SAQ: Stabilizer-Aware Quantum Error Correction Decoder
Authors: David Zenati
Eliya Nachmani
Venue: ICLR 2026 Poster (2026)
Keywords: Deep learning
Quantum Information
Error Correcting Codes
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)

Abstract: Quantum Error Correction (QEC) decoding faces a fundamental accuracy-efficiency tradeoff. Classical methods like Minimum Weight Perfect Matching (MWPM) exhibit variable performance across noise models and suffer from polynomial complexity, while tensor network decoders achieve high accuracy but at prohibitively high computational cost. Recent neural decoders reduce complexity but lack the accuracy needed to compete with computationally expensive classical methods. We introduce SAQ-Decoder, a unified framework combining transformer-based learning with constraint aware post-processing that achieves both near Maximum Likelihood (ML) accuracy and linear computational scalability with respect to the syndrome size. Our approach combines a dual-stream transformer architecture that processes syndromes and logical information with asymmetric attention patterns, and a novel differentiable logical loss that directly optimizes Logical Error Rates (LER) through smooth approximations over finite fields. 
SAQ-Decoder achieves high accuracy decoding, with error thresholds of 10.99\% (independent noise) and 18.6\% (depolarizing noise) on toric codes that closely approach the theoretical ML bounds of 11.0\% and 18.9\% while outperforming existing neural and classical baselines in accuracy, complexity, and parameter efficiency. Our findings establish that learned decoders can simultaneously achieve competitive decoding accuracy and computational efficiency, addressing key requirements for practical fault-tolerant quantum computing systems.

Published as a conference paper at ICLR 2026
SAQ: STABILIZER-AWARE QUANTUM ERROR COR- RECTION DECODER
David Zenati, Eliya Nachmani School of Electrical and Computer Engineering Ben-Gurion University of the Negev
znat ihatid@post aquuacc.l... eliyanacebgu.ac.
ABSTRACT
Quantum Error Correction (QEC) decoding faces fundamental accuracy- efficiency tradeoff. Classical methods like Minimum Weight Perfect Matching (MWPM) exhibit variable performance across noise models and suffer from poly- nomial complexity, while tensor network decoders achieve high accuracy but at prohibitively high computational cost. Recent neural decoders reduce com- plexity but lack the accuracy needed to compete with computationally expensive classical methods. We introduce SAQ-Decoder, a unified framework combining transformer-based learning with constraint aware post-processing that achieves both near Maximum Likelihood (ML) accuracy and linear computational scala- bility with respect to the syndrome size. Our approach combines dual-stream transformer architecture that processes syndromes and logical information with asymmetric attention patterns, and a novel differentiable logical loss that directly optimizes Logical Error Rates (LER) through smooth approximations over finite fields. SAQ-Decoder achieves high accuracy decoding, with error thresholds of 10.99% (independent noise) and 18.6% (depolarizing noise) on toric codes that closely approach the theoretical ML bounds 1.0% and 18.9% while outper- forming existing neural and classical baselines in accuracy, complexity, and pa- rameter efficiency. Our findings establish that learned decoders can simultane- ously achieve competitive decoding accuracy and computational efficiency, ad- dressing key requirements for practical fault-tolerant quantum computing systems.
INTRODUCTION
Since Feynman's 's 1982 vision of quantum computation Feynman (2018). significant progress has demonstrated that quantum computers can leverage quantum mechanical principles to achieve fun- damental computational advantages over classical methods (Steane 1998 Ladd al. 2010 Preskill
2012 deMarti iOlius et al. 2024). Landmark quantum algorithms have demonstrated computational advantages, including exponential speedup for factoring (Shor) 1994) and quadratic search improve- ment (Grover 1996). Recent experimental demonstrations of quantum supremacy have further vali-
dated quantum computing's ' potential across diverse domains (Arute et al. 2019) [hong et al.
2020
Wu et al. |2021] Huang al., 2022; Madsen et al. 2022 Bao et al. 2023 Bluvstein et 2024
Aghaee Rad et al. 2025). These advances promise to revolutionize cryptography (Ekert) 1991
Ben-
nett Brassard 2014). optimization (Kadowaki & Nishimori 1998) Bharti al. (002). materials
science (Lloyd 1996), and machine learning (Huang et al. 2022 Cerezo et al. (002)).
Yet. for practical quantum computation to become a reality, errors on the physical level must be corrected with high confidence. Despite recent advances, quantum noise remains a major obstacle to practical quantum computing (goo) 2023). These errors arise through numerous mechanisms: quantum gates cause unwanted errors due to imprecise implementation (Fowler et al. (012a)). while additional errors stem from imperfections
the equipment (Preskill 2018), interaction with the sur- rounding environment (Burnett et al.) 2019 Etxezarreta Martinez et al. 2021), measuring quan- tum systems (Fowler et al. 22). W While fault-tolerant quantum computation can theoretically be achieved through redundancy by combining multiple physical qubits into logical qubit (Shor, 1995 Nielsen & Chuang) 2010), this approach creates critical computational bottleneck: QEC requires real-time decoding algorithms that must process syndrome measurements and determine

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Published as a conference paper at ICLR 2026
corrections within microsecond timescales while maintaining near-optimal accuracy (Terhal 2015 Higgott 2022). Current decoding methods face fundamental trade-off between computational eff-
eiency and error-correcton performance, methods like MWPM (Fowler) 2013). Belief Propagation with Ordered Statistics Decoding (BP-OSD) (Roffe et al. 2020) and tensor network decoder (Bravyi et al. 2014) scaling prohibitively with code distance while faster heuristics sacrifice the accuracy essential for fault-toierant operation (deMarti iOlius et al. 2024). The field of QEC has advanced significantly, with several families of QEC codes proposed, including topological codes (Kitaev
et al. 2012a Chamberland et al.) 2020). Quantum Kalachev [021] Breuckmann & Eberhardt 202I). and quantum turbo codes (Poulin et al. 2009). Recently, there
2003 Bombin & Martin-Delgade
Low-Density Parity Check (QLDPC) codes (MacKay al.
2004] Panteleee
has been significant growth in machine learning techniques applied to quantum decoding (Wang & Tang. 2024. Klusch et al.) (024). However, existing neural decoders typically fail to achieve near-
optimal error thresholds, creating a gap between the theoretical potential of learned approaches and the performance requirements of fault-tolerant quantum computing. We address this challenge by introducing a unified framework that combines transformerbbaee neural decoding with specialized architectural innovations. Our approach leverages neural networks to learn syndrome-to-ent to-error map- pings while employing dual-stream processing and logical-centric loss design to directly optimize logical error suppression. To achieve this, our framework introduces several key innovations:
A novel dual-stream transformer architecture (Vaswani et al., 2017) that simultaneously processes syndrome and logical information streams with specialized attention mecha- nisms, featuring global tokens (Zaheer et al. 2020) and structured masking patterns that capture the geometric constraints and local correlations inherent in stabilizer codes.
A novel logical-centrie multi objective loss, including differentiable minimum entropy loss that directly optimizes LER through smooth approximations of discrete GF(2) constraints, enabling end-to-end training that circumvents the non-differentiabilit challenges in QEC.
Constraint- Projeccte Nullspace Descent (CPND). novel deterministie post processing al- gorithm that leverages transformer probabilities as reliability weights to construct recovery operators with exact syndrome consistency.
Near-optimal error thresholds of 10.99% and 18.6% for toric codes under independent and depolarizing noise, approaching ML bounds 11.0% and 18.9%, with linear scalability in syndrome size and general applicability across stabilizer code families, contrasting favor- ably with polynomial-scaling classical methods.
Our results significantly outperform existing neural decoders like QEC Transformer (QECCT) (Choukroun & Wolf, 2024) and classical methods like MWPM, while matching the performance of computationally expensive approaches across both toric and rotated surface codes.
The remainder of this paper is organized as follows. Section 2 surveys related work in QEC. Sec- tion 3 provides essential background on the quantum decoding problem. Our unified framework is presented in Section 4 where we detail the transformer architecture and dual-stream design and our loss formulation. Section presents comprehensive experimental evaluation. Finally, Section 6 summarizes our contributions and discusses implications for fault-tolerant quantum computing.
