Evolutionary BP+OSD Decoding for Low-Latency Quantum Error Correction

Hee-Youl Kwak; Seong-Joon Park; Hyunwoo Jung; Jeongseok Ha; Jae-Won Kim

quant-phDec 20, 2025

Evolutionary BP+OSD Decoding for Low-Latency Quantum Error Correction

Authors: Hee-Youl Kwak, Seong-Joon Park, Hyunwoo Jung, Jeongseok Ha, Jae-Won Kim

Organizations: Department of Electrical, Electronic and Computer Engineering, University of Ulsan, Ulsan 44610, South Korea · Department of Electrical Engineering at Pohang University of Science and Technology (POSTECH), Pohang, Gyeongbuk 37673, South Korea · School of Electrical Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, South Korea · Department of Electronic Engineering, Gyeongsang National University, Jinju, South Korea

Abstract

Quantum error correction (QEC) for fault-tolerant quantum computing requires a balanced decoding solution that offers high performance, low complexity, and low latency. However, the de facto standard - belief propagation (BP) combined with ordered statistics decoding (OSD) - suffers from excessive iterations in the BP stage and high complexity in the OSD stage. To address these challenges, we propose an evolutionary BP (EBP) decoder optimized via a differential evolution (DE) algorithm. By leveraging the gradient-free nature of DE, we enable end-to-end optimization of the EBP+OSD structure to maximize overall performance. In addition, a multi-objective selection rule is introduced to suppress frequent OSD activation, significantly reducing complexity overhead. Simulation results on surface codes and quantum low-density parity-check (QLDPC) codes demonstrate that EBP+OSD simultaneously achieves both superior decoding performance and substantially lower complexity compared to conventional BP+OSD, particularly in stringent low-latency regimes.

Figures & tables

Panel of Figure 1 — Fig. 1: (a) Topological lattice of a $[[9,1,3]]$ surface code, where $n=9$ data qubits are positioned at the vertices and $m=8$ stabilizers are associated with the plaquettes. (b) Syndrome extraction circuits for $Z$ -type and $X$ -type stabilizers, where ancilla qubits are measured to obtain syndrome bits. (c) The corresponding Tanner graph representation, where the measured syndrome vector $\mathbf{s}$ is mapped onto CNs for the BP-based decoding.

\overline{\mathcal{W}}_{t}=\mathcal{W}_{t_{1}}+F(\mathcal{W}_{t_{2}}-\mathcal{W}_{t_{3}})

Algorithm 1 DE for the EBP Decoder

Panel of Figure 3 — Fig. 2: (a) DE algorithm for optimizing the weight set $\mathcal{W}$ in the EBP decoder, where the objective directly targets the overall EBP+OSD performance rather than the standalone EBP performance, by incorporating the EBP+OSD decoder into the optimization loop. (b) The proposed weight-sharing method for the $d=3$ surface code, where edges are assigned identical weights based on their colors.

Panel of Figure 4 — Fig. 3: Effect of the proposed weight-sharing method and end-to-end optimization: (a) The number of weights is significantly reduced and remains constant across different $d$ in surface codes. (b) The proposed sharing method of the EBP+OSD decoder (Shared) accelerates DE convergence compared with the full parameterization case (Full) for the surface code with $d=7$ , achieving a lower ${\rm LER}_{+}$ at the final generation and outperforming the baseline decoders BP+OSD and NBP+OSD. When the selection is instead driven by the standalone pre-decoder performance ${\rm LER}_{\rm P}$ ( ${\rm LER}_{\rm P}$ -Select), the overall ${\rm LER}_{+}$ may even degrade.

Decoder	Shared	Multi.	Adap.	${\rm UFR}_{\rm P}$	${\rm FFR}_{\rm P}$	${\rm UFR}_{\rm O}$	${\rm LER}_{\rm+}$
Decoder	Shared	Multi.	Adap.	$10^{-4}$	$10^{-1}$	$10^{-2}$	$10^{-3}$
EBP+OSD	X	X	X	$2.70$	$1.46$	$3.16$	$4.89$
EBP+OSD	O	X	X	$3.50$	$1.26$	$2.96$	$4.08$
EBP+OSD	O	O	X	$4.80$	$0.95$	$4.03$	$4.31$
EBP+OSD	O	O	O	$5.60$	$0.77$	$4.52$	$4.05$
BP+OSD	-	-	-	$1.60$	$3.74$	$1.32$	$5.10$

