Efficient and accurate decoders are essential for quantum error correction. Surface codes are due to their high error threshold and scalability. The Minimum Weight Perfect Matching (MWPM) decoder and Union-Find (UF) decoder are the two most commonly used decoders for quantum surface codes, known for their accuracy and decoding speed, respectively. In this paper, we design an iteratively reweighted UF (IRUF) decoder considering the correlation of bit flip error and phase flip error in the depolarization noise channel. It has higher decoding accuracy than the MWPM decoder and UF decoder, and the decoding threshold is 11.5% and 6.3% higher respectively. Notably the IRUF decoder almost retains the same linear time complexity as the UF decoder, which is far better than the MWPM decoder.
Efficient and accurate decoders are essential for quantum error correction. Surface codes are due to their high error threshold and scalability. The Minimum Weight Perfect Matching (MWPM) decoder and Union-Find (UF) decoder are the two most commonly used decoders for quantum surface codes, known for their accuracy and decoding speed, respectively. In this paper, we design an iteratively reweighted UF (IRUF) decoder considering the correlation of bit flip error and phase flip error in the depolarization noise channel. It has higher decoding accuracy than the MWPM decoder and UF decoder, and the decoding threshold is 11.5% and 6.3% higher respectively. Notably the IRUF decoder almost retains the same linear time complexity as the UF decoder, which is far better than the MWPM decoder.
Efficient and realistic error decoding is crucial for fault-tolerant quantum computation (FTQC) on near-term devices. While decoding is a classical post-processing task, its effectiveness depends on accurately modeling quantum noise, which is hardware-dependent. In particular, correlated bit-flip (X) and phase-flip (Z) errors often arise under circuit-level noise. We introduce the Iterative Reweighting Minimum-Weight Perfect Matching (IRMWPM) decoder, which systematically incorporates such correlations to enhance quantum error correction. Our method leverages fault-detection patterns to guide reweighting: correlated X and Z detection events are identified, and their conditional probabilities update weights on the primal and dual lattices. This iterative procedure improves handling of realistic error propagation in a hardware-agnostic yet noise-aware manner. We prove that IRMWPM converges in finite time while preserving the distance guarantee of MWPM. Numerical results under circuit-level noise show substantial improvements. For distances ≥17 and physical error rates ≤0.001, IRMWPM reduces logical error rates by over 20x with only a few iterations. It also raises the accuracy threshold from 1% to 1.16%, making it practical for near-term real-time decoding. Extrapolated estimates suggest that to reach logical error rate 10−16, IRMWPM requires distance d=31, while standard MWPM needs d=50, implying a major reduction in qubit overhead.
Union-Find (UF) and Minimum-Weight Perfect Matching (MWPM) are popular decoder designs for surface codes. The former has significantly lower time complexity than the latter but is considered somewhat inferior, in terms of decoding accuracy. In this work we present an interpretation of UF decoders that explains why UF and MWPM decoders perform closely in some cases: the UF decoder is an approximate implementation of the blossom algorithm used for MWPM. This interpretation allows a generalization of UF decoders for weighted decoding graphs and explains why UF decoders achieve high accuracy for certain surface codes.
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.
Quantum error correction is needed for future quantum computers. Classical error correcting codes are not suitable for this due to the nature of quantum mechanics. Therefore, new codes need to be developed. A promising candidate is the toric code, a surface code, because of its locality and its high error correcting capability and thresholds (the error probability below which increasing the size of the code decreases the failure rate). This thesis provides an introduction to quantum error correction and the stabilizer formalism, which is used to introduce the toric code. Several decoders are looked at, including the Minimum Weight Perfect Matching (MWPM) decoder and a recently developed decoder, the Union-Find (UF) decoder. The UF decoder is useful because of its almost-linear time complexity, while only reducing the threshold by a marginal amount compared to the MWPM decoder. In this thesis, the time complexity of the weighted growth version of the UF decoder is analysed. The toric code and the decoders have been implemented and simulated, and the thresholds have been determined. For the MWPM decoder, the threshold was determined to be 11.5±0.2%, which is not in agreement with the threshold in literature of 10.3%, and should not be possible according to the optimal threshold of about 11%. This probably is a result of the low grid sizes (up to a gridsize of 11) of the code used due to the time needed to simulate the MWPM decoder. For the UF decoder, the threshold found was 9.7 ± 0.9%, which is in agreement with the threshold of 9.9% found by N. Delfosse and N. Nickerson. The time complexity of the UF decoder has also been determined, and is indeed almost-linear as expected by the analysis.
Real-time Quantum Error Correction (QEC) for Fault-Tolerant Quantum Computing (FTQC) demands immediate surface code decoding with both high accuracy and low latency. The highly accurate Union-Find (UF) algorithm has traditionally been limited to batch-processing, as its iterative cluster growth is incompatible with the concurrent initiation of decoding with each syndrome measurement round, required for real-time QEC. This research presents a novel online UF decoder microarchitecture designed to overcome these limitations. Our key innovation decomposes UF cluster growth into incremental, per-cycle steps managed by a dedicated SHIFT stage. ASIC evaluation of our implementation demonstrates a102.38ns latency for a surface code with a distance of 23 and an approximate 1.4% threshold, meeting critical QEC performance targets. By demonstrating a viable online UF decoder with competitive performance, this research offers a crucial building block for the practical realization of scalable FTQC.
Fast decoders that achieve strong error suppression are essential for fault-tolerant quantum computation (FTQC) from both practical and theoretical perspectives. The union-find (UF) decoder for the surface code is widely regarded as a promising candidate, offering almost-linear time complexity and favorable empirical error suppression supported by numerical evidence. However, the lack of a rigorous threshold theorem has left open whether the UF decoder can achieve fault tolerance beyond the error models and parameter regimes tested in numerical simulations. Here, we provide a rigorous proof of a finite threshold for the UF decoder on the surface code under the circuit-level local stochastic error model. To this end, we develop a refined error-clustering framework that extends techniques previously used to analyze cellular-automaton and renormalization-group decoders, by showing that error clusters can be separated by substantially larger buffers, thereby enabling analytical control over the behavior of the UF decoder. Using this guarantee, we further prove a quasi-polylogarithmic upper bound on the average runtime of a parallel UF decoder in terms of the code size. We also show that this framework yields a finite threshold for the greedy decoder, a simpler decoder with lower complexity but weaker empirical error suppression. These results provide a solid theoretical foundation for the practical use of UF-based decoders in the development of fault-tolerant quantum computers, while offering a unified framework for studying fault tolerance across these practical decoders.