Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2603.25757

Decoder Dependence in Surface-Code Threshold Estimation with Native Gottesman-Kitaev-Preskill Digitization and Parallelized Sampling

Dennis Delali Kwesi Wayo, Chinonso Onah, Leonardo Goliatt, Sven Groppe

quant-phcs.ITSubmitted Mar 25, 2026• Announced Mar 27, 2026• 0 citations• Revised Mar 31, 2026 (v2)
v1 · Mar 25, 2026v2 · Mar 31, 2026
arXivHTMLPDF

Abstract

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 $σ^\star_{3,5}=0.10$ (1911/2000 valid) and $σ^\star_{5,7}=0.1375$ (1941/2000 valid), while other decoders show no valid crossing samples. Dense-window scanning over $σ\in [0.08,0.24]$ 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 ($Δ_{\mathrm{MWPM-UF}}=-0.00383$, 95\% interval $[-0.0104,0.00329]$) across $σ\in [0.05,0.35]$. Threaded execution preserves statistical fidelity while improving throughput: $1.34\times$ speedup in Pauli mode and $1.94\times$ in native GKP mode, with mean $|Δ\mathrm{LER}|$ $6.07\times10^{-3}$ and $5.20\times10^{-3}$, respectively. We therefore recommend estimator-conditional threshold reporting coupled to runtime-fidelity checks for reproducible hardware-facing practical future decoder benchmarking workflows.

Related papers

Decoder Dependence in Surface-Code Threshold Estimation under Digitized Hybrid Continuous-Variable and Discrete Noise

Dennis Delali Kwesi Wayo, Chinonso Onah, Leonardo Goliatt +1 more

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.

quant-pharXiv:2603.06730•Fortschritte der Physik•Mar 9, 2026•0 citations

Decoder Comparability Across Quantum Software Stacks: Repeated-Round Surface and Digitized-GKP Syndrome Replay

Dennis Delali Kwesi Wayo, Chinonso Onah, Rodrigo Alves Dias +2 more

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: $24\,000$ request lines, $24\,000$ 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 ($\mathrm{BP}<\mathrm{MWPM}<\mathrm{UF}$); 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.

quant-pharXiv:2607.19446•Jul 22, 2026•0 citations

Achieving Thresholds via Standalone Belief Propagation on Surface Codes

Pedro Hack, Luca Menti, Francisco Lazaro +1 more

The standard approach to Belief Propagation (BP) decoding passes messages on the Tanner graph of the quantum error-correcting (QEC) code, and is known to lack a threshold for the surface code. We propose novel BP decoders that exchange messages on the decoding graph and obtain thresholds (both for code capacity and circuit-level noise) via standalone BP for the surface code under depolarizing noise. Our approach, similarly to the minimum weight perfect matching (MWPM) decoder, is applicable to any graphlike QEC code. The thresholds observed with our decoders are close to those obtained by MWPM. This result opens the path towards scalable hardware-accelerated implementations of MWPM-compatible decoders.

quant-pharXiv:2603.05381•Mar 6, 2026•0 citations

Decoder Dependence in Surface-Code Threshold Estimation with Native Gottesman-Kitaev-Preskill Digitization and Parallelized Sampling

Dennis Delali Kwesi Wayo, Chinonso Onah, Leonardo Goliatt +1 more

•Jan 1, 2026•0 citations

Multidimensional Bose quantum error correction based on neural network decoder

Haowen Wang, Yun-Jia Xue, Yingjie Qu +2 more

Abstract Boson quantum error correction is an important means to realize quantum error correction information processing. In this paper, we consider the connection of a single-mode Gottesman-Kitaev-Preskill (GKP) code with a two-dimensional (2D) surface (surface-GKP code) on a triangular quadrilateral lattice. On the one hand, we use a Steane-type scheme with maximum likelihood estimation for surface-GKP code error correction. On the other hand, the minimum-weight perfect matching (MWPM) algorithm is used to decode surface-GKP codes. In the case where only the data GKP qubits are noisy, the threshold reaches σ ≈ 0.5 ( $$\bar{p}\approx 12.3 \%$$ p ¯ ≈ 12.3 % ). If the measurement is also noisy, the threshold is reached σ ≈ 0.25 ( $$\bar{p}\approx 10.02 \%$$ p ¯ ≈ 10.02 % ). More importantly, we introduce a neural network decoder. When the measurements in GKP error correction are noise-free, the threshold reaches σ ≈ 0.78 ( $$\bar{p}\approx 15.12 \%$$ p ¯ ≈ 15.12 % ). The threshold reaches σ ≈ 0.34 ( $$\bar{p}\approx 11.37 \%$$ p ¯ ≈ 11.37 % ) when all measurements are noisy. Through the above optimization method, multi-party quantum error correction will achieve a better guarantee effect in fault-tolerant quantum computing.

