Surface-code threshold estimates depend on the inference pipeline, including decoder and estimator choices. We compare decoders within a single LiDMaS+ workflow under Pauli-reference and digitized hybrid continuous-variable/discrete sweeps. In the Pauli-reference mode, the matching-style backend outperforms Union-Find and yields crossing median $p_c=0.0531$ (bootstrap interval $[0.0415,0.0572]$) and collapse fit $p_c=0.052$ ($ν=1.35$). For the hybrid mode, a dense transition-window sweep at $d=3,5,7$ uses $σ\in[0.30,0.50]$ with step $0.01$ and $3000$ trials per point. After the initial exact-zero plateau is excluded from crossing localization, the matching-style backend gives interior crossing estimates $σ_c=0.4707$ for $(d=3,5)$ and $σ_c=0.3275$ for $(d=5,7)$; the latter lies in a low-LER region and remains estimator-sensitive. A targeted $d=9$ extension shows larger Union-Find LER at moderate-to-high $σ$ and matching-fallback rates up to $0.747$ at $σ=0.50$. In a $d=5$ neural-guidance sensitivity sweep, full learned reweighting reduces the sampled mean LER from $0.1773$ to $0.1663$ over $σ\in[0.35,0.55]$. These results show that estimator resolution and backend fallback diagnostics are part of an auditable decoder comparison.
Surface-code threshold estimates depend on the inference pipeline, including decoder and estimator choices. We compare decoders within a single LiDMaS+ workflow under Pauli-reference and digitized hybrid continuous-variable/discrete sweeps. In the Pauli-reference mode, the matching-style backend outperforms Union-Find and yields crossing median pc=0.0531 (bootstrap interval [0.0415,0.0572]) and collapse fit pc=0.052 (ν=1.35). For the hybrid mode, a dense transition-window sweep at d=3,5,7 uses σ∈[0.30,0.50] with step 0.01 and 3000 trials per point. After the initial exact-zero plateau is excluded from crossing localization, the matching-style backend gives interior crossing estimates σc=0.4707 for (d=3,5) and σc=0.3275 for (d=5,7); the latter lies in a low-LER region and remains estimator-sensitive. A targeted d=9 extension shows larger Union-Find LER at moderate-to-high σ and matching-fallback rates up to 0.747 at σ=0.50. In a d=5 neural-guidance sensitivity sweep, full learned reweighting reduces the sampled mean LER from 0.1773 to 0.1663 over σ∈[0.35,0.55]. These results show that estimator resolution and backend fallback diagnostics are part of an auditable decoder comparison.
We quantify decoder dependence in surface-code threshold studies under two matched regimes: Pauli noise and native GKP-style Gaussian displacement digitization. Using LiDMaS+ v1.1.0, we benchmark MWPM, Union-Find (UF), Belief Propagation (BP), and neural-guided MWPM with fixed seeds, identical sweep grids, and unified reporting across runs 06--14. At d=5 and σ=0.20, MWPM and UF define the Pareto frontier, with (runtime, LER) = (1.341 s, 0.2273) and (1.332 s, 0.2303); neural-guided MWPM is slower and less accurate (1.396 s, 0.3730), and BP is dominated (7.640 s, 0.6107). Crossing-bootstrap diagnostics are stable only for MWPM, with median σ3,5⋆=0.10 (1911/2000 valid) and σ5,7⋆=0.1375 (1941/2000 valid), while other decoders show no valid crossing samples. Dense-window scanning over σ∈[0.08,0.24] returns NaN crossings for all decoders, confirming estimator- and window-sensitive threshold localization. Rank-stability and effect-size bootstrap analyses reinforce ordering robustness: BP remains rank 4, neural-guided MWPM rank 3, and MWPM-UF differences are small (ΔMWPM−UF=−0.00383, 95\% interval [−0.0104,0.00329]) across σ∈[0.05,0.35]. Threaded execution preserves statistical fidelity while improving throughput: 1.34× speedup in Pauli mode and 1.94× in native GKP mode, with mean ∣ΔLER∣6.07×10−3 and 5.20×10−3, respectively. We therefore recommend estimator-conditional threshold reporting coupled to runtime-fidelity checks for reproducible hardware-facing practical future decoder benchmarking workflows.