RELATED WORKS
A broad suite of QEC codes has been devised protect quantum information from decoherence, noise, and gate imperfections. Extracting the underlying logical state from these codes requires dedicated decoders that infer the likely errors from the measured syndromes and prescribe correc- tions (Dennis et al. 2002). However, ML decoding for quantum codes is NP-hard (Kuo & Luj 2020 prompting the adoption of approximate methods that trade optimality for computational tractability (deMarti iOlius et al., 2024). Classical quantum decoding approaches include MWPM, which achieves near-optimal thresholds under independent noise but suffers from poor scaling even with practical approximations (Edmonds) 1965 Fowler et al. 2012b: Meinerz et al. 2022 ; belief propagation, effective for sparse parity-check check codes but impeded by quantum degeneracy (Peall 2022 Panteleev & Kalachev 2021 Wang & Tang 2024); union-fina decoders that map syndromes to graph problems but achieve lower thresholds than MWPM (Delfosse & Nickerson, 202I); and
tensor-network decoders that attain the highest accuracy steep computational cost (Bravyi al.
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Despite their foundational role, these conventional approaches exhibit inherent limitations that impede practical deployment in large-scale, fault-tolerant quantum systems (Krenn et al. 2023 deMarti iOlius et al. 2024). Machine learning has emerged compelling alternative, with various architectures that demonstrate accuracy and speed gains over classical baselines while allowing adaptation to device-specific, correlated noise processes that challenge traditional decoders [Wang& & Tang 2024] deMarti iOlius et al. 2022|] Varsamopoulos et al. 2017 2019; Harper et al.)
2014 goo 2023
2020 Magesan & Gambetta [020 Liu & Poulin 2019). Specifically, these architectures are em-
ployed in reinforcement learning, (Colomer et al. [2020] Sweke et al. 2020; Fitzek et al., 2020
Çelikkanat al. 2022] Veeresh et al., 2024). and supervised learning (Bishop & Nasrabadi 2006
Goodfellow et al. 2016). where models are trained on labeled datasets to map measured syndromes
to recovery operations (deMarti iOlius et al. 2024; Wang Tang. 2024). Early approaches included feedforward networks (Torlai & Melko 2017), neural decoders learning error distributions (Kras-) tanov & Jiang 2017). and quantum autoencoders (Locher et al. 2023), demonstrating generalization while reducing complexity and adapting to noise. CNN-based decoders achieve strong performance on topological codes via spatial correlations (Maskara et al. [019 Meinerz et al. (022). More re- cently, transformer- -easoe architectures have been explored, most notably the QECCT (Choukroun) & Wolf, 2024), outperforming MWPM across topological codes. While dual-stream transformers like CPossMPT (Park et al., 2024 have been developed for classical codes, they rely on symmetric, locally-constre constrecon attention to emulate belief propagation, lacking the asymmetric global context necessary address quantum degeneracy. Another innovative Al-based decoder is AlphaQubit (Bausch et al., 22244, Semior et al.) 2025). which represents a major milestone in QEC decoding but employs a recurrent structure and processes analog measurement data, unlike our feed-forward architecture which utilizes discrete binary syndrome inputs.
3 BACKGROUND
A binary linear code C S GF(()) is defined as the nullspace of parity-check matrix H E (F(2))(1-)xx where E N physical bits encode h E N logical (message) bits. For error vector e E GF(()", the syndrome s = He serves as a sufficient statistic for ML decoding. The transition to QEC introduces fundamental complications absent in classical settings. Unlike clas- sical bits that exist in definite states {0, 1}, quantum information is encoded in qubits two-evel quantum systems that exist in coherent superpositions:
|+) = (0 + 311), where a, C,
|a|2 + 18
(1)
This quantum nature creates fundamental challenges: quantum errors form a continuous group, and the no-cloning theorem eliminates classical redundancy. Fortunately, the Pauli channel provides tractable error model. Any single-qubit error can be decomposed in the Pauli basis (,, Z}:
T = a0 + 11; x(v> = al1) + 310): Ylv) = -ia 11 + i,800;; Z1y) = al0) - -11) (2)
A general single-qubit Pauli channel applies error P E (1, Z} with probability dp, where Es Op 1. For qubits, errors are tensor products E P.. P... leading to 4" possible error patterns. The exponential growth in error patterns (4" vs. 2 classically) creates rich combinatorial optimization problem well which suited to neural approaches.
Stabilizer Formalism. The stabilizer framework, (Gottesman) 1997). addresses QEC challenges by discretizing the error space while preserving quantum coherence. This formalism exploits the algebraic structure of the Pauli group construct quantum codes syndrome extraction. The Pauli group foundation. The n-qubit Pauli group captures all local quantum errors:
P.
{ [ @ & Pr : w E (+1, ti), P, E (I,X,Y for lorj=,...,,n .
(3)
The global phases leave syndrome measurements invariant and can be quotiented out.
A stabilizer group S forms an abelian subgroup of P, with - e S. The abelian structure guar- antees that all stabilizer elements commute, enabling simultaneous measurability. An [[,, k, Leote]] stabilizer code with distance Leode uses m = n k independent generators { whose joint +1 eigenspace defines the codespace:
Cs (1w) E H2 se(bt) =1,), for i 1,....,, m)
(4)
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For eπror E E Pn. the syndrome s(S,, E) indicates whether stabilizer S; commutes (0) or ticommutes (1) with E. The full syndrome vector s(E) = (s(S,,))..... E).... (SS.. m. E)) E {0, 1 provides a classical signature of the quantum emror. Crucially, measuring these stabilizers is non de- molition, i.e., extracting error information without disturbing the encoded quantum state. Quantum degeneracy occurs when multiple distinct errors produce identical syndromes because they differ by logical operators that commute with all stabilizers yet act nontrivially on the codespace.
Quantum degeneracy creates prediction problem: given syndrome S, determine which logical coset contains the true error. This presents formidable computational challenges with exponential syndrome spaces (20(Lan) for surface codes), making neural approaches particularly attractive for learning optimal syndrome-to-coset mappings. Surface codes possess inherent geometric structure ideal for neural learning, with local syndrome correlations and hierarchical error patterns that align perfectly with attention mechanisms capable of capturing both local and global correlations.
SAQ DECODER
We address QEC problem: given syndrome measurements, predict recovery operations that restore correct logical states. Due to degeneracy. multiple errors yield identical syndromes, requiring de- coders that find logically equivalent recovery operations. To tackle this challenge, we propose a novel architecture consists of three sequential stages: (i) dual-stream representation construction, (ii) Syndrome-LLogica Transformer Decoder (SLTD), (iii) the post-processing CPND stage and (iv) novel differentiable logical centric loss.
The dual-stream representation construction stage takes syndrome measurements as input and gener- ates initial logical class estimates. These estimates, along with the original syndrome measurements, are then transformed into two token streams that serve as input to the SLTD. Using shared trans- former weights, the SLTD processes these streams with distinct attention patterns tailored for QEC: syndrome tokens capture local correlations between neighboring stabilizer measurements, while logical tokens integrate information globally determine error classes. The dual-stream architec- ture explicitly models the asymmetric information flow in quantum decoding, from local syndrome violations global logical error determination.
The SLTD outputs logical class predictions and qubit flip predictions, which are trained using differ- entiable logical centric losses that approximate discrete GF(2), before being fed to the CPND stage. Subsequently, the CPND enforces syndrome consistency while preserving the transformer's 's learned representations, ensuring valid QEC.
4.1 STAGE 1: DUAL- REPRESENTATION CONSTRUCTION.
Given syndrome vector E (-1,+11 we first obtain an initial logical class estimate ė E R
through a shallow MLP by (-1,+13m E expressed
i (s))
(5)
where 4 represents the total number of logical equivalence classes for & logical qubits. The shallow MLP (s) provides informed priors about the most likely logical class, enabling the SLTD to refine these estimates rather than exploring the entire logical space from scratch. Such a mapping, where syndrome input processed by shallow feed-forward network, appeared in earlier works, notably the FFN layer in (Meinerz et al. (022) and the initial noise estimator in QECCT (Choukroun & Wolf 2024). Crucially, in our work, by performs global estimation of the logical class (E), serving as a global prior input to the Logical Stream (Tt). This contrasts with related approaches that utilize these initial layers primarily for generating local physical error probabilities extracting an initial prediction of the recovery operator. The design choice aligns with the stabilizer formalism, where correction decisions made purely based syndrome information, independent of the protected quantum information. With both syndrome measurements s and initial logical class estimates i available, we construct two complementary token streams that capture different aspects of the quantum decoding problem:
Syndrome Stream Construction. Each syndrome measurement 8; E (-1,1) for i = 11.... m.mmmmm is mapped learned embedding t. s.w? E Rdd using learnable positional embeddings w IRd
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T
T
T
T
uady
FF
W
Noen
Nora
proji
Manked MISS
Nerm
T SAQ Decoder Layer
T
T
Initial Embedding Layer
Output Layer
Figure 1: Architecture of Eeo-Decoder.
(where dl is the embedding dimension), which collectively form the learnable syndrome embedding matrix W = [wi;... :.i:...:... Rmxd A learnable global token g E Rd is then prepended
to enable synss-sy information exchange, forming the complete syndrome stream: T
[g: [0] ........ ml (m++)xxd The global token enables efficient information aggregation across distant syndrome regions essential for handling correlated noise and large error clusters.
Logical Stream Construction. Predicted logical class logits are embedded as 5.1. Rdd for j = 1* using learnable class-specific representations w/ Rdd. These embeddings form the matrix WL [wf::..:w/4 R xd and yield the logical token sequence T [t i....... 10 Rd The dual-stream design reflects that syndrome measurements encode local constraint violations while logical estimates capture global degeneracy patterns.
[0 L
4.2 STAGE 2: SYNDROME-LOGICAM TRANSFORMER DECODER (SLTD)
Having constructed dual token streams, T and T!o the SLTD refines these representations through N transformer layers with asymmetric attention. We detail each computational step for an arbitrary layer I. Both token streams first undergo layer normalization (Ba et al., 2016) before attention computation. The normalized tokens are processed using asymmetric attention mecha- nism that captures the fundamental information flow in QEC: syndrome measurements reflect local physical constraints, while logical error determination requires global integration. This design stricts syndrome attention to topological neighborhoods while allowing logical tokens global access, enabling efficient local-global information processing. Syndrome self-attention processes syndrome tokens through:
I
[-1]
T
1--11
W!ll Q:
K!! ---l- W..
V V
- "w
=
A Softmax
(a-1/2 QK
+Ms)
[-1]
We introduce a novel syndrome attention mask Ms that enforces topological constraints:
Msli.j] - {2 1-o0 otherwise
if (HH + Im)).. > or = 0 or
(
where H 10,1)mxn is the parity-check matrix and I,, is the identity matrix. The mask permits attention between: (i) each syndrome and itself (I,), (ii) syndrome pairs that share physical qubits (HH > 0), and (iii) all syndromes with the global aggregation token (corresponding to i.j 0). For the logical stream, logical cross-attention enables logical tokens to attend to the updated syndrome representations.
Q - T Wil
K! T W!
v = T w!!

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Al Softmax dd11/2 "!!
(10)
Logical tokens employ unrestricted attention patterns, enabling global syndrome integration. Fol- lowing attention computation, residual connections combine the attention outputs with input tokens. Both streams pass through standard FFNs with 4x expansion and GELU activation (Hendrycks & Cimpel, 2016). Finally, a second residual connection yields the layer outputs. This process trans- forms initial token representations into refined syndrome and logical embeddings that capture both local correlations and global quantum code structure.
Output Generation. Final token representations are normalized and projected to outputs, where syndrome tokens (excluding the global token) produce physical error predictions = Wouns (T
S,no-global (Woool,) and logical tokens generate class = WouLL (T/)) Wpool,L).
4.3 STAGE CONSTRAINT- PROJECTED NULLSPACE DESCENT (CPND)
Neural decoders face a constraint challenge: networks learn correlations but cannot guarantee recov- ery operators satisfy syndrome consistency over GF(2). CPND bridges this gap through constraint enforcement preserving learned representations. It operates via (i) exact projection ensuring syn- drome consistency, and (ii) greedy descent using transformer probabilities to guide optimization toward lower-weight solutions. The prediction è (its hard decision, core)) is not guaranteed to satisfy the input syndrome constraint. Furthermore, the direct logical class prediction é provides a slightly superior estimate of the logical class than the class implied by the raw error prediction. The definitive output, e(s), is therefore produced by the CPND stage, which enforces
criti- cal constraints (i) the syndrome constraint s He(s) and (ii) the target logical class i Le(ss) where L {0, 1] 1 encodes the logical operators. This stage uses the transformer outputs è as informative priors. complete derivation and description the method is provided in Appenddi
4.4 LOGICAL-CENTRIC LOss DESIGN
Our training objective combines three loss terms addressing quantum degeneracy by minimizing LER via informed priors, direct classification, and differentiable constraint approximation.
Informed logical priors loss trains the auxiliary MLP (s) map syndromes to logical classes:
LLP = CE( é, yelass)
(11)
where Blaass encodes the true logical syndrome as a class index, providing informed priors to guide transformer processing.
Logical class prediction loss supervises the transformer's refined logical output:
LL CE(ê, Jetnss)
(12)
ensuring accurate logical classification after artss-att processing.
Logical-minimum entropy loss. key challenge in neural quantum decoding is enforeing the dis- crete constraint that recovery operators must preserve logical information. Specifically, we require the true error eie and the recovery operator ered satisfy L tre <b prod) epred is hard decision on the logits è and the residual eтror ene + pred must be stabilizer for successful error recovery. Our contribution constraint. We model the probability that each residual bit is flipped as Pr(r; 1|e("e) qi o((1 )é.) for = 1..... n and o is the sigmoid function. For each logical operator L,, the probability of violating the logical constraint is:
over GF(2), where
differentiable approximation to this discrete
Pr (L. r
Pr
A JEXI
LISTTs=1 =1
(IL-24)) JEXI
(13)
where Xi are the non zero elements set in L;- Our logical-minimun entropy loss minimizes the expected number of logical violations (full derivation is provided in Appendix A}:
2
Limropy =
log
Pr(L, r= 1))
(14)
2k

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MWMM
MWTM
MWWM
BPOSD-2
RPOSD-2
GEOCT
OECCT
SAQ (No CPND:
SAQ (No CPND:
CPND)
SAO
6.2 Hyydkal
(a) Leqade
Luods
(c) Leude 10
Figure 2: Toric code depolarizing noise model results.
5.94
XWNM
3ONPM
BPOSD-2
BPOSD-2
QECCT
QECCT
QECCT
SAO CRND)
SAD (No CINT)
SAD CPND).
SAO
SAO
SAO
c1c Hipkal
(a) Lequde
Luude
(c) Tecude 10
Figure 3: Toric code independent noise model results.
The combined objective is L = ALPLLP + ALeLIC + AEnroop Entrroy.
EXPERIMENTS AND RESULTS
To empirically validate the adaptability of the SAQ-Decoder distinet lattice geometries, eval- uated our method across diverse code families and noise types. We primarily focused on topological codes due to their prominence in fault-tolerant quantum computing (deMarti iOlius al. 2024)
specifically toric codes (Kitaey 1997) and rotated surface codes (Bombín & Martin-Delgado 2007
To further demonstrate the framework's broad applicability, we included an evaluation of the Repeti- tion Code (Peres) 1985) and the Color Code (Bombin& & Martin-Delgade 2006) using stim (Gidney) 2021. We evaluated our method under three well-studied noise models: independent noise, de- polarizing noise and circuit noise. Detailed descriptions of the code constructions provided in Appendix [ training details and hyperparameters are provided Reproducibility Statement. We evaluate our approach against three key baselines: the QECCT (Choukroun & Wolf 2024), state- of-the-art neural decoder that outperforms classical methods, Belief Propagation with Order-2 Or- dered Statisties Decoder (BPOSD-2) (Roffe al., 2020). and MWPM algorithm (Fowler 2013). the gold standard classical decoder for surface codes. The BPOSD family of decoders is widely used, (
although its worst-case complexity scales as Onn (deMart iOlius al. 2024). our implementa-
tion leverages the optimized implementation (Roffe (022) to provide a strong, practical classical benchmark for quantum codes. Although MWPM has worst-case complexity O(n log n) (de- Marti 1Olius et al. 2024). we use the optimized implementation from Higgott (2022) which achieves near-quadratic complexity and serves the primary classical benchmark for topological codes. We also consider the performance of the raw Syndrome Stream prediction, termed SAQ-Decoder (No CPND), as an architectural baseline. As our primary evaluation metric, we use the LER, which mea- sures the probability that the QEC process fails to properly recover the encoded logical information. We also evaluate code threshold, i.e., the critical noise rate below which increasing code distance improves performance.
5.1 EXPERIMENTAL RESULTS
The experiments span code lengths from Laske = 3 to 11, comparable those evaluated in QECCT. These results demonstrate that SAQ-Decoder achieves superior decoding performance compared to taat--of-the-art baselines across diverse QEC scenarios. Figure presents the performance of toric

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KOREM
NIOSA B
Viar OPcET SAD Dacice
BrozD22
Ang tNe PPPi 2012 brcoado
Bing piu CPRIO Xng Deadde
NMNN Aantr 1ORLD
Pyone
24
(a) Dep, Leude
(b) Dep, Lonke 11
(c) Ind, Loode
(d) Ind, Loode -1
Figure 4: Rotated surface code results.
codes under depolarizing noise, where SAQ-Decoder exhibits striking advantages. Our method demonstrates consistent superiority across all code distances, with particularly dramatic improve- ments at Leode = 10 where SAQ-Decoder achieves 25 50% lower LER compared to MWPM and BPOSD-2 at physical error rates above 0.15. The scalability benefits are clearly evident as the performance gap widens from Leode = 6 to Loode = 10. Similarly, under independent noise in Figure 3] SAQ-Decoder maintains robust performance across all code sizes, with particularly notable advantages at Lende = 8 and Tewode = 10. Conversely, QECCT exhibits minimal perfor- method's performance on rotated surface codes under independent and depolarizing noise models. Under depolarization noise in Figures 4ab SAQ-Decoder consistently maintains lower LER than QECCT, BPOSD-2 and MWPM across the entire noise range. Under independent noise conditions in Figures 4c SAQ-Decoder demonstrates even more substantial improvements, while QECCT exhibits inferior performance compared to MWPM and BPOSD-2. These results validate the effectiveness of our learned decoder with post-processing approach across the spectrum of topological QEC codes and noise models. We attribute SAQ-Deccder'ss superior performance over QECCT to two key factors: (i while QECCT focuses on reducing Bit Error Rate (BER), which is not the primary objective in QEC, SAQ-Decoder directly optimizes for logical error suppression; and (ūi) novel architecture combines direct logical class prediction with qubit flip priors, integrating these observations a post-processing stage that guarantees syndrome consistency-uniee qubit-level flips without ensuring this crucial constraint. Crucially, the SAQ-Decoder (No CPND) variant consistently outperforms all classical and neural baselines across the entire noise spectrum, demonstrating the power of the dual-stream architecture alone. The SAQ-Decoder achieves error thresholds of 10.99% and 10.7% for toric and rotated surface codes respectively under independent noise, and error thresholds of 18.6% and 18.3% under depolarizing noises, For the toric code under depolarizing noise (Figure Sa). this threshold of 18.6% approaches the MI. bound of 18.9% (Bombin) et al. 2012) while maintaining linear complexity in syndrome size. This significantly outperforms BPOSD-2 and MWPM (16%) (Wang et al. 2009) and exceeds the previous QECCT result (17.5 (1). For toric codes under independent noise (Figure 5b we achieve a threshold of 10.99% while signif- icantly outperforming BPOSD-2 (10.8%) and MWPM (10.3%) (Wang et al. 2003 Higgoti, 2022). essentially reaching the ML threshold estimated between 10.9% and 11.0% (ecz 2024p. For rotated surface codes, SAQ-Decoder demonstrates remarkable consistency with the toric code performance. Under depolarizing noise, we achieve threshold of 18.3% (Figure 5c) significantly exceeding both QECCT (17.2%), BPOSD-2 (14.1 %, based on our experiments) and MWPM (14.0%, (deMarti iOlus et al., 20244), while under independent noise, the threshold reaches 10.7% (Figure where QECCT achieved 10.3%, BPOSD-2 10.2% and MWPM 10.6%, based on our experiments. These findings indicate that our approach generalizes effectively across different topological code geome- tries without compromising performance. A detailed comparison of our depolarizing noise threshold against other neural and classical decoders is provided in Appendix
mance gains relative MWPM and BPOSD-2. Figure T demonstrates
QECCT, which only predicts
To rigorously validate the generalizability of our SAQ-Decoder framework beyond surface codes, we conducted new experiments on two distinet code families: the color code (Teode = 3, 5) and the repetition code (Leode = 3, 5). Critically, both experiments were performed under realistic, multi- round circuit-level noise models. These experiments demonstrate robustness confirm our frame- it is fundamentally agnostic to the code family. Figure 66a| illustrates
work's claim of generality, the LER performance on a distance 3 color code with rounds of circuit noise, demonstrating robustness with high marginal gaps from the baselines. In Figure 6b| SAQ-Decoder significantly outperforms all baselines across the entire range of physical error rates. For example, at the highest analyzed rate of p- 0.02, SAQ-Decoder achieves LER that is 17.0% lower than QECCT and
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CA8 80 8.
038 64 C20
XB
120
Plysocal Ecre
Playiical Paat
(a) Toric (dep)
(b) Toric (ind)
(c) Rot. surface (dep)
(d) Rot. surface (ind)
Figure 5: Error threshold analysis across topological codes and noise models.
CEOCE
BAY elle
arosp2
OBOCT
CDCET
Premect Bnee
Prytes
(a) Leode
(b)
Leode
Leade
(d) Lexte 5
Figure 6: Color code and repetition code with circuit noise results. (a)-(b) are color code results and (c) are repetition code results.
64.2% lower than the MWPM baseline. Figure 6c presents the LER results for distance 3 repeti- tion code with rounds of circuit noise, showing that the decoder remains robust with performance gaps relative to the baseline methods. Similarly, Figure shows the results for distance 5 repe- tition code with 3 rounds of circuit noise. While the performance of all decoders is closer on this code, SAQ-Decoder consistently maintains the lowest logical error rate. At p 0.25, SAQ-Decoder achieves a LER that is 1.83% lower than QECCT and 2.61% % lower than MWPM.
shows
5.2 ABLATION STUDIES AND ANALYSIS
To understand the contribution of individual architectural components in our framework, we conduct comprehensive ablation studies on toric code with distance T = 6 under depolarizing noise.
Dual-Stream Study. To validate our dual-stream design, we conducted an ablation study exam- ining four architectural variants: (1) separate weights per stream within layer instead of weight sharing across the decoder stack, (2) symmetric cross-attention where both streams attend to each other rather than asymmetric design, (3) segical-streamm-nnie architecture removing syndrome processing, and (4) sundrome-strcam--nn architecture removing logical processing, as shown in Figure 7a The results reveal clear performance hierarchy from worst best: logical-stream-onnly bidirectional cross-attention, syndrome-stream--
no weight sharing, and our full SAQ-Decodee architecture. These ablations demonstrate that single-stream variants perform poorly, weight shar- ing across layers slightly improves efficiency while halving the parameter count, and asymmetric information flow from syndromes logical inference outperforms symmetric attention, validating the importance of our specialized dual-stream stream processing design for effective QEC.
Multi-Loss Ablation. To investigate the relative importance of different training objectives in our framework, we conduct an ablation study on the loss function components, varying the weighting parameters ALP. ALC, and Entropy for logical prior, logical classification, and en- tropy regularization respectively, as shown in Figure [bb The full multi-objective formulation (ALP, XEntropy = 0.2, 1.0, 1.0) achieves the lowest final average LER of 1.972e-01. System- atic removal of individual components reveals measurable performance degradation: removing logi- cal classification increases LER to 2. 113e-01 (+7.2%). removing logical prior to 2.055e-01 (+4.2%), and removing entropy regularization to 2.047e-01 (+3.8%). These results confirm that all three ob- jectives contribute mcaningfully to the model's QEC performance, with logical classification being the most critical component.

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Published as a conference paper at ICLR 2026
Ardtoodard Ab-ilen Stady
Impect of Diffron Oiapitiers
Nasiday ane Setel hhan Adaibe Stary
Porenuder Seete cobd
Sigle Retirctieed Craa-Ai
Buacken the Musk dbbul
Hus only Die (aase)
Lingle-Sie Siran SAC Dudider
Aur 18..1.4 1-11-
SeciSansder (Makk Rlbed)
(a) Architecture study.
(b) Loss ablation.
(c) Global token ablation.
(d) Param. comp.
Figure 7: Ablation studies results.
Effect of Global Token. The inclusion of global token (SAQ-Decoder) improves both training dynamics and final performance compared to the masked architecture without a global token (Mask Only), as shown in Figure 7c The full SAQ-Decoder achieves faster convergence and lower final LER (~ 0.19 versus ~ 0.21 average LER). Notably, attention masking itself provides substantial benefits, with the mask-only architecture significantly outperforming the unmasked baseline (Base- line, neither mask nor global token). The global token acts as a syndrome-level aggregator, enabling the model to capture global syndrome patterns that local interactions might miss.
Computational Complexity. Our model achieves O(Nmd²) time complexity per forward pass by exploiting sparse attention patterns, avoiding the naive dN complexity of dense attention. The subsequent CPND refinement requires ((m) time for online inference. This yields optimal linear scaling in the syndrome length and quadratic scaling in code distance. The 22t embedding term is negligible since A is small in practice (e.g.. k 1 for surface codes and k = 2 for toric codes). Our numerical comparison in Table demonstrates that the SAQ-Decoder achieves significantly lower FLOOPs and faster inference time compared QECCT, validating its suitability for real-time QEC decoding. Extended data is available in Appendix F
Table I: Abridged Numerical Complexity Comparison (Toric Code)
Metric
L=6 (Depol)
L=10 (Depol)
SAQ-Decoder
QECCT
SAQ-Decoder
QECCT
Total FLOPs [G 4]
0.21
1.05
0.80
4.10
Inference Time [ms J]
1.2
7.0
4.5
20.1
Parameter Efficiency. Figure 7d demonstrates that SAQ-Decoder for toric code maintains near- constant parameter count ( [1. 1.9M) across code distances Leade = Loode = 10 for both noise models, exhibiting excellent scalability. In contrast, QECCT suffers from significant param- eter growth, reaching M.71M parameters Leode = 10 under depolarizing noise, which is a 3.5x increase over our method. This difference stems from fundamental architectural choices: while QECCT processes both qubit and syndrome information through transformer layers, our approach leverages logical class prediction decouple syndrome processing from the full qubit space di- mensionality. Although both methods employ sparse attention patterns, SAQ-Decoder's logical embedding strategy avoids the quadratic scaling in the qubit-syndrome space that affects QECCT.
CONCLUSION
We introduced EAQ-Decoder, unified QEC framework that combines learned syndrome-to-ende mappings with exact syndrome constraint satisfaction. Our dual-stream architecture with asymmet- ric attention captures the geometric structure of stabilizer codes while maintaining linear computa- tional complexity in syndrome size. Experimental results demonstrate error thresholds of 10.99% and 18.6% for toric codes under independent and depolarizing noise respectively, essentially ap- proaching ML bounds while significantly outperforming existing neural and classical decoders. The framework's 's parameter efficiency and general applicability to any stabilizer code family make it practical solution for scaling fault-tolerant quantum computation, bridging the critical gap between neural pattern recognition and the structured optimization requirements of QEC.
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REPRODUCIBILITY STATEMENT
Logical-minimun entropy loss. Due to page limitations, we provide the full derivation and expo- sition of the logical-minimum entropy loss Appendix A
CPND stage. A detailed derivation and exposition of the CPND stage is presented in Appendix B
Training Details. For reprodueibility. we detail our complete training methodology and hyper- parameters in Appendix [ with full source code provided in the Supplementary Materials. Our source code is publicly available on GitHub https: /ttt://githb.... com.DavidZenatt1. SAQ-Decoder/treet /main
ACKNOWLEDGEMENT
The contribution of David Zenati is part of Ph.D. thesis research conducted at Ben Gurion Univer- sity of the Negev.
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A LOGICAL-MINIMUM ENTROPY LOSS DERIVATION
The goal of decoding in stabilizer codes is to produce a correction eored (hard decision on the logits
é) such that the combined eπrоr = e!THe + e is a stabilizer, hence acts trivially all logical
qubits. Equivalently, with denoting the logical-operator matrix, the logical-coset constraint is
L (e @ = over GF(()
(15)
so decoding succeeds if no logical parity flips. Optimizing this condition directly is difficult because XOR is non-differentiable. We therefore derive a smooth, probability-cclibrated surrogate that re- places hard parities by differentiable sign-expectations of Bernoulli logits. The resulting logical minimum-entropy loss maximizes the probability that each logical parity is zero while preserving the exact coset semantics in expectation and providing stable, well-aligned gradients for end-to-end training.
For each qubit let é E IR be the model logit and
A (e:)
(16)
1+e-a
the corresponding flip probability, where o.)) is the sigmoide function. We model the predicted error bit as
epred Bernoull(pp.).
(17)
while the ground-truth bit eiuue {0, 1) is fixed for the given sample. Define the per-qubit XOR
P equu + pred E 10.
(18)
Conditioning r; on emme, we have
Bernoulli (o())
Teeue
rele tnee
(19)
Bernoulli (1 - ((i))
if eeue
holds since for r; = @ epese
pred and ri = 1 + .rod
prod
We focus on the non-parity conditional error probability
4: Pr(r: = 1|e""=) = (1 c (a((s) + (1 - o (
(20)
which our loss is designed to minimize; driving q: + forces the prediction to agree with the ground truth modulo stabilizers (i.e., no logical flip).
We now simplify qi. starting from
1 6, tuee
exp (-ê:) true
f: = Pr(r; 1|c
+
(21)
1 1 cxp (-ē;)
+ cxp (-ē:)
1 equu + exp
(-ê:) c,
(22)
+ exp (-ē;)
Let 1 2e E {11, 1), so e
(1 22/ focusing on the numerator
1 eiee + exp (-éz) c me
+ e (exp (-é:) - 1)
(23)
- a
1 +
(exp (-ê,) -1)
(24)
2
- + 2
(exp (-(;) -1) -1)- a
(exp (-2:) - -1)
(25)
(1+ exp (-é,) - a. (exp (-ê,) 1)
(26)
2
Plugging this back to equation 21 gives
(1 + exp (-ē:) a (exp (-):)= 1)
q: Pr(r:: 1/c,u0)
(27)
2 (1 exp 22))
1
exp (-è:) 1
1
(28)
2
exp (-ē;)
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Using
exp (-p)-))-1 tanh (z/2) with é exp (x
q: 1 + tanh
(29)
Here Lanh (+) is the hyperbolic tangent. From the previous result,
Pr(ri 11 e b [ (e/2()
(rue)
2e {+1}.
(30)
Substituting true (so a -1) gives
1
e!Tu 1- =1)= = 2 Pr(r: = l
(anh()/2))
(31)
while etrue = (so a +1) gives
Pr(r;=1| e j.rue = =0)= 1[ (ēs
(32)
Since tanh odd, i.e., a tanh(x) = tanh(ax) for a {=1}.
1 1 + tanh (a (2)
Pr(r: (ere))
(33)
2
To convert the tanh form back probability, we the sigmoid-tam identity (as shown in Propo- sition A.I).
proposition A.1 (Sigmoid-tane tanh identity). For all y E R,
1
1 [ + tanh(y/2)
(y) + exp y)
(34)
2
Proof. Start from sigmoid (y), multiplying numerator and denominator by exp(y/2):
cxp(y/2)
oly) 1 + exp( -yy (xxp//))xx((u/////
(35)
Let A exp(y/2) and B exp( 1//2).Them
A
1 (A + B) + (A B)
A B
(36)
A+
B
2
A+ B
2
+ B
But
A B
exp(y/2) exp(- exp(-y/2)) tanh
(37)
A+ B
explu/2) + exp(- 22
Hence (yy) = [1 + tanh(g))
Therefore,
1 (anh(a,./2) =(aa;) d(1 (1-22eruuc 2e.rue))e) e.)
= Pr(r, = =1 ( 1!
(38)
We now pass to logical parities: each of the logical operator matrix L (which correspond to distinct logical operator) induces a parity check over r;
L r LisTy
(39)
JEXi
where X are the non zero elements set in L; the following proposition provides L, r distribution in closed form.
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proposition A.2 (Bernoulli parity distribution). Let {X,}}=1 be independent Bernoulli random
variables with Pr(X, 1) q1 [0, 1], and define the GF(2) parity Q +i X, {0, 1}. Then
Pr(Q = 1)= =
II
-24))
(40)
Pr(Q 0) 1+
I 224))
(41)
Equivalently, [B[(-1) -I,= [T 24²)
Proof. Consider the following product identity
(-1)² = - (-1) (-1)xxxBx II (-1)x.
(42)
This identity holds because:
= (-1 (-1))1((11001))) = --1)0 (-1)² and
1(-1)101 = = (-1)² (-1)²
Taking expectations both sides yields
B[(-1) E II (-1)
(43)
and, by expanding the definition of expectation with respect to Q,
B[(-1)9) = (-1)². Pr(Q = 0) + (-1)' Pr(Q 1) -Pr(Q =)
(44)
Under the independence assumption.
E
x
(45)
and each factor evaluates to B[(-1)*:]= - (-1) - (-1)² Pr(X, = 1) = 9.) 9:
2g:
(46)
Thus,
E (-11 II (1 - -2g)
(47)
Equating equation and and equation 47 and solving for Pr(Q = 1) gives
Pr(Q = = {[ II - 2q:)
(48)
The key insight is that the XOR operation in GF(2) corresponds to multiplication in the group ((-1,1),:) which enables closed-form expression for the parity distribution under independence.
Therefore, follows from Proposition A.2 and from equation 38 that
Pr(L, r 1)
II JEX:
24,)
II JEXi
(1-20 (1- ( ē,))
(49)
Since L, r = corresponds to logical error detected by the i-th logical operator (i.e., a bit flip or phase flip logical qubit), successful QEC requires minimizing this probability across all logical operators. To achieve this objective, we employ minimum entropy loss that directly optimizes the negative likeliiold of the desired outcome, namely that logical emors occur. This approach
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concentrates the probability mass the correct logical state while heavily penalizing configurations that lead to logical failures.
