LiftDiff: Guided Discrete Diffusion for Quantum Code Construction

Yotam Peled; Eliya Nachmani

quant-phOct 18, 2026

LiftDiff: Guided Discrete Diffusion for Quantum Code Construction

Authors: Yotam Peled, Eliya Nachmani

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

Abstract

The realization of fault-tolerant quantum computation relies on Quantum Error-Correcting Codes (QECC). Of particular interest are codes that combine a high encoding rate with sufficient distance. Existing algebraic constructions, such as lifted product (LP) codes, build a large quantum low-density parity-check (qLDPC) code from a classical base matrix. In this work we introduce LiftDiff, a generative approach to the discovery of QECC. A discrete denoising diffusion model is trained to generate base matrices over the group algebra $\mathbb{F}_2[\mathbb{Z}_\ell]$ , which are then expanded into LP codes. The reverse diffusion process is guided by an energy function that evaluates quantum code properties, including girth, degree, sparsity and decoding performance, steering the sampler toward matrices that yield LP codes with favorable parameters. Our method discovers codes with parameters such as $[[169,13,11]]$ , $[[143,15,9]]$ , and $[[260,24,10]]$ , including a $[[247,19,14]]$ code that achieves the highest ${kd^2}/{n}$ among the baselines. Moreover, several discovered codes outperform the Gross code across a broad range of the tested physical error rates under both code-capacity and circuit-level noise.

Figures & tables

Figure 1: Overview of the inference process. The matrix

M_{t}

and timestep

t

are input to the model, producing logits

z_{t}

. The logits are passed through a sigmoid function and fed into the guidance mechanism, whose output adjusts the prediction to obtain

z_{t}^{g}

. The guided logits are passed through a second sigmoid function, and the resulting output is used to produce

M_{t-1}

#	$[[n,k,d]]$	$k/n$	$w_{\mathrm{chk}}$	#	$[[n,k,d]]$	$k/n$	$w_{\mathrm{chk}}$
1	$[[260,20,6]]$	$0.077$	$5$	17	$[[156,20,8]]$	$0.128$	$8$
2	$[[260,20,8]]$	$0.077$	$6$	18	[[247, 19, 14]]	$0.077$	$8$
3	$[[260,20,11]]$	$0.077$	$7$	19	$[[325,45,8]]$	$0.138$	$9$
4	$[[182,14,11]]$	$0.077$	$7$	20	$[[400,80,6]]$	$0.200$	$9$
5	$[[247,19,11]]$	$0.077$	$7$	21	$[[312,24,10]]$	$0.077$	$9$
6	[[169, 13, 11]]	$0.077$	$7$	22	$[[169,13,12]]$	$0.077$	$9$

Table 1: The strongest codes generated by the model.

k/n

is the encoding rate and

w_{\mathrm{chk}}

the maximum check weight, the largest row weight over both

H_{X}

and

H_{Z}

Figure 2: Selected codes from Table 1 against the baselines. Each marker is one code, colored by its

w_{chk}

. Circles are codes from this work, tagged with their row number in Table 1 . Squares are BB codes of Bravyi et al. (2024) and diamonds the CSS codes of Cruz-Benito et al. (2026) . (a) Encoding rate

k/n

against distance

d

. (b) Block length

n

against

kd^{2}/n

. Codes sharing a point exactly are offset by a small fraction of the axis. Baseline key: B1

[[288,12,18]]

, B2

[[144,12,12]]

, B3

[[108,8,10]]

, B4

[[72,12,6]]

; C1

[[360,20,\leq 14]]

, C2

[[360,16,\leq 14]]

, C3

[[288,24,12]]

, C4

[[288,16,12]]

, C5

[[144,8,12]]

, C6

[[288,50,8]]

, C7

[[288,32,6]]

, C8

[[144,24,6]]

Panel of Figure 3 — Figure 3: (a) Code-capacity block LER vs. physical error rate $p$ for four LiftDiff codes and the [[144,12,12]] BB code. Each label’s index identifies the code in Table 1 . (b) Share of codes that differ from their unguided counterpart per denoising step. (c) Weight-distance cells where guidance produces more codes (blue) or fewer (red) than unguided generation. Distances are upper bounds.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Figure 4: Circuit-level LER per logical qubit per round vs. physical error rate

p

for two codes produced by LiftDiff and the

[[144,12,12]]

BB code. Number of rounds equal to each code’s distance. The open marker denotes a point with no observed failures, showing the

95\%

upper bound. The index beside each label identifies the code in Table 1 .

Quantum Error Correction

Quantum error correction detects and corrects qubit noise so quantum information and computations remain reliable.

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Quantum computers could outperform classical machines on important problems, but only if the errors that pervade quantum hardware can be corrected at scale. Quantum low-density parity-check (qLDPC) codes offer a promising route to this goal by combining sparse parity checks with finite encoding rate and growing distance, but their construction remains a challenging discrete design problem. Here we introduce structured concept evolution (SCE), a search framework that pairs a large language model with a structured algebraic mutation grammar to discover lifted-product code families, a class of CSS qLDPC codes. Instead of asking the LLM to design codes from first principles, SCE evolves structured concepts consisting of algebraic specifications paired with executable programs that realize them, using hierarchical mutations that modify the group algebra, protograph geometry, or base space. Running SCE, we discover a diverse set of competitive code families, ranging from abelian constructions to families over non-abelian groups beyond those underlying standard designs such as bivariate-bicycle codes, and characterize them under code-capacity depolarizing noise with BP+OSD decoding. These results are obtained with lightweight models (GPT-5.4-mini and GPT-5.4-nano).

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