Intrinsic Heralding and Optimal Decoders for Non-Abelian Topological Order

2025/07/31 by Jing, Dian, Sala, Pablo, Jiang, Liang +1 · 4 citations
#FOS: Physical sciences #Quantum Physics (quant-ph) #Statistical Mechanics (cond-mat.stat-mech) #Strongly Correlated Electrons (cond-mat.str-el)

paper · doi:10.48550/arxiv.2507.23765

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 D₄ ≅ \mathbb Z₄ \rtimes \mathbb Z₂ 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.

Citations

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• PyMatching: A Python package for decoding quantum codes with minimum-weight perfect matching
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Cited by

• Universal Gates from Braiding and Fusing Anyons on Quantum Hardware
• Spacetime Spins: Statistical mechanics for error correction with stabilizer circuits
• Fractional quantum Hall states under density decoherence
• Spectral properties and coding transitions of Haar-random quantum codes

Related

• Stability and Loop Models from Decohering Non-Abelian Topological Order
• Decoherence and Wave-Function Deformation of D₄ Non-Abelian Topological Order
• Mixed-state topological order and error-correction thresholds in non-Abelian codes: rigorous results
• Non-Abelian anyons and topological quantum computation
• A Universal Circuit Set Using the S₃ Quantum Double