Substituting the expression for Pr(L, 1) into the entropy objective yields
2h Pr(L; -r- =-=1)
Lentropy =
log
(50)
2k t=1
log 1 + II (1-2 2- 20 )e,))))))))))))))){))
(51)
2k
jEXi
B CONSTRAINT-PROIECTED NULLSPACE DESCENT (CPND)
Neural decoders face constraint challenge: networks learn correlations but cannot guarantee recov- ery operators satisfy syndrome consistency over GF(2). CPND bridges this gap through constraint enforcement preserving learned representations. It operates via (i) exact projection ensuring syn- drome consistency, and (ii) greedy descent using transformer probabilities guide optimization toward lower-weight solutions.
The augmented matrix H [H:L] L] {0, (0.1)(m+26)xx) is constructed by vertically stacking the parity-check matrix H {0, 1) m (whose rows represent stabilizer generators that, when mea- sured, produce the syndrome) above the logical operator matrix E {0, 1]2kx= (whose rows encode logical operators that, when applied the error vector, determine the logical + k Z-type logical operators). Since the stabilizer generators are linearly independent over GF(2) and the logical operators lie in the normalizer but not in the stabilizer subgroup, the rows of H are linearly independent. Therefore, (H) + 2k.
class, X-type
c(s)
CPND
ND
proje
proj.
TIN
T
Figure 8: CPND.
The constraint vector is b = |s; E E {0, 1 (mm+ where is the syndrome and & is the binarized logical class prediction. We define the feasible recovery operator c(s) set
F = e((s E 10" He(s) = =b}
(52)
We precompute a left inverse B E {0, mm with HB Im+ proof of existence of such B is given in Proposition B.I
proposition B.1 (Existence and constructive computation of left inverse GF(2)). Let A (0, 11rxxx with r < n. There exists X E {0, 1]mxr such that AX = I, if and only if rank A) = r. Moreover, when A has full row rank, the r column-wise systems
Ax; = Ci
i= 1=
(where e; is the i-th standard basis vector of GF are all consistent, and any selection of solu- tions {xr) Yi stacked X [2n x.) satisfies AX T,
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Proof. (=) AX = I,, then rank(A) rank(I,) r, hence rank(A) = (+) rfank r, then the column space im(A) _ GF(()" has dimension r and therefore equals GF(()). Thus each e lies in im), there exists 2; with Az; ex- Stacking these solutions yields AX [Az] ... Ax,] = (1 ... e,] I,-
Given eProd we compute the residual y b ( Heare and apply the projection e' = e ( By. By construction: Hc' Herre @ y b. ensuring c' E F. The projected solution c' satisfies all constraints but is suboptimal in sense of minimum weight recovery operation, the left inverse B is constructed purely algebraically and ignores the transformer's learned qubit flip probability. Since the constraint set forms an affine space e' @ ker (H), we traverse this space to find lower- cost solutions while preserving feasibility. Let N [Vi,..., Val E {0, 1|aog span ker(H) with 9 n (m + 2k). We extract qubit flip probability from transformer predictions: Py =((e) and define weights as log-likelihood ratios W/ = (og(p,/( - py]). The objective is to minimize weighted Hamming cost: wtw (e) Σq=1 Wyeg where E 10,1)"
Having e', we want to descend in the nullspace to find minimal weight solution. First we convert the binary solution e' to signs o' = (1 - 2e') E (+1, -+,", where o' = +1 if eg = 0 and o -1 if ea for q 1..., n. The main loop performs single pass over the 9 nullspace generators {v,} ) For each generator v,. we identify its support Xj = {4 vig 1} and compute the cost change A, Enex, wooy If A < O, we accept the move: update e' +- e' v. and flip signs of + -o' for all g Xj.
Since Hv, 0 for j 1 1...9. g. the nullspace descent preserves constraint satisfaction while achieving monotonic cost reduction, terminating at locally optimal solution. Algorithm T the complete method.
presents
Algorithm Constraint-Projectee Nullspace Descent (CPND)
Require: Å, B, N; b; eorod; weights w E R
Ensure: c(s) E (0, 13" with Ho(s) = b, and reduced weighted cost wt (c(s)) = Eg CgCg(s)
I: y b + Hoard
2: e' +- ered @ B
3: o' + (1 2e') E (+1,-1))
4: for j to do
5:
Xi + { q:: U/ = 1}
6:
A Σ
Wg o'
nEX;
7:
if A < then
8:
e' -- e' @ Vj
9:
0' + of for all q E X
10:
end if
11:
end for
12:
return c(s) c c
Since stabilizer generators commute, yielding HHT = O over GF(2), the columns of HT span subspace of ker(H), providing an approximated basis for nullspace descent.
Figure 9 presents decoder recovery operator weights as a function of physical error rate e (0.05, 10, 0.15, 0.20) for the torie code under independent noise (Tuudt = 4). We compare three post-processing approaches that operate on SAQ-Decoder representations: the projection baseline (e' = e @ By), CPND and OSD-0 (Fossorier Lin 1995; Roffe et al. |2020). OSD-0 is post- processing algorithm that achieves minimum weight solutions while maintaining syndrome consis- tency, but requires matrix inversion operations that scale cubically in complexity, compared to our method's 's linear complexity. Across all error rates, the methods exhibit consistent performance order- ing: OSD-O achieves the lowest weights, CPND performs comparably to OSD-O, while the projec- tion baseline consistently yields the highest weights. The performance gaps increase monotonically
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Decoder minimum weight
Projection
OSD-6
-10
Plysicel Birer Bace
Figure 9: Comparison of recovery operator weights.
with p. demonstrating that CPND consistently approaches minimum-weige solutions. These re- sults highlight that structure-aware post-processing methods (CPND and OSD-0) achieve uniformly superior weight minimization compared to naive projection approaches.
C SURFACE CODES
We evaluate our method on surface codes due their prominence in fault-tolerant quantum comput- ing, specifically toric codes (Kitaev 1997) which encode k = 2 logical qubits in = 2L physical qubits, and rotated surface codes (Bombin & Martin-Delgado 2007) which encode k logical qubit in 22 physical qubits. Toric codes utilize periodic boundary conditions with qubits on lattice edges, while rotated surface codes employ a lattice geometry with qubits on vertices. The stabilizer generators organized into two distinct groups based on lattice geometry, with different implementations for each code family. For toric codes, vertex stabilizers are constructed as products of Pauli-X operators acting all qubits adjacent to each lattice vertex, while plaquette stabilizers consist of products of Pauli-Z operators acting on qubits surrounding each lattice face, yielding a total of m 21 stabilizers corresponding to the vertices and faces of the Lende X Leade toric lattice). Rotated surface codes employ a fundamentally different geometry where all stabilizer generators are placed on lat- tice faces rather than being split between vertices and plaquettes. These based stabilizers come two alternating types: X-type stabilizers (tensor products of Pauli-X operators on qubits surround- ing a face) and Z-type stabilizers (tensor products of Pauli-Z. operators on qubits surrounding face). The lattice arrangement creates a natural checkerboard pattern where X- and Z-type stabilizers alter- vertex of the rotated lattice participates
stabilizer generators (L 1 vertex stabilizers and L 1 plaquette
nate, ensuring that every physical qubit qubit-positioned in exactly two X-type and two Z-type stabilizer measurements. For an Leede x Leode rotated surface code, this checkerboard arrangement yields = 2L 1 independent stabilizer generators, with L2
generators of each type, when L is always odd. Figure 10 presents both code geometries.
2
Two standard noise models are examined: independent noise and depolarizing noise. Under the independent (uncorrelated) noise model, bit-flip (X) and phase-flip (7) errors occur independently with equal error probability, allowing the decoding of X and Z syndromes to be treated separately. In contrast, the depolarizing noise model assigns equal probability p/3 to the non-identity Pauli operators, i.e., Pr(X)= = Pr(Z)= = Pr(Y)= = P.
3. Pr(T) = 1- 1-p. p. where iXZ.
D TRAINING DETAILS
Our training methodology randomly samples noise within the physical error rate testing range to en- sure robust generalization across different noise regimes. The model architecture employs N 6-8 transformer layers with shared parameters across dual token streams and an embedding dimension of d = 128 and h 16 attention heads. The multi-component loss function uses weighting param- eters ALP = 0.2, Are = 1.0, and Arm = 1.0 for informed logical priors loss, logical class prediction loss, and minimum entropy loss, respectively.
We optimize using the Adam optimizer (Kingma & Ba 2014) with mini-batches of 128 512 samples over 200 600 epochs, processing 5, 000
20, 000 mini-batches per epoch for total of
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Vertex
Z
Qubit
X
Z
Plaquette
Figure 10: Surface codes: (left) Toric code with Loode = 4, where gray qubits represent boundary conditions with periodic boundary conditions (top row connects bottom row, left column connects to right column). (right) Rotated surface code with Leade 5. Data qubits adjacent to red faces correspond to Z-type stabilizer generators, while those adjacent to blue faces correspond X-type stabilizer generators.
approximately 2.56 x 10 error samples per training run. The initial learning rate is set between 3 x 10-1 and 1 10-4, with cosine annealing decay to 1 10-6 6 by the end of training (Loshchilov & Hutter 2016). Detailed experimental configurations are presented Table 2 with all experiments conducted on 48GB NVIDIA L40S GPU.
We initialized our development from the QECCT implementation. While longer training and alter- native configurations may yield further improvements, time and computational constraints limited our exploration of the hyperparameter space. We use the toric code implementation from Krastanov & Jiang (2017), while the rotated surface code is implemented from scratch.
To simulate circuit-level noise, we utilized the stim package. were constructed using stim. Circuit......mm generated, with the noise parameters after.clifford www.foorddeepllarzaaiiooo
Quantum circuits
before round_danadd deaataadpplarizatiioo. all set to the physical error rate, p. parameter was explicitly set to zero. To
and letoore.measureefeee flip wwt.:lppppppbliiii.eoappp The wtw.reeet.llip.robblle generate decoding dataset, we extracted the detector error model (DEM)
For the Minimum Weight Perfect Matching (MWPM) baseline, we employ the PyMatching library Higgott (022). For the Belief Propagation with Ordered Statistics Decoding (BP-OSD) decoder, utilize the Idpe Python package Roffe (002)). The Belief Propagation stage is configured with the product-sum algorithm using serial schedule and maximum of 100 iterations. This is followed by an Exhaustive Ordered Statistics Decoding (OSD-E) post-processing stage with order 2, initialized with prior channel error probability derived from the mean of the tested physical error rates.
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Table 2: Experimental configuration across different code distances and noise models.
Code Distance
Code Type
Noise Type
Learning Rate
Epochs
Batch Size
SAQ-Decoder Layers
Model Params
Physical Error Rate
Epoch Time
[sec]
Independent
2.5c-4
200
512
6
1.2M
0.01-0.20
192
Toric
Depolarizing
200
128
1.2M
0:05-0.20
283
Rolated Surface
Independent Depolarzzig
le4
200
128
6
1.2M
0.01-0.20
153
300
128
1.2M
Color
Circuit
2e44
200
512
6
1.23M
0.001-0.02
217
Toric
Depolarizing
Independent
2.504
3e44
400
400
512
512
1,23M
1.2M
0.05-0.20
0.01-0.20
478
Rotated Surface
Independent
1c44
500
512
1.2M
0.01-0.20
245
Depolarizing
36-4
500
512
1.21M
0.05-0.20
584
Color
Circuit
2e-4
200
512
1.29M
0.001-0.02
438
Turic
Depolarizing
Independent
1,5e4
26-4
600
600
128
128
1.22M
1.7M
0.05-0.20
0.01
2464
386
Rotated Surface
Independent
le4
600
512
1.2M
0.01 0.20
285
Depolarzing
le44
600
128
1.63
0.05-0.20
641
Torie
Depolanzing
Independent
3e44
600
600
512
512
L85M
1,26M
0.01-0.20
0.20
3090
683
Rotated Surface
Independent Depolarizing
600
512
1.21M
0.01-0.20
le44
600
512
1.71M
0.05 0.20
1282
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E COMPARISON WITH LEADING NEURAL DECODERS
This appendix provides detailed comparison of the SAQ-Decoder's performance threshold under depolarizing noise against recent neural decoders and aigh-performance classical decoders. Our numerical comparison in Table B] the results confirm that the EAQ-Decoder's is achieved threshold of 18.6% approaches the theoretical Maximum Likelihood (ML) bound of 18.9% (Bombin al., 2012) and represents the highest value reported for practical, scalable decoder.
Table 3: Comparative Thresholds under Depolarizing Noise
Method
Toric
Rotated
SAQ-Decoder (Ours)
18.6% %
18.3%
QECCT (Choukroun & Wolf
2024)
17.8%
17.2%
Astra (Maan & Paler 2025
17.0%
SU-NeIQD (Zhang et al. 2025
16.3%
ML+UF (Memner al. 2022
16.2%
UIUF (Lin & Lai 2025
15.5%
15.6%
To rigorously evaluate the performance of the proposed SAQ decoder, we benchmark it against the latest advancements in neural decoding architectures. The field has recently moved towards so- phisticated deep learning models that aim to exploit the geometric structure of topological codes better than standard generic networks. In this section, compare method with two lead- ing state-of-the-as approaches: the vision- transformer- and the U-Net-style SU-NetQD (Zhang et al. 2025). Our analysis demonstrates that SAQ consis- tently achieves superior logical error suppression compared to these strong baselines. By effectively modeling both local stabilizer constraints and global error patterns, the SAQ decoder establishes new performance standard, particularly in the low-error regimes essential for practical fault-tolerant quantum computation.
QuantumSMoE (Nguyen et al.) 2026)
Recently, QuantumSMoF has emerged as promising decoding framework that effectively adapts vision transformer architectures to the geometric constraints of topological codes. This method explicitly incorporates code structure through specialized plus-shaped embeddings that capture lo- cal stabilizer interactions, adaptive masking to enforce lattice connectivity, and a Soft Mixture-of- Experts layer that enhances model capacity through sparse, conditional computation. While Quan- tumSMoE demonstrates the utility of spatial inductive biases, our proposed SAQ decoder achieves superior logical error suppression across the examined error regimes, as shown in Table 4 For in- stance, on the Leode
torie code a physical error rate of p 0.09, SAQ reduces the logical error rate 3.63 10 compared to 4.92 x 10 for QuantumSMoE, demonstrating robust per- formance improvements in the critical low-error regime and suggesting that SAQ more effectively captures the global topological correlations necessary for high-fidelity decoding.
This performance advantage extends to comparisons with other leading architectures, such as SU- NetQD. As summarized in Table []] the proposed SAQ decoder consistently achieves the lowest logical error rate among all considered decoders across toric code with distances Loode E [5,7] and physical error rates p E {0.09, 0. 0.13, 0.20]. Compared the SU-NetQD architecture, SAQ yields systematically smaller LERs, while the classical BP-OSD-2 and MWPM decoders exhibit substantially higher error rates in the same regime. particular, for distance Leode at p 0.09, SAQ attains logical error rate of 1.95 x 10 whereas SU-NetQD, BP-OSD-2, and MWPM reach 2.76x 10-2, 7.20 10-2, and 6.90 10-2, respectively. This corresponds a relative LER reduction of about 29% with respect to SU-NetQD, and roughly 73% and 72% reductions compared to the BP- OSD-2 and MWPM baselines, respectively, highlighting the substantial performance advantage of the proposed SAQ decoder.
F DETAILED COMPUTATIONAL METRICS
This appendix provides the detailed numerical comparisons of computational complexity and effi- ciency metrics for the SAQ-Decoder against the QECCT baseline. Metrics include the total number of floating-poins operations (FLOPs), the total number of trainable parameters (Params), and the
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Table 4: LER comparison of SAQ, QuantumSMoE, and baseline decoders for toric code with dis- lance Leade at selected physical error rates p.
SAQ
QuantumSMoE
BP-OSD-2
MWPM
0.09
0.0363
0.0492
0.1143
0.1238
0.11
0.0812
0.0985
0.2010
0.2161
Loode 6
0.13
0.1555
0.1690
0.3118
0.3343
0.15
0.2489
0.2560
0.4216
0.4443
Table 5: LER comparison of SAQ ,SU-NeiQD, and baseline decoders for toric code with distances Lode 5, 7 selected physical error rates p.
p
SAQ
SU-NeIQD
BP-OSD-2
MWPM
0.09
0.051
0.057
0.107
0.108
Leode
0.13
0.177
0.191
0.280
0.279
0.20
0.519
0.534
0.623
0.624
0.09
0.019
0.028
0.072
0.069
Lecade
0.13
0.139
0.149
0.262
0.253
0.20
0.548
0.581
0.680
0.676
inference time per sample (Time). Our numerical comparison in Table 6 consistently demonstrates the superior efficiency and scalability of the SAQ-Decoder across all tested code distances (L) and noise models (independent 'ind' and depolarizing 'dep').
G THE USE OF LARGE LANGUAGE MODELS (LLMS)
LLMs were employed assist in several aspects of this research and manuscript preparation. For the literature review, LLMs aided in the identification and sourcing of relevant related works to en- sure comprehensive coverage of the field. During the research process, LLMs were consulted for research ideation, though these explorations did not yield beneficial outcomes that influenced the final work. manuscript preparation, LLMs assisted with improving grammar, enhancing textual transitions between sections, and refining the exposition of technical concepts for better clarity and readability. For software development, LLMs provided assistance in code writing, GPU accelcration optimizations, and debugging code issues. Despite these auxiliary uses, all core research contribu- tions, experimental design, theoretical insights, and scientific conclusions presented in this work are entirely the product of the authors' original research and analysis.
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Table 6: Comparison of SAQ and QECCT Decoder Computational Metrics
Code Type
Loode
Noise
Decoder
FLOPs [M 4]
Params [M 4]
Time [sec
Toric
ind
SAQ
27.48
1.20
2.13e-05
QECCT
57.01
1.23
2.61e-05
dep
SAQ
61.51
1.20
2.59c-05
QECCT
114.09
1.33
8.13e-05
ind
SAQ
55.16
1.20
2.27e-05
QECCT
128.37
1.37
1.01e-04
dep
SAQ
116.88
1.23
6.39e-05
QECCT
257.08
1.90
3.41e-04
ind
SAQ
93.92
1.22
5,01c-05
QECCT
228.45
1.75
2.73e-04
dep
SAQ
259.15
1.70
2.15c-04
QECCT
458.01
3.42
9.72e-04
10
ind
SAQ
143.76
1.26
9.72e-05
QECCT
357.44
2.55
6.28e-04
dep
SAQ
392.09
1.85
4.50e-04
QECCT
717.61
6.64
2.33e-03
Rot. Surf.
SAQ
19.97
1.20
6.63e-06
ind
QECCT
43.94
1.21
1.78e-05
dep
SAQ
38.55
1.20
1.50e-05
QECCT
87.92
1.28
5.49e-05
ind
SAQ
36.57
1.20
1.46e-05
QECCT
86.73
1.27
5.37e-0S
dep
SAQ
71.77
1.21
3.23e-05
QECCT
173.62
1.52
1.75e-04
ind
SAQ
58.72
1.20
2.35e-05
QECCT
143.84
1.41
1.23e-04
dep
SAQ
154.71
1.63
9.15e-05
QECCT
288.13
2.08
4.16e-04
11
ind
SAQ
86.40
1.22
4.49c-05
QECCT
215.33
1.69
2.51e-04
dep
SAQ
228.54
1.68
1.73e-04
QECCT
431.66
3.18
8.87e-04
26


Reviews:
[{"review_id": "a2x4YK3LQS", "reviewer": "Reviewer_H6iu", "summary": "SAQ-Decoder proposes a stabilizer-aware decoder for Quantum error codes, that combines existing transformer-based decoding approaches with a constraint-inducing post-processing. The key idea is to use a dual-stream representation - one for syndromes and one for logical information. These streams are then processed by a transformer, which is followed by a lightweight constraint-preserving post-processing stage that enforces exact syndrome consistency.\n\nThe architecture attains near-ML thresholds on toric codes, while maintaining linear scaling in syndrome size and strong parameter efficiency. The method outperforms MWPM and prior neural baseline QECCT in both accuracy and scalability. Similar gains hold on rotated surface codes, indicating generalization across codes.", "questions": "-", "limitations": "", "rating": 6, "confidence": 1, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The method is well-motivated and seems technically sound, combining learned decoding with constraint-preserving post-processing is a clear and sensible idea.\n\n2. The results shown are strong: the decoder achieves near-ML thresholds on toric and rotated surface codes while maintaining linear scaling and good parameter efficiency.\n\n3. The approach seems to generalize well across different noise models and code families.", "weaknesses": "I do not have sufficient background in quantum error correction to fully assess the novelty of the proposed method relative to prior decoders.\nWhile the approach appears reasonable and the empirical performance is impressive, I am unable to evaluate the theoretical aspects of stabilizer enforcement or the loss formulation.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Y3sYQhClS3", "reviewer": "Reviewer_ehw9", "summary": "The authors propose a transformer-based quantum decoder, called the SAQ-decoder, with several innovative features: dual-stream transformers, a differentiable logical loss, and constraint-aware post-processing. The proposed decoder achieves near-optimal accuracy for topological codes.", "questions": "In Fig. 6, are the authors testing the effect of both masking and the global token? Does the \"no mask\" label mean only the global token is used, and \"mask only\" means only masking is applied? Is SAQ-decoder the version with both features? The labeling is somewhat confusing. Also, it would be helpful to include results where neither masking nor the global token is used.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "The dual-stream transformer architecture is intriguing, and the experimental results appear very promising.", "weaknesses": "1.\tIt’s unclear what the final output of the proposed decoder actually is. Is it \\hat{l}, \\hat{e}, or the output of the CPND? Based on the appendix, it seems the final output used for evaluating LER is from the CPND, but this should be clarified more clearly in the main text. Also, e^{pred} above Eq. (13) appears to represent the final output, but this is not explicitly stated.\n2.\tTo my knowledge, other transformer-based decoders for QEC already exist, such as alphaQubit. These should be cited, and the differences explained. Additionally, the dual-stream transformer architecture seems similar to CrossMPT for classical codes, which also uses cross-attention. That work is worth referencing as well.\n3.\tThe MLP ϕb_\\phi seems technically identical to the FFN in Kai (2022, PRL), as it takes the syndrome vector and outputs logical information. This similarity should be acknowledged.\n4.\tThe asymmetric structure of the dual streams is not intuitive. Why does the syndrome stream (which contains local information) influence the logical stream (which contains global information)? The opposite direction seems more meaningful. Without empirical results, can the authors illustrate the reasoning behind this design? Moreover, since the syndrome stream is not affected by the logical stream, does this mean the error vector is derived directly from the input syndrome? If so, how does this differ from QECCT?\n5.\tIs the gain over QECCT mainly due to the novel transformer architecture or the post-processing via CPND? To make a fair comparison, the SAQ-decoder should also be compared to QECCT without CPND.\n6.\tThe non-differentiability of the loss function L(e_{\\text{true}} + e_{pred}) has already been addressed by Liu (2019, PRL). Could you explain the difference?\n7.\tThe authors use MWPM as the standard classical decoder. However, BP+OSD is also a well-known and widely used decoder for quantum codes and should be included in the comparison.\n8.\tThe computational complexity is only discussed using Big-O notation. It would be more informative to also provide numerical comparisons, such as FLOPs or inference time.\n9.\tRegarding parameter efficiency, the SAQ-decoder still relies on syndrome information, which scales quadratically with the lattice size LLL. Therefore, I think it cannot avoid quadratic scaling.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "erN2ot01s3", "reviewer": "Reviewer_wwv5", "summary": "The paper introduces SAQ-Decoder. It is a neural network approach for QEC that addresses the fundamental accuracy efficiency trade off in QEC decoding. The method combines a dual-stream transformer architecture with a novel differentiable logical loss function and a post-processing algorithm, constraint projected null space descent. The approach achieves near optimal error thresholds while maintaining linear computational complexity.", "questions": "1. The ablation studies don’t clearly show CPND’s contribution versus simpler projection. Can you provide a direct comparison of logical error rates with/without CPND?\n2. Does decoder require retraining for different physical error models?\n3. What are the convergence guarantees for CPND?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1.  Strong empirical performance. The decoder achieves very good results on toric codes, also approaching maximum bounds while outperforming both classical methods and recent neural approaches.\n2. Novel architectural design. The dual-stream transformer with asymmetric attention patterns is well-motivated for QEC.\n3. Differentiable logical loss. Paper provides a rigorous mathematical derivation of a differentiable approximation to the discrete GF(2) constraints, enabling end to end training that directly optimizes logical error rates rather than bit error rates.", "weaknesses": "1. The experiment limited to code distances up to 10. For practical fault-tolerant quantum computing should be larger.\n2. The paper lacks comparison with recent strong baselines. \n3. While the method clams applicability to any stabilizer code family, but experiments are limited to surface codes.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "jtlN8dWH3q", "reviewer": "Reviewer_xQqG", "summary": "I would like to note that I am not familiar with the quantum computing or quantum machine learning domain, and therefore I do not have the necessary background to properly evaluate this submission. While I can assess general aspects of clarity and structure, I am not qualified to judge the technical novelty, correctness, or significance of the paper within the context of quantum research.\n\nI respectfully suggest that this paper be reassigned to a reviewer with expertise in quantum algorithms or quantum information, to ensure a fair and accurate evaluation.", "questions": "See Summary", "limitations": "", "rating": 4, "confidence": 1, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "See Summary", "weaknesses": "See Summary", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)