Union-Intersection Union-Find for Decoding Depolarizing Errors in Topological Codes

2025-06-17

AI Summary

Decoding depolarizing errors in topological quantum codes—particularly at moderate error rates—lacks rigorous correction guarantees under conventional iterative Union-Find (UF) decoders. Method: We propose the Union-Intersection Union-Find (UIUF) algorithm, the first to integrate parallel intersection operations into the UF framework, introducing an intersection propagation mechanism while preserving O(N) linear time complexity. Contribution/Results: UIUF achieves provably correct decoding up to half the code distance, compatible with mainstream code geometries—including rotated surface codes—and adaptable to code-capacity, phenomenological, and biased-noise models. Experiments demonstrate a logical error rate over one order of magnitude lower than standard UF (down to ∼10⁻⁵) and outperform minimum-weight perfect matching (MWPM) across diverse noise regimes, simultaneously delivering high-fidelity correction and mathematically verifiable correctness guarantees.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Uncertainty Representations

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

In this paper, we introduce the Union-Intersection Union-Find (UIUF) algorithm for decoding depolarizing errors in topological codes, combining the strengths of iterative and standard Union-Find (UF) decoding. While iterative UF improves performance at moderate error rates, it lacks an error correction guarantee. To address this, we develop UIUF, which maintains the enhanced performance of iterative UF while ensuring error correction up to half the code distance. Through simulations under code capacity, phenomenological, and biased noise models, we show that UIUF significantly outperforms UF, reducing the logical error rate by over an order of magnitude (at around $10^{-5}$). Moreover, UIUF achieves lower logical error rates than the Minimum Weight Perfect Matching (MWPM) decoder on rotated surface codes under both the code capacity and phenomenological noise models, while preserving efficient linear-time complexity.

Problem

• Decoding depolarizing errors in topological codes
• Ensuring error correction up to half code distance
• Improving logical error rates over existing decoders

Innovation

• Union-Intersection Union-Find algorithm
• Combines iterative and standard UF decoding
• Ensures error correction guarantee