• Flow to Nishimori universality in weakly monitored quantum circuits with qubit loss

Cited by in corpus (1)

• Spectral properties and coding transitions of Haar-random quantum codes