Topological quantum memories need decoders that are both reliable and scalable, but these goals compete: globally informed decoders are accurate near threshold yet expensive, while strictly local rules are fast but can miss long-range structure. Motivated by recent recoverability and mixed-state viewpoints, we make this tradeoff operational for the dephased toric code through a decoder-level local recoverability diagnostic. We compare global MWPM corrections to quasi-local corrections inside a target region and define a matching ratio $R_{\mathrm{match}}$, with mismatch $ε_{\mathrm{match}}=1-R_{\mathrm{match}}$. Across geometry families, $ε_{\mathrm{match}}$ shows strong buffer-controlled suppression and is well organized by a two-geometry scaling form. For scaled families, especially $a=b=d/8$, $R_{\mathrm{match}}$ exhibits a clear crossing and finite-size collapse near $p\!\sim\!0.09$, consistent with a growing recoverability scale near the decoding transition. We use this scaling to formulate an adaptive buffer-selection rule and to identify a distance-scaled initialization for a composite quasi-local RG decoder on the full torus. In the tuned family $a_0=b_0=d/8$, the resulting logical-failure curves show an apparent finite-size crossing at $p\simeq0.09$--$0.10$, below the conventional MWPM threshold scale. We focus on dephasing noise with perfect syndrome measurements to cleanly isolate the underlying behavior.
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14 pages, 8 figures. Extended version of a poster presented at QEC25: Fast Quasi-local Decoders for Topological Codes (https://qec25.yalepages.org/assets/images/QEC25-Posters.pdf)
Topological quantum memories need decoders that are both reliable and scalable, but these goals compete: globally informed decoders are accurate near threshold yet expensive, while strictly local rules are fast but can miss long-range structure. Motivated by recent recoverability and mixed-state viewpoints, we make this tradeoff operational for the dephased toric code through a decoder-level local recoverability diagnostic. We compare global MWPM corrections to quasi-local corrections inside a target region and define a matching ratio Rmatch, with mismatch εmatch=1−Rmatch. Across geometry families, εmatch shows strong buffer-controlled suppression and is well organized by a two-geometry scaling form. For scaled families, especially a=b=d/8, Rmatch exhibits a clear crossing and finite-size collapse near p∼0.09, consistent with a growing recoverability scale near the decoding transition. We use this scaling to formulate an adaptive buffer-selection rule and to identify a distance-scaled initialization for a composite quasi-local RG decoder on the full torus. In the tuned family a0=b0=d/8, the resulting logical-failure curves show an apparent finite-size crossing at p≃0.09--0.10, below the conventional MWPM threshold scale. We focus on dephasing noise with perfect syndrome measurements to cleanly isolate the underlying behavior.
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.
Topological codes have many desirable properties that allow fault-tolerant quantum computation with relatively low overhead. A core challenge for these codes, however, is to achieve a low-overhead universal gate set with limited connectivity. In this work, we explore a non-Pauli stabilizer code that can be used to complete a universal gate set on topological toric and surface codes in strictly two dimensions. Fault-tolerant syndrome extraction for the non-Pauli code requires mid-circuit X corrections, a key difference to conventional Pauli codes. We construct and benchmark a just-in-time (JIT) matching decoder to reliably decide these corrections. Under a phenomenological error model with equally likely physical and measurement errors, we find a high threshold of ≈2.5%, close to the ≈2.9% of a decoder with access to the full syndrome history. We also perform a finite-size scaling analysis to estimate how the logical error rate scales below threshold and verify an exponential suppression in both physical error rate and in the system size. A second global decoding step for Z errors is required and the non-Clifford gates in the circuit reduce the threshold from ≈2.9% to ≈1.8% with a naive decoder. We show how Z decoding can be improved using knowledge of the X corrections, pushing the threshold to ≈2.2%. Our results suggest non-Clifford logic in 2D codes could perform comparably to 2D quantum memory. Our formalism for efficient benchmarking and decoding directly generalizes to a broader family of CSS codes whose X stabilizers are twisted by diagonal Clifford operators, and spacetime versions thereof, defined by CSS-like circuits enriched by CCZ, CS, and T gates.
Realizing the full potential of quantum computation requires Quantum Error Correction (QEC). QEC reduces error rates by encoding logical information across redundant physical qubits, enabling errors to be detected and corrected. A common decoder used for this task is Minimum Weight Perfect Matching (MWPM) a graph-based algorithm that relies on edge weights to identify the most likely error chains. In this work, we propose a data-driven decoder named Neural Minimum Weight Perfect Matching (NMWPM). Our decoder utilizes a hybrid architecture that integrates Graph Neural Networks (GNNs) to extract local syndrome features and Transformers to capture long-range global dependencies, which are then used to predict dynamic edge weights for the MWPM decoder. To facilitate training through the non-differentiable MWPM algorithm, we formulate a novel proxy loss function that enables end-to-end optimization. Our findings on the toric code under depolarizing noise demonstrate thresholds of 17.9% and 10.95%, nearing the 18.9% and 11.0% maximum likelihood bounds, highlighting the advantage of hybrid decoders that combine the predictive capabilities of neural networks with the algorithmic structure of classical matching.