npj Quantum Information•Nov 18, 2022•26 citations

Circuit-level benchmarks of GKP-concatenated qLDPC Codes

Yuan Yao, Ruipeng Xing, Jianshuo Gao +1 more

Scalable fault-tolerant quantum error correction based on Gottesman-Kitaev-Preskill (GKP) codes requires finite-rate outer codes that can exploit analog GKP information under realistic circuit noise. However, quantum low-density parity-check (qLDPC) outer codes have not been systematically compared as candidates for circuit-level GKP concatenation. Here, we benchmark BB and tricycle codes using a unified simulation framework that progresses from code-capacity noise to repeated noisy syndrome extraction and schedule-resolved displacement propagation. Across all three noise models, analog-informed belief-propagation with ordered-statistics decoding (BP-OSD) yields higher finite-size crossing estimates than hard-decision decoding. In the full circuit model, the analog-informed crossings are \(σ_{\mathrm{th}}\simeq0.212\) for the selected BB sequence and \(0.142\) for the tricycle sequence, corresponding to squeezing requirements of approximately \(10.46\,\mathrm{dB}\) and \(13.94\,\mathrm{dB}\), respectively. These results show that analog GKP information continues to improve decoding under circuit-level noise. Under the common simulation assumptions used here, the selected BB sequence also yields a higher finite-size crossing estimate than the selected tricycle sequence.

