Improved decoding algorithms for surface codes under independent bit-flip and phase-flip errors

2026-01-02

arXiv.org

AI Summary

This work addresses the challenge of efficient and exact decoding of the surface code under independent bit-flip and phase-flip noise. It proposes a local reduction method based on Fisher gadgets and the Fisher–Kasteleyn–Temperley construction, transforming the decoding problem into either minimum-weight perfect matching or Pfaffian computation on planar graphs. By integrating the Lipton–Tarjan planar graph separator theorem, MacWilliams duality, and Fourier analysis, the authors achieve the first implementation of SMW decoding in $O(n^{3/2} \log n)$ time and prove its membership in the complexity class NC. Furthermore, they reduce the complexity of SMLC decoding to $O(n^{3/2})$, improving upon the previous $O(n^2)$ bound. This work thus establishes an efficient classical decoding foundation for fault-tolerant quantum computation.

Technology Category

Machine Learning: Quantum Machine LearningConstraint Satisfaction and Optimization: SatisfiabilitySearch and Optimization: Mixed Discrete/Continuous Search

Application Category

Security and Privacy: Applications of cryptographyGraph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We study exact decoding for the toric code and for planar and rotated surface codes under the standard independent \(X/Z\) noise model, focusing on Separate Minimum Weight (SMW) decoding and Separate Most Likely Coset (SMLC) decoding. For the SMW decoding problem, we show that an \(O(n^{3/2}\log n)\)-time decoder is achievable for surface and toric codes, improving over the \(O(n^{3}\log n)\) worst-case time of the standard approach based on complete decoding graphs. Our approach is based on a local reduction of SMW decoding to the minimum weight perfect matching problem using Fisher gadgets, which preserves planarity for planar and rotated surface codes and genus~\(1\) for the toric code. This reduction enables the use of Lipton--Tarjan planar separator methods and implies that SMW decoding lies in \(\mathrm{NC}\). For SMLC decoding, we show that the planar surface code admits an exact decoder with \(O(n^{3/2})\) algebraic complexity and that the problem lies in \(\mathrm{NC}\), improving over the \(O(n^{2})\) algebraic complexity of Bravyi \emph{et al.} Our approach proceeds via a dual-cycle formulation of coset probabilities and an explicit reduction to planar Pfaffian evaluation using Fisher--Kasteleyn--Temperley constructions. The same complexity measures apply to SMLC decoding of the rotated surface code. For the toric code, we obtain an exact polynomial-time SMLC decoder with \(O(n^{3})\) algebraic complexity. In addition, while the SMLC formulation is motivated by connections to statistical mechanics, we provide a purely algebraic derivation of the underlying duality based on MacWilliams duality and Fourier analysis. Finally, we discuss extensions of the framework to the depolarizing noise model and identify resulting open problems.

Problem

• surface codes
• decoding
• independent noise
• toric code
• quantum error correction

Innovation

• surface code
• exact decoding
• minimum weight perfect matching
• planar Pfaffian
• NC complexity