We present a contract-preserving, family-aware comparison of decoder behavior across four syndrome-generation stacks (PennyLane, Qiskit, Cirq, and a LiDMaS+ reference) under a fixed replay interface. Request streams from repeated-round surface-code and digitized-GKP circuits are replayed through BP, MWPM, and UF with matched controls. The unified matrix spans 24 cells and achieves line-level integrity: 24000 request lines, 24000 response lines, response ratio =1.0 in every cell, zero parse failures, and zero decoder-name mismatches. Fifteen warning-no-syndrome events occur only in GKP-Cirq rows. Within-family ordering is stable in both families (BP<MWPM<UF); source-averaged flip counts are 2.435, 4.768, and 5.870 for surface and 1.595, 2.908, and 3.681 for GKP. Relative to MWPM, BP reduces mean intervention volume by 48.9% in surface and 45.1% in GKP. Source-vs-reference effects are family dependent, with larger coherent shifts in GKP. Source-bootstrap ranks remain unchanged, and hidden-truth sidecars add an outer-code logical-parity error-rate check in which UF has the largest source-mean rate in both families. The comparison is stack-aware, contract-verified, and avoids raw cross-family threshold-equivalence claims.
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.
We present an architecture-level hardware-to-logical-to-decoder execution stack for hybrid continuous-variable and discrete-variable quantum error correction in LiDMaS+. Provider-native records are normalized into a single decoder IO contract and replayed under fixed controls across MWPM, UF, BP, and neural-MWPM. In a Xanadu case study using fixture inputs and sampled public datasets, replay integrity was complete: 108/108 fixture and 4000/4000 real-slice request-response lines, with zero request-parse errors, zero response-parse errors, and zero decoder-name mismatches. Under matched inputs, decoder behavior is clearly regime-dependent. For weighted fixture summaries, average flip count was 1.296 (MWPM), 1.296 (UF), 0.667 (BP), and 1.296 (neural-MWPM). For weighted real-data summaries, average flip count was 0.641 (MWPM), 0.741 (UF), 0.318 (BP), and 0.641 (neural-MWPM); corresponding nonempty-flip rates were 0.490, 0.490, 0.318, and 0.490. Across fixture data, BP reduced weighted correction volume by 48.6\% versus MWPM; across real slices, BP reduced weighted correction volume by 50.4\% versus MWPM and 57.1\% versus UF. Quality controls show the central interpretability tradeoff: BP is intervention-conservative but leaves higher residual burden, while MWPM-family decoders intervene more aggressively and clear more syndrome. Warning-no-syndrome rates remained decoder-invariant and dataset-driven (fixture weighted 0.259; real weighted 0.510), confirming preserved sparsity semantics from hardware input to logical correction. Re-running analysis stages reproduced identical SHA-256 artifacts, enabling deterministic study iteration. These results establish a practical benchmarking foundation for photonic GKP-oriented hardware programs where decoder policy must be selected as a function of operating regime.
The typical model for measurement noise in quantum error correction is to randomly flip the binary measurement outcome. In experiments, measurements yield much richer information - e.g., continuous current values, discrete photon counts - which is then mapped into binary outcomes by discarding some of this information. In this work, we consider methods to incorporate all of this richer information, typically called soft information, into the decoding of quantum error correction codes, and in particular the surface code. We describe how to modify both the Minimum Weight Perfect Matching and Union-Find decoders to leverage soft information, and demonstrate these soft decoders outperform the standard (hard) decoders that can only access the binary measurement outcomes. Moreover, we observe that the soft decoder achieves a threshold 25\% higher than any hard decoder for phenomenological noise with Gaussian soft measurement outcomes. We also introduce a soft measurement error model with amplitude damping, in which measurement time leads to a trade-off between measurement resolution and additional disturbance of the qubits. Under this model we observe that the performance of the surface code is very sensitive to the choice of the measurement time - for a distance-19 surface code, a five-fold increase in measurement time can lead to a thousand-fold increase in logical error rate. Moreover, the measurement time that minimizes the physical error rate is distinct from the one that minimizes the logical performance, pointing to the benefits of jointly optimizing the physical and quantum error correction layers.
We extend the circuit-level erasure noise model and Wang et al.\ quadratic expansion fitting of Chang et al.\ from planar surface codes to hyperbolic CSS surface codes. Under Chang et al.'s noise models, the {8,3} Bolza fine-grained family threshold reaches 3.6% under the general-Pauli models (which coincide at temporal resolution η=1) and 4.7% under the tailored spatially perfect model at Re=1 (pure erasure). The corresponding erasure-to-Pauli ratios (5.0× and 6.5×) match the surface code values to within 5%. Per-observable crossing-point analysis at Re=1 (Models 1--3) independently yields an erasure-to-Pauli ratio of 5.3×. These results establish that the erasure advantage extends to hyperbolic codes.