Trapping Sets of Detector Error Models

Abstract: Message-passing decoders are among the most promising candidates for scalable quantum error correction, yet their behavior in the low-error-rate regime remains poorly understood under realistic circuit-level noise. In this work, we introduce a systematic framework for identifying the graph structures that govern decoder failures and for using them to predict the resulting error floor. We apply exhaustive trapping-set enumeration directly to the detector error model of a bivariate bicycle code and test all low-weight fault configurations supported on the resulting structures. This converts the analysis of extremely rare logical failures into a finite structural search, avoiding the prohibitive cost of direct Monte Carlo simulation. We evaluate the framework on three iterative decoders with substantially different architectures and decoding heuristics. Remarkably, for \texttt{RelayBP}, the resulting prediction accurately reproduces the simulated error floor; for the others, it remains within the same order of magnitude. Despite their differences, leafless elementary trapping sets capture a substantial part of the low-weight error-floor contribution for all three decoders. Moreover, each decoder admits failures caused by fault configurations well below the correction capability implied by the circuit-level distance, revealing a substantial gap between code distance and practical iterative-decoding performance. These results establish trapping-set analysis as a practical framework for predicting error floors, exposing the structural weaknesses of iterative decoders, and guiding the joint design of decoding algorithms.

• The paper develops a systematic framework that enumerates leafless elementary trapping sets in circuit-level detector error models and tests their contribution to QLDPC decoder failures.
• Applied to the [[144,12,12]] bivariate bicycle code, the method identifies 92,088,583 trapping-set instances and finds no failures below weight three, while decoder-dependent failures begin at weight three or four.
• The analysis closely predicts RelayBP’s simulated error floor and shows that a small set of graph topologies causes most failures, offering targets for trapping-set-aware decoder design despite incomplete coverage of larger or non-elementary structures.

This paper develops a systematic framework for characterizing the failure mechanisms of message-passing decoders for quantum low-density parity-check (QLDPC) codes under circuit-level noise, by transplanting the classical trapping-set (TS) methodology to detector error models (DEMs). The authors apply exhaustive leafless elementary trapping set (LETS) enumeration directly to the DEM of the $[[144,12,12]]$ bivariate bicycle (BB) code, exhaustively test low-weight fault patterns supported on these structures for three architecturally distinct iterative decoders, and derive a lower bound on the error floor that closely matches Monte Carlo simulation for one decoder and remains within an order of magnitude for the others.

Motivation and positioning

Iterative decoders are attractive for QLDPC codes because of their low complexity and parallelism, but they generally fail to correct all errors up to the code's minimum distance and exhibit error floors caused by small-weight trapping sets. Prior TS analyses of QLDPC codes were restricted to code-capacity noise models, where Tanner graphs are small and regular; circuit-level DEMs present large, highly irregular Tanner graphs that had resisted such analysis. The paper positions itself against two complementary lines of work: analytical characterizations such as absorbing-set conditions for Gallager-B decoding (Morris et al., 2023), which do not extend to decoders with randomized schedules, ensembles, and re-initialization heuristics; and the failure-spectrum approach of Beverland et al. (Beverland et al., 19 Nov 2025), which estimates rare-event logical rates without retaining structural information about the failing patterns. The TS-based approach retains both: a topological catalog of harmful structures reusable across decoders, and a computationally efficient route to error-floor prediction.

Trapping sets in detector error models

The framework rests on extending classical TS definitions to the DEM setting. An $(a,b)$ trapping set is a subgraph induced by $a$ variable nodes whose neighborhood contains $b$ odd-degree checks. The paper emphasizes leafless elementary trapping sets, in which every check has induced degree one or two and every variable node connects to at least two degree-two checks—a class known in classical LDPC theory to dominate low-weight failures. Quantum trapping sets (QTSs), defined as supports of stabilizers, are handled uniformly as $(a,0)$ structures; crucially, the definition carries over to DEMs: a fault configuration $F$ is a stabilizer of the DEM if it produces neither detector events nor observable flips, i.e., it lies in the kernel of the combined map $$\begin{bmatrix} D & L \end{bmatrix}$$.

Enumeration uses the dot-path-lollipop (dpl-search) algorithm of Hashemi and Banihashemi [bani_irregular], which grows LETSs from short cycles via three complete expansion primitives—dot, path, and lollipop—guided by an expansion table. Applied to the DEM of the BB code with syndrome-extraction circuits from Bravyi et al., the search yields 92,088,583 LETS instances for $a_{\max}=5$, $b_{\max}=5$. The search is capped at this depth because larger parameters overflow memory; whether the resulting truncation suffices is validated empirically rather than proven.

Error floor estimation methodology

After enumeration, fault patterns of weight one through four are exhaustively injected on the union of LETS supports (an upper bound of roughly 2.69 billion patterns before deduplication) and decoded. No weight-one or weight-two failures occur for any decoder; RelayBP and ImpulseBP first fail at weight four, ELMS at weight three. For deterministic decoders, summing the probabilities of failing supports gives an exact evaluation of the identified contribution and a lower bound on the full logical error rate. For randomized decoders (RelayBP, ELMS), each support is decoded once and the binary outcome weighted by its probability; the estimator is unbiased since $\mathbb{E}[X_S]=q_S$, with variance bounded by $(a,b)$0. Decoder priors are fixed at $(a,b)$1 across the sweep, and experiment-level probabilities are converted to per-round rates via $(a,b)$2 with $(a,b)$3 rounds.

The three benchmarked decoders are deliberately heterogeneous: RelayBP (randomized sequential re-initializations, local), ImpulseBP (deterministic ensemble with global prior modification), and ELMS, a new ensemble of 20 layered min-sum decoders with randomized layer ordering and damped messages ($(a,b)$4), introduced primarily as a simpler benchmark.

Results

For RelayBP, the LETS-based estimate closely reproduces the Monte Carlo error floor; for ImpulseBP and ELMS the estimate lies below simulation but within one order of magnitude, indicating that some failures escape the enumerated structure space (possibly requiring $(a,b)$5, ETSs with leaves, or non-elementary TSs).

Class-level statistics reveal strong non-monotonicity: failures concentrate in the $(a,b)$6 and $(a,b)$7 classes while $(a,b)$8, $(a,b)$9, $a$0, $a$1, and $a$2 produce none. Topology-level analysis (isomorphism classes of induced subgraphs) shows extreme concentration:

Decoder	Failing instances	Harmful topologies	Share of top 3
RelayBP	152,857	71 / 2851 (2.49%)	76.29%
ImpulseBP	42,624	28 / 2851 (0.98%)	68.07%
ELMS	3,166,542	647 / 2851 (22.69%)	30.93%

RelayBP and ImpulseBP share an identical top-three topology set ($a$3 in $a$4, $a$5 in $a$6, $a$7 in $a$8), with $a$9 alone accounting for over 42% of failures for both. These structures remain among ELMS's most harmful (ranks 6, 10, 17). Concentration cannot be explained by abundance: $b$0 constitutes only 0.107% of $b$1 instances yet accounts for 53.8% of RelayBP's failures in that class. Most harmful topologies are "mixed" (containing both failing and succeeding instances), so topology indicates vulnerability but does not deterministically predict outcome—location in the Tanner graph, priors, and injected pattern also matter.

A notable finding is that all three decoders fail on certain weight-four (and for ELMS, weight-three) fault patterns, despite the code's circuit-level distance of 12 implying correction of all faults up to weight five. This exposes a substantial gap between bounded-distance capability and practical iterative-decoding performance.

Limitations and open questions

The paper concedes several constraints explicitly. The exