Integer Linear Programming Decoder for Abelian and Non-Abelian Topological Codes

Abstract: Topological orders (TOs) are widely used as quantum error-correcting codes, with anyon excitations serving as error syndromes. For certain Abelian TOs, decoding can be performed by independently matching particle-antiparticle pairs of each species. However, matching-based decoders cannot handle more general fusion rules in either Abelian or non-Abelian TOs, nor account for noise that correlates different anyon species. While clustering decoders are more broadly applicable, they typically neglect anyon data and fusion properties, leading to poor performance in practice. In this work, we introduce a fundamentally different decoder for arbitrary TOs based on integer linear programming (ILP). The ILP formulation linearizes the error-correction problem through the introduction of auxiliary variables and encodes fusion rules as linear constraints. Classical optimization then identifies the minimum-weight error configuration. As concrete examples, we determine error-correction thresholds for three TOs: the Abelian $\mathbb{Z}_2$ TO under depolarizing noise, where charge and flux errors are correlated; the Abelian $\mathbb{Z}_3$ TO, which does not admit a pairwise matching decoder; and the non-Abelian $D_4$ TO under noise channels that generate all anyon species. We demonstrate the versatility of the ILP decoder by showing a clear performance advantage over most existing decoders in all three cases. We further extend the method to incorporate noisy syndrome measurements and propose a just-in-time variant for continuous error correction. Our results establish ILP as a natural framework for handling correlated errors and general anyon fusion rules, and as a powerful and flexible general-purpose decoder for incoherent anyon noise in arbitrary TOs, with applications to fault-tolerant quantum computation.

• The paper introduces a sparse integer linear programming decoder that directly encodes anyon fusion rules, correlated error channels, and homology constraints for arbitrary Abelian and non-Abelian topological orders.
• The decoder reaches thresholds of 18.039% for the depolarizing-noise toric code and 15.346% for the Z3 quantum double, closely approaching estimated optimal performance and exceeding matching or renormalization-group methods.
• Benchmarks on non-Abelian D4 topological order show that effective weight tuning outperforms two-step MWPM, while spacetime and just-in-time extensions remain promising but require numerical validation and stronger runtime analysis.

The paper introduces a minimum-weight decoder for topological quantum error correction based on integer linear programming (ILP), applicable to arbitrary topological orders (TOs), Abelian and non-Abelian alike (2608.18512). Its central observation is that anyon fusion rules, which are the structural obstacle for matching-based decoders and are ignored by clustering decoders, can be encoded directly as linear constraints on binary decision variables. Classical optimization then identifies the most probable error configuration consistent with the measured syndrome, yielding the non-Abelian generalization of minimum-weight perfect matching (MWPM). The authors benchmark this approach on three models — the toric ($\mathbb{Z}_2$) code under depolarizing noise, the $\mathbb{Z}_3$ quantum double under incoherent charge noise, and the non-Abelian $D_4$ topological order on the three-colorable kagome lattice under single-qubit Pauli noise generating all anyon species.

Motivation and relation to prior decoding approaches

Existing decoders divide into two families, each with a structural deficiency. Matching-based decoders exploit anyon data but apply only when anyons are self-antiparticles with acyclic fusion rules; they fail entirely for cyclic fusion rules and cannot represent noise correlating different anyon species. Clustering (renormalization-group) decoders apply to arbitrary TOs but discard fusion and braiding information, resulting in numerical thresholds well below those achieved by MWPM for Abelian codes. Linear programming has previously appeared mainly in the qLDPC literature, where syndromes behave like Abelian anyons with trivial particle–antiparticle fusion; there its expressive power is largely unused and belief propagation is typically preferred.

The ILP framework addresses both deficiencies simultaneously. Error strings may terminate not only on measured syndromes but also on other error strings according to the fusion rules of the underlying TO — a configuration class inaccessible to matching or clustering decoders. Correlated errors across species (e.g., Pauli $\hat{Y}$ creating $e$ and $m$ anyons jointly) enter naturally through constraints imposed by the noise model.

The general ILP formulation

Decoding proceeds from Bayes' rule, approximating $P(h|\bm{\sigma})$ by its dominant contribution within each homology class $h$, i.e., maximizing $P(\bm{\sigma}|E)P(E)$. The objective function is chosen as $\ln[P(\bm{\sigma}|E)P(E)]$, so that products of local probabilities become additive weights $\mathbb{Z}_3$0 for physical errors and $\mathbb{Z}_3$1 for nondeterministic fusion channels at site $\mathbb{Z}_3$2.

Three classes of binary decision variables suffice: (i) error-activation variables $\mathbb{Z}_3$3 in one-to-one correspondence with local incoherent anyon-creation events; (ii) fusion variables $\mathbb{Z}_3$4 selecting, at each site with measured anyon $\mathbb{Z}_3$5, one allowed fusion channel $\mathbb{Z}_3$6; and (iii) auxiliary integer variables implementing parity-like conditions that linear algebra cannot express via modulo arithmetic. The key constraints enforce uniqueness of the selected fusion channel per site,

$\mathbb{Z}_3$7

and consistency between incident error-generated anyons and the multiplicities $\mathbb{Z}_3$8 of the selected channel,

$\mathbb{Z}_3$9

Both the number of variables and constraints scale approximately linearly with code distance times the number of fusion channels, so the formulation remains sparse. The authors note candidly that for non-Abelian TOs, $D_4$0 is not exactly a product of local factors: global consistency conditions requiring isolated homologically trivial components to fuse to vacuum are neglected, though such configurations occur rarely and have minimal effect on performance.

Results for Abelian codes

For the $D_4$1 toric code under single-qubit depolarizing noise ($D_4$2), the ILP decoder achieves a threshold of 18.039(7)%, exceeding uncorrelated MWPM's 15.5(5)% and outperforming RG-BP, MPS, BP-MWPM, BP-ADOSD, deep Q-learning, and UIUF decoders, while approaching the optimal value of 18.9(3)% obtained from Monte Carlo sampling and duality arguments. The gain over MWPM stems precisely from exploiting $D_4$3/$D_4$4 correlations induced by $D_4$5 errors, which pairwise matching cannot represent. Runtime analysis shows that although minimum-weight decoding here is NP-hard, both mean and median solver runtimes scale consistently polynomially below threshold over accessible distances, with near-threshold scaling inconsistent with simple exponential dependence on system size.

For the $D_4$6 TO charge sector, where fusion rules include $D_4$7, so no pairwise matching decoder exists, the ILP decoder achieves 15.346(5)%, close to the optimal 15.8(2)% and substantially above all reported renormalization-group thresholds, none exceeding 13%. This demonstrates that the framework captures nontrivial deterministic fusion without sacrificing near-optimal performance.

Results for the non-Abelian $D_4$8 topological order

The $D_4$9 TO contains three non-Abelian $\hat{Y}$0 and three Abelian $\hat{Y}$1 anyons ($\hat{Y}$2) with fusion rules such as $\hat{Y}$3. The authors first establish the phase diagram: unlike its Abelian $\hat{Y}$4 parent on the same lattice, which exhibits two distinct classical-memory phases, the $\hat{Y}$5 TO has only one classical-memory phase, since proliferation of non-Abelian $\hat{Y}$6-anyons necessarily induces proliferation of Abelian $\hat{Y}$7-anyons through their fusion rules. This phase structure holds for all decoder variants tested, indicating it is intrinsic to the TO rather than decoder-dependent.

Against the two-step MWPM baseline, three ILP variants are compared:

Decoder variant	Behavior
Maximize $\hat{Y}$8 only	Best at small $\hat{Y}$9 via intrinsic heralding; degrades sharply at large $e$0
Maximize $e$1	Robust at large $e$2; slightly better than MWPM everywhere
Effective weight ratio $e$3	Outperforms all variants across the full parameter range

The physics behind these differences is instructive. At small $e$4, Abelian $e$5-anyons appear predominantly along $e$6-anyon error strings and provide reliable intrinsic heralding; maximizing only $e$7 exploits this aggressively. As $e$8 grows, heralding becomes unreliable and misguides correction. Including $e$9 acts as an entropic penalty favoring shorter $m$0-anyon strings, weakening heralding appropriately. Tuning the single ratio $m$1 interpolates between regimes; because the ILP solution changes discretely, optimal ratios lie in finite intervals and require no fine tuning. The effective-weight variant achieves the highest thresholds among all tested decoders, establishing ILP as superior to currently available alternatives for this non-Abelian TO with perfect syndromes.

Extension to noisy measurements and just-in-time decoding

The framework extends to spacetime decoding under imperfect syndrome measurements by lifting the lattice to $m$2 sites and introducing binary measurement-error variables $m$3. A notable feature specific to non-Abelian TOs is that defects alone no longer suffice to specify allowed configurations: because fusions such as $m$4 can create an unreported anyon without a defect, both the defect pattern and the measured syndrome must be provided as input. The spacetime constraints generalize the spatial ones with separate bookkeeping for spatial string multiplicities and incoming/outgoing temporal strings representing measurement errors.

Building on this, the authors propose a just-in-time ILP algorithm for continuous error correction: clusters of uncorrected defects are decoded against the syndrome history, and a cluster is corrected only once its age exceeds its spacetime extent, ensuring measurement errors can be inferred reliably. This mirrors just-in-time protocols developed for the $m$5 TO with matching decoders. However, this component remains a proof of principle: the numerical implementation and performance benchmarking of the just-in-time ILP decoder are left open, as is a rigorous threshold proof, which would require showing that hierarchical cluster correction does not spread temporally into higher hierarchy levels.

Limitations and open questions

Several limitations deserve emphasis. First, worst-case runtime guarantees are unavailable: minimum-weight decoding of the $m$6 TO under Pauli noise is NP-hard, so polynomial scaling is observed empirically in the subthreshold regime but not guaranteed, and the near-threshold behavior could still deteriorate beyond accessible sizes. Second, the weight assignment neglects global fusion-consistency conditions for isolated homologically trivial components of non-Abelian error strings, justified only empirically by their rarity. Third, the $m$7 benchmarks assume perfect syndrome measurements; the spacetime and just-in-time formulations have not been numerically validated. Fourth, the effective-weight decoder requires learning $m$8 from device data, albeit over robust finite intervals rather than fine-tuned points. Finally, information-theoretic analyses suggest local anyon-syndrome measurements may not capture all recoverable information in decohered non-Abelian TOs, raising the question of whether collective or adaptive measurements should be integrated with optimization-based decoders — particularly relevant for continuum and chiral phases lacking commuting local syndrome observables.

Conclusion

This work establishes integer linear programming as a general-purpose minimum-weight decoder for arbitrary topological orders. By encoding fusion rules and inter-species noise correlations as linear constraints, it achieves thresholds approaching optimality for Abelian codes under correlated noise (18.039% for $m$9, 15.346% for $P(h|\bm{\sigma})$0) and outperforming existing decoders for the non-Abelian $P(h|\bm{\sigma})$1 TO under noise generating all anyon species. The remaining gaps — rigorous complexity analysis, validated spacetime decoding, and integration with more expressive measurement schemes — define the concrete questions the paper leaves open.