GFlowQEC: Generative Flow Networks that Learn to Sample Quantum LDPC Codes

Nuno Batista; Gabriel Falcao; Penousal Machado

quant-phNov 4, 2026

GFlowQEC: Generative Flow Networks that Learn to Sample Quantum LDPC Codes

Authors: Nuno Batista, Gabriel Falcao, Penousal Machado

Organizations: Department of Informatics Engineering, University of Coimbra, Coimbra, Portugal · Instituto de Telecomunicações, Department of Electrical and Computer Engineering, University of Coimbra, Coimbra, Portugal · New York University Abu Dhabi, Abu Dhabi, United Arab Emirates · University of Coimbra, CISUC/LASI – Centre for Informatics and Systems of the University of Coimbra, Department of Informatics Engineering

Abstract

Choosing a quantum error-correcting code for a real device means scoring candidates with circuit-level simulations so slow that only a few hundred can be run. So every evaluation counts, and a search is judged by how many good codes it finds per paid evaluation. Quantum LDPC codes built from a few group elements protect many logical qubits, but good ones are rare in a huge space. We introduce GFlowQEC, to our knowledge the first generative flow network (GFlowNet) for quantum code design. It builds a code one group element at a time and learns to propose codes in proportion to a power of their score. We train it once on scores from earlier searches, freeze it, and let it propose lifted-product codes with 180 physical qubits, scored by their distances. Ranking ten of its draws per paid call, it finds more than twice as many new top-scoring codes (inequivalent to any it was trained on) as tuned parallel tempering, even when tempering starts from GFlowQEC's training codes. It finds more than three and a half times as many as a supervised screen ranking ten random candidates per call. When both rank a hundred, the screen beats GFlowQEC, and does best ranking GFlowQEC's draws. Soft Q-learning, which shares the GFlowNet's optimal policy, does as well at one noise bias and better at the other, so the gain does not depend on the GFlowNet's training loss. Trained proposers can make a budget of expensive evaluations go further when good codes are rare.

Figures & tables

Work	Family	Score	Returns	Matched budget	Exact distances
Chengyu et al. [ 10 ] , Bayesian optimisation	BB, any weight	Rate $k/n$ and error rate	one code per weighting	yes	no
Cruz-Benito et al. [ 11 ] , LLM	BB	$kd^{2}/n$	a catalogue	partly	some
Liu and Marquardt [ 24 ] , LLM	LP	Rate $k/n$ and error rate	an archive	no	no
Bhardwaj et al. [ 4 ] , filtered sampling	LP, non-abelian	Distance upper bounds	one code per group	no	some
Sella et al. [ 34 ] , reinforcement learning	BB as defined by Bravyi et al. [ 5 ]	Decoder success rate	best code so far	no	no
Freire et al. [ 13 ] , three searches	Hypergraph product	Erasure failure rate	one best code	yes	not stated

Table 1: Prior code searches compared with this paper. LP stands for lifted product and BB for bivariate bicycle codes, two-block codes built from two sums of elements of

Z_{l}\times Z_{m}

, the pairs of integers mod

l

and mod

m

(Bravyi et al. [ 5 ] allow only pairs with one entry zero).

Figure 1: GFlowQEC (

n=180

). Rows of slots are tuples, with grey slots fixed to the identity, and Tanner graphs are codes. (a) Training, with no oracle call: (1) a stored form

x

with its score

s

and a bias

\gamma

, which give its reward

R(x,\gamma)

, (2) augmented and, with probability

1/2

, mirrored by

M

with

\gamma\mapsto-\gamma

, (3) read by the transformer (three blocks, causal mask shaded), which takes the bias as a condition

z

on its start token, (4) the trajectory-balance loss, weighing

R(x,\gamma)

against

Z_{\theta}(z)\,p_{\theta}

, and its gradient step (dashed). (b) Deployment: (1) the frozen policy at

\gamma=\pm 1

, (2)

\mu N

drawn forms (Sec. 5.4 ), (3) the

N

with the highest

\log p_{\theta}(x\mid z)

, (4) the oracle’s bound score and SAT certification, (5) the top-scoring codes outside the stored forms.

	Certified top-scoring codes		Inequivalent
Method	$\gamma=+1$	$\gamma=-1$	$\gamma=+1$	$\gamma=-1$
Trained on, or started from, the stored forms
GFlowQEC	$313$	$317$	$221$	$222$
Supervised screen	$87$	$84$	$62$	$59$
Soft Q-learning, trained offline	$317$	$364$	$225$	$259$
Tuned PT from the stored forms	$119$	$123$	$79$	$84$

Table 2: Results at

180

physical qubits: certified top-scoring codes in each run’s first

1{,}000

paid calls, mean over ten seeds rounded to whole codes, at noise bias

\gamma=+1

and

\gamma=-1

. In the first group of rows only novel codes count. Inequivalent: Sec. 6.2 . Ratios in the text pool the seeds both methods ran.

Quantum Error Correction

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

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OmniQEC: discovering practical quantum error-correcting codes by an AI scientist

Ge Yan, Shanchuan Li, Pengyue Ma, Qixin Zhang+4

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AI Agents for Scientific Discovery Quantum LDPC Codes Quantum Error Correction Large Language Model-Guided Scientific Discovery Error-Correcting Codes

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Quantum error correction (QEC) is indispensable for scalable fault-tolerant quantum computing. However, discovering QEC codes that remain effective is challenging, as logical performance depends on the interplay between code structure, hardware, syndrome extraction, and decoding, which often impose competing requirements. Here we introduce OmniQEC, an efficient AI scientist for discovering QEC codes suited to deployment on modern quantum processors. OmniQEC formulates QEC design as an iterative discovery process in which an orchestrator, implemented by advanced large language models (LLMs), coordinates code generation, code-level screening, syndrome-extraction synthesis, and decoder-based circuit evaluation. At its core, OmniQEC combines a self-evolving reasoning mechanism with a slow--fast synergistic workflow: a fast loop explores candidates using inexpensive code-level proxies, whereas a slow loop performs physically grounded circuit-level evaluation and feeds the resulting evidence back into the search. We evaluate OmniQEC across four qLDPC construction families, three LLM backends, and

14

total-physical-qubit budgets per backend. The discovered codes show steadily improving logical-error suppression with increasing physical-qubit budgets and outperform the BB codes with

[\![72,12,6]\!]

and

[\![144,12,12]\!]

under complete-implementation budgets of 98 and 240 physical qubits, respectively. The discovered codes are hardware-friendly and may be of independent interest for practical QEC implementation. These findings pave the way towards LLM-assisted QEC discovery grounded in physically informed code--circuit--decoder co-design.

Abstract

Quantum Error Correction

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