Intrinsic Heralding and Optimal Decoders for Non-Abelian Topological Order

TL;DR

This paper explores how non-Abelian topological order can intrinsically herald errors and improve quantum error correction thresholds through novel decoding strategies and Bayesian inference.

Contribution

It introduces a new error correction approach leveraging non-Abelian anyon fusion and derives optimal thresholds using Bayesian models, outperforming traditional methods.

Findings

Intrinsic heralding improves error thresholds in non-Abelian topological codes.

Bayesian inference yields the optimal noise threshold conditioned on syndrome measurements.

Numerical results show the proposed decoder surpasses standard MWPM in threshold performance.

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.