We measure the depolarizing-channel threshold of the 4D cubic toric code on the 4-torus T^4 under the BP + OSD-CS-7 decoder (ldpc 2.3.9, min-sum beta=0.625, 50 iterations, parallel schedule), at three sizes: [[768, 6, 8]] (sides 2,4,4,4), [[1152, 6, 12]] (sides 3,4,4,4), and [[1536, 6, 16]] (sides 4,4,4,4). For each (d, p) with p in {0.005, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08}, we sample 10,000 independent X-error vectors on the face qubits, decode the Z-syndrome with BP + OSD-CS-7, and classify the residual. Total: 270,000 trials. Across all three sizes and all nine rates the decoder cleared every syndrome (zero uncleared in 270,000 trials). Standard finite-size-scaling extraction places the BP + OSD threshold at p_th ≈ 0.10, 95% CI [0.09, 0.13] (consensus across four independent estimators: deep-below-threshold power-law fit, quadratic FSS restricted to the universal regime, quadratic FSS on all data, and three pairwise crossing-point extrapolations). This is roughly an order of magnitude above the surface-code MWPM threshold (~0.01) and is achieved without single-shot, repeated measurement, or correlated-decoder structure beyond what the off-the-shelf ldpc package provides. The empirical logical-failure curves cleanly separate by code distance up to p ≈ 0.06, with the larger codes suppressing failures by 50–500× at the matched physical rate. Mean correction weights match expected error weights to < 1% at every (d, p), confirming BP + OSD is finding genuine minimum-weight corrections rather than gaming the classifier. This is the stochastic-noise companion to the structured-burst result of DOI 10.5281/zenodo.20301574 on the same code family, which together establish that the 4D cubic toric family on T^4 is well below threshold across the entire near-term physical error-rate regime under a standard decoder. The bundle contains the source (torus_code_4d.py, burst_recheck_v2.py, noise_sweep_v2.py, noise_sweep_v3_high_p.py, fit_threshold_final.py), all per-rate raw counts as JSON (noise_sweep_v2_*.json and noise_sweep_v3_*.json), per-sweep stdout logs, the fit summary (threshold_fit_final.json), the threshold figure (threshold_curve_final.png), and the paper (THRESHOLD_PAPER.md). Reproducible in about 11 minutes on a single x86 VPS core.
The central prerequisite of any fault-tolerant quantum architecture is a quantum memory: a block of encoded physical qubits whose logical state is actively preserved against noise across many rounds of error correction. In neutral-atom Rydberg arrays, realizing such a memory is obstructed not by the entangling gates themselves, which are already fast and high-fidelity, but by the auxiliary operations that a conventional error-correction cycle requires: mid-circuit fluorescence measurement, inter-zone atom transport, and locally focused single-qubit addressing. Each of these introduces latency, atom loss, or optical crosstalk that exceeds the cost of the underlying gates by orders of magnitude. These costs accumulate cycle after cycle, progressively degrading the very logical information the code is meant to protect. Here we propose a protocol that stabilizes a toric-code quantum memory without moving, measuring or local addressing atoms. The key is to use a three-species Rydberg atom array for the complete stabilizer cycle, including syndrome extraction, coherent correction, and ancilla reset, under global, species-selective laser pulses. Numerical simulation of a 4×4 rotated toric code shows a longer qubit lifetime when the physical error rate is below a pseudo-threshold p⋆≈0.034. The scheme offers a concrete, hardware-efficient route to topological quantum memory in neutral-atom platforms.
Local decoders provide a promising approach to real-time quantum error-correction by replacing centralized classical decoding, with significant hardware constraints, by a fully distributed architecture based on a simple, local update rule. We propose a new local decoder for Kitaev's toric code: the 2D signal-rule, that interprets odd parity stabilizer measurements as defects, attracted to each other via the exchange of binary signals. We present numerical evidence of exponential suppression of the logical error rate with system size below a threshold, under a phenomenological noise model with data and measurement errors at each iteration. The construction achieves a significantly improved threshold and optimal finite-size scaling relative to hierarchical schemes. It also provides a lightweight alternative to windowed local decoder constructions while maintaining strong performance, thus enabling a streamlined architecture for a two-dimensional local quantum memory.