quant-pharXiv:2609.35282•Sep 29, 2026•0 citations
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=5d=5d=5 and σ=0.20σ=0.20σ=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⋆=0.10σ^\star_{3,5}=0.10σ3,5⋆​=0.10 (1911/2000 valid) and σ5,7⋆=0.1375σ^\star_{5,7}=0.1375σ5,7⋆​=0.1375 (1941/2000 valid), while other decoders show no valid crossing samples. Dense-window scanning over σ∈[0.08,0.24]σ\in [0.08,0.24]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=−0.00383Δ_{\mathrm{MWPM-UF}}=-0.00383ΔMWPM−UF​=−0.00383, 95% interval [−0.0104,0.00329][-0.0104,0.00329][−0.0104,0.00329]) across σ∈[0.05,0.35]σ\in [0.05,0.35]σ∈[0.05,0.35]. Threaded execution preserves statistical fidelity while improving throughput: 1.34×1.34\times1.34× speedup in Pauli mode and 1.94×1.94\times1.94× in native GKP mode, with mean ∣ΔLER∣|Δ\mathrm{LER}|∣ΔLER∣ 6.07×10−36.07\times10^{-3}6.07×10−3 and 5.20×10−35.20\times10^{-3}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.
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=0.0531p_c=0.0531pc​=0.0531 (bootstrap interval [0.0415,0.0572][0.0415,0.0572][0.0415,0.0572]) and collapse fit pc=0.052p_c=0.052pc​=0.052 (ν=1.35ν=1.35ν=1.35). For the hybrid mode, a dense transition-window sweep at d=3,5,7d=3,5,7d=3,5,7 uses σ∈[0.30,0.50]σ\in[0.30,0.50]σ∈[0.30,0.50] with step 0.010.010.01 and 300030003000 trials per point. After the initial exact-zero plateau is excluded from crossing localization, the matching-style backend gives interior crossing estimates σc=0.4707σ_c=0.4707σc​=0.4707 for (d=3,5)(d=3,5)(d=3,5) and σc=0.3275σ_c=0.3275σc​=0.3275 for (d=5,7)(d=5,7)(d=5,7); the latter lies in a low-LER region and remains estimator-sensitive. A targeted d=9d=9d=9 extension shows larger Union-Find LER at moderate-to-high σσσ and matching-fallback rates up to 0.7470.7470.747 at σ=0.50σ=0.50σ=0.50. In a d=5d=5d=5 neural-guidance sensitivity sweep, full learned reweighting reduces the sampled mean LER from 0.17730.17730.1773 to 0.16630.16630.1663 over σ∈[0.35,0.55]σ\in[0.35,0.55]σ∈[0.35,0.55]. These results show that estimator resolution and backend fallback diagnostics are part of an auditable decoder comparison.
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: 24 00024\,00024000 request lines, 24 00024\,00024000 response lines, response ratio =1.0=1.0=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\mathrm{BP}<\mathrm{MWPM}<\mathrm{UF}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%48.9\%48.9% in surface and 45.1%45.1\%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.
The standard approach to Belief Propagation (BP) decoding passes messages on the Tanner graph of the quantum error-correcting (QEC) code, and is known to lack a threshold for the surface code. We propose novel BP decoders that exchange messages on the decoding graph and obtain thresholds (both for code capacity and circuit-level noise) via standalone BP for the surface code under depolarizing noise. Our approach, similarly to the minimum weight perfect matching (MWPM) decoder, is applicable to any graphlike QEC code. The thresholds observed with our decoders are close to those obtained by MWPM. This result opens the path towards scalable hardware-accelerated implementations of MWPM-compatible decoders.
Abstract Boson quantum error correction is an important means to realize quantum error correction information processing. In this paper, we consider the connection of a single-mode Gottesman-Kitaev-Preskill (GKP) code with a two-dimensional (2D) surface (surface-GKP code) on a triangular quadrilateral lattice. On the one hand, we use a Steane-type scheme with maximum likelihood estimation for surface-GKP code error correction. On the other hand, the minimum-weight perfect matching (MWPM) algorithm is used to decode surface-GKP codes. In the case where only the data GKP qubits are noisy, the threshold reaches σ ≈ 0.5 ( pˉ≈12.3%\bar{p}\approx 12.3 \%pˉ​≈12.3% p ¯ ≈ 12.3 % ). If the measurement is also noisy, the threshold is reached σ ≈ 0.25 ( pˉ≈10.02%\bar{p}\approx 10.02 \%pˉ​≈10.02% p ¯ ≈ 10.02 % ). More importantly, we introduce a neural network decoder. When the measurements in GKP error correction are noise-free, the threshold reaches σ ≈ 0.78 ( pˉ≈15.12%\bar{p}\approx 15.12 \%pˉ​≈15.12% p ¯ ≈ 15.12 % ). The threshold reaches σ ≈ 0.34 ( pˉ≈11.37%\bar{p}\approx 11.37 \%pˉ​≈11.37% p ¯ ≈ 11.37 % ) when all measurements are noisy. Through the above optimization method, multi-party quantum error correction will achieve a better guarantee effect in fault-tolerant quantum computing.
Scalable fault-tolerant quantum error correction based on Gottesman-Kitaev-Preskill (GKP) codes requires finite-rate outer codes that can exploit analog GKP information under realistic circuit noise. However, quantum low-density parity-check (qLDPC) outer codes have not been systematically compared as candidates for circuit-level GKP concatenation. Here, we benchmark BB and tricycle codes using a unified simulation framework that progresses from code-capacity noise to repeated noisy syndrome extraction and schedule-resolved displacement propagation. Across all three noise models, analog-informed belief-propagation with ordered-statistics decoding (BP-OSD) yields higher finite-size crossing estimates than hard-decision decoding. In the full circuit model, the analog-informed crossings are σth≃0.212σ_{\mathrm{th}}\simeq0.212σth​≃0.212 for the selected BB sequence and 0.1420.1420.142 for the tricycle sequence, corresponding to squeezing requirements of approximately 10.46 dB10.46\,\mathrm{dB}10.46dB and 13.94 dB13.94\,\mathrm{dB}13.94dB, respectively. These results show that analog GKP information continues to improve decoding under circuit-level noise. Under the common simulation assumptions used here, the selected BB sequence also yields a higher finite-size crossing estimate than the selected tricycle sequence.