TABLE I: Performance analysis of EBP+OSD with various optimization configurations against BP+OSD and NBP+OSD

Fig. 4: Empirical probability mass functions (PMFs) of the residual error

\mathcal{E}\hat{\mathcal{E}}

given a flagged failure of the pre-decoders (NBP and EBP) for the

d=7

surface code at

p=0.05

Panel of Figure 7 — Fig. 5: Performance comparison between BP+OSD and EBP+OSD for surface codes (left column) and QLDPC codes (right column) in terms of LER (upper row) and threshold (lower row).

Decoder	Iteration $\overline{\ell}$	Threshold	Pseudo-Threshold
		Surface	Surface	QLDPC
			$d=7$	$n=72$	$n=144$
MWPM [ 45 ]	–	14.7%	11.9%	–	–
FFNN [ 28 ]	–	–	12.4%	–	–
CNN [ 29 ]	–	–	11.9%	–	–
CNN [ 30 ]	–	–	13.2%	–	–

TABLE II: Threshold comparison of various decoders

Panel of Figure 9 — Fig. 6: Comparison of decoding complexity between BP+OSD and EBP+OSD decoders: (a) EBP consistently reduces the average iteration count and ${\rm FFR}_{\rm P}$ for surface and QLDPC codes. (b) EBP+OSD achieves a 34–62% reduction in total complexity across all codes.

Panel of Figure 10 — Fig. 7: Additional benchmark comparisons. (a) LER comparison between MWPM, BP+OSD, and EBP+OSD for surface codes with distances $d=7,9,11$ . (b) LER comparison for the $[[145,5,6]]$ hypergraph-product code and the $[[48,6,8]]$ generalized bicycle code. (c)–(d) LER comparison with relay BP, restart BP, and BP+OSD-CS for (c) the $[[144,12,12]]$ QLDPC code and (d) the $[[85,1,7]]$ planar surface code.

Panel of Figure 11 — Fig. 8: Circuit-level noise results, with LER normalized per syndrome-extraction round for the (a) surface codes and (b) $[[72,12,6]]$ QLDPC code.

Fig. 9: Optimization convergence for the

[[72,12,6]]

QLDPC code. As in the surface code, weight sharing is required for effective DE optimization, and replacing DE with PSO yields comparable performance.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Decoder	$\epsilon$	${\rm UFR}_{\rm P}$	${\rm FFR}_{\rm P}$	${\rm UFR}_{\rm O}$	${\rm LER}_{\rm+}$
Decoder	$\epsilon$	$10^{-4}$	$10^{-1}$	$10^{-2}$	$10^{-4}$
EBP+OSD	0	0.22	2.35	2.69	0.86
EBP+OSD	0.01	0.25	1.85	3.04	0.81
EBP+OSD	0.05	0.37	1.12	4.38	0.86

Appendix

TABLE III: Performance analysis of EBP+OSD with various

\epsilon

values.

Fig. 10: LER of the

[[144,12,12]]

QLDPC code, with error bars indicating

95\%

Wilson binomial confidence intervals.

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Quantum error correction (QEC) is essential for scalable quantum computing. However, it requires classical decoders that are fast and accurate enough to keep pace with quantum hardware. While quantum low-density parity-check codes have recently emerged as a promising route to efficient fault tolerance, current decoding algorithms do not allow one to realize the full potential of these codes in practical settings. Here, we introduce a convolutional neural network decoder that exploits the geometric structure of QEC codes, and use it to probe a novel "waterfall" regime of error suppression, demonstrating that the logical error rates required for large-scale fault-tolerant algorithms are attainable with modest code sizes at current physical error rates, and with latencies within the real-time budgets of several leading hardware platforms. For example, for the

[144, 12, 12]

Gross code, the decoder achieves logical error rates up to

\sim 17

x below existing decoders - reaching logical error rates

\sim 10^{-10}

at physical error

p=0.1\%

- with 3-5 orders of magnitude higher throughput. This decoder also produces well-calibrated confidence estimates that can significantly reduce the time overhead of repeat-until-success protocols. Taken together, these results suggest that the space-time costs associated with fault-tolerant quantum computation may be significantly lower than previously anticipated.

Abstract

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