Distance Prediction for Quantum Codes

Nadav Schneider; Eliya Nachmani

quant-phOct 13, 2026

Distance Prediction for Quantum Codes

Authors: Nadav Schneider, Eliya Nachmani

Organizations: School of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Israel

Abstract

The distance of a quantum code is a fundamental metric that dictates its error-correction capability. While quantum low-density parity-check (qLDPC) families such as Bivariate Bicycle (BB) codes offer a promising pathway to low-overhead fault-tolerant quantum computing, computing their distance is challenging. Current exact solvers and even search heuristics that provide a distance upper bound suffer from severe computational bottlenecks, restricting the discovery of optimal codes. In this paper, we construct a dataset of BB codes with corresponding distance upper bounds obtained through high-budget heuristic searches. We introduce BB-Net, an attention-based neural network architecture for predicting the distance of BB codes. It achieves 96.39% accuracy on the reference distances in the test set, compared with 94.86% for state-of-the-art heuristics with a budget of up to $10^{4}$ times more computational runtime. On a subset with certified exact distances, BB-Net achieves 98.94%. By relieving the distance-calculation bottleneck, BB-Net enables faster screening of large code spaces, particularly in ML-driven search pipelines, and thereby can accelerate the discovery of novel quantum error-correcting codes, an important capability for advancing scalable fault-tolerant quantum computing. Our code is publicly available: https://github.com/NadavSc/BBNet

Figures & tables

Figure 1: Distance estimation for bivariate bicycle codes through two approaches: computationally intensive heuristic search and orders of magnitude more efficient inference using BB-Net. Numbers refer to a full benchmark evaluation.

Figure 2: BB-Net architecture . A BB code is processed by two encoders; The polynomial encoder applies self-attention to monomial tokens and is conditioned on the code parameters through FiLM layers. In parallel, the Tanner spectral encoder processes Fourier-character tokens using attention as well. Each branch produces a code latent representation, which is combined through a learned gated residual connection. The fused representation then passed to a classification head.

Figure 3: BB-Net tokenization and embedding. A BB code specified by sparse polynomials

A

and

B

is mapped to two complementary representations: (i) polynomial tokens encode monomial supports on

\mathbb{Z}_{\ell}\times\mathbb{Z}_{m}

with the type (A,B) indicator as a one-hot vector and the first eight Fourier harmonics, represented by sine and cosine components along each lattice dimension, yielding a 32-dimensional positional representation. (ii) Tanner spectral tokens encode the responses of the full polynomials at each group character

\chi

using 18-dimensional character features. Learned CLS tokens are used as latent vector aggregators for the downstream classification task.

	QDistRnd	QDistEvol	BP-OSD	GEStim	CCStim	UEStim	BB-Net (ours)
Trials Per Code	200	200	200	1	1	3	1
Time Per Code [s] ( $\downarrow$ )	3.3	1.0	19.1	1.2	1.2	13.1	0.005
Total Time [s] ( $\downarrow$ )	19,383	6,057	112,275	6,940	7,164	76,775	26.8
RMSE ( $\downarrow$ )	2.319	1.485	23.322	4.950	9.309	1.933	1.131
Accuracy ( $\uparrow$ )	92.20%	94.86%	82.23%	0.02%	34.60%	61.23%	96.39%

Table 1: Test benchmark across common heuristic methods (5,870 codes)

Panel of Figure 4 — Figure 4: The $x$ -axis denotes the minimum distance threshold $t$ , and the $y$ -axis reports the accuracy computed over codes whose distance satisfies $d\geq t$ . Thus, moving to the right restricts the evaluation to increasingly high-distance codes. This was examined on three subsets: the whole benchmark, certified exact distances, and upper bound, unresolved distances.

	QDistRnd	QDistEvol	BP-OSD	GEStim	CCStim	UEStim	BB-Net (ours)
Trials Per Code	200	200	200	1	1	3	1
Time Per Code [s] ( $\downarrow$ )	5.0	1.6	3.8	1.0	1.0	2.1	0.005
Total Time [s] ( $\downarrow$ )	11,887	3,883	9,032	2,406	2,407	4,898	10.8
RMSE ( $\downarrow$ )	0.000	0.000	0.699	11.453	7.434	2.641	0.594
Accuracy ( $\uparrow$ )	100.00%	100.00%	99.96%	0.04%	64.03%	98.44%	98.94%

Table 2: Certified distance codes,

d\leq 14

(2,366 codes)

	QDistRnd	QDistEvol	BP-OSD	GEStim	CCStim	UEStim	BB-Net (ours)
Trials Per Code	200	200	200	1	1	3	1
Time Per Code [s] ( $\downarrow$ )	2.1	0.6	29.5	1.3	1.4	20.5	0.005
Total Time [s] ( $\downarrow$ )	7,496	2,175	103,243	4,535	4,758	71,877	16.0
RMSE ( $\downarrow$ )	3.002	1.923	30.181	42.355	35.362	37.190	1.380
Accuracy ( $\uparrow$ )	86.93%	91.38%	70.26%	0.00%	14.73%	36.10%	94.66%

Table 3: Upper bound distance codes (3,504 codes)

	$d\!\geq\!4$	$d\!\geq\!8$	$d\!\geq\!10$	$d\!\geq\!12$	$d\!\geq\!14$	$d\!\geq\!16$	$d\!\geq\!18$	$d\!\geq\!20$	$d\!\geq\!22$	$d\!\geq\!24$
BB-Net	1.000	0.998	0.996	0.993	0.989	0.989	0.988	0.987	0.990	0.991
QDistEvol	1.000	1.000	1.000	1.000	0.999	0.998	0.998	0.995	0.991	0.992
QDistRnd	1.000	1.000	1.000	1.000	1.000	0.998	0.997	0.993	0.991	0.990
BP-OSD	1.000	1.000	1.000	0.997	0.987	0.967	0.950	0.936	0.931	0.926
UEStim	0.996	0.996	0.997	0.992	0.974	0.944	0.917	0.900	0.890	0.883
GEStim	0.998	0.946	0.913	0.892	0.876	0.871	0.855	0.849	0.845	0.845

Table 4: AUC for distance screening over the test benchmark. Best value in each column in bold, second best underlined (higher is better).

Panel of Figure 5 — Figure 5: Bold curves denote the best configuration, including augmentation, shift canonicalisation, and FiLM conditioning; lighter curves show the corresponding model with the component named in each panel removed.

Component			NLL $\downarrow$
Aug.	Shift-Canon	FiLM	BB-Net	BB-Net-Tanner
$\times$	$\times$	$\checkmark$	$1.496\pm 0.014$	$1.039\pm 0.019$
$\checkmark$	$\times$	$\checkmark$	$1.436\pm 0.014$	$0.762\pm 0.012$
$\times$	$\checkmark$	$\checkmark$	$1.409\pm 0.013$	$0.924\pm 0.026$
$\checkmark$	$\checkmark$	$\times$	$1.217\pm 0.006$	$0.712\pm 0.025$
$\checkmark$	$\checkmark$	$\checkmark$	$1.157\pm 0.001$	$\mathbf{0.683\pm 0.010}$

Table 5: Ablation study of BB-Net on the validation set. Best-epoch NLL is reported.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Params	Values
$L$	8
$d_{\mathrm{model}}$	256
lr	3e-4
Dropout	0.1
Epochs	90
Warmup Epochs	9

Appendix

Table 6: Hyperparameters used for training.

$(N,\ell,m)$	$A(x,y)$	$B(x,y)$	$C(x,y)$	$D(x,y)$	GT	BP-OSD	QDistEvol	QDistRnd	BB-Net (ours)
$(324,18,9)$	$x^{2}y^{2}+x^{9}y+x^{9}y^{8}+x^{13}y$	$x^{4}y^{8}+x^{8}y+x^{12}y^{4}+x^{16}y$	–	–	18	44	45	18	18
$(270,27,5)$	$y^{2}+x^{5}+x^{16}y+x^{26}y^{3}$	$x^{8}y^{3}+x^{10}y^{4}+x^{24}y^{3}+x^{26}y^{2}$	–	–	25	35	38	37	25
$(342,19,9)$	$x^{8}y^{2}+x^{16}y+x^{16}y^{3}$	$y^{5}+x^{2}y^{7}+x^{12}y^{3}$	$x^{12}y^{5}+x^{12}y^{7}$	$x^{6}y^{3}+x^{6}y^{5}$	24	Fail	36	26	24
$(348,29,6)$	$x^{8}y^{2}+x^{13}y+x^{25}y^{3}$	$x^{2}y^{5}+x^{12}y^{4}+x^{16}y^{3}$	$x^{11}y^{4}+x^{22}y$	$x^{18}+x^{25}y^{3}$	24	Fail	36	44	24
$(272,17,8)$	$xy^{2}+x^{4}y^{2}+x^{8}y^{3}+x^{9}y^{7}$	$y^{4}+x^{7}y^{2}+x^{8}y^{2}+x^{12}y^{2}$	–	–	25	37	36	28	25
$(360,30,6)$	$x^{2}+x^{7}+x^{19}y^{5}$	$x^{5}y^{2}+x^{12}y^{2}+x^{26}y$	–	–	20	30	30	22	20

Appendix

Table 7: Selected test codes on which the heuristics overestimate the reference distance and BB-Net matches it. The reference is the minimum found by the same heuristics at 10,000 trials. GEStim, CCStim and UEStim failed to output a result.

Panel of Figure 6 — Figure 6: Corpus statistics. (a) distribution of the code distance; (b) distribution of block length $n=2\ell m$ ; (c) mean logical dimension $k$ per block length, with $\pm$ one standard deviation. Throughout, the distance of a code without an exact solution is its best available upper bound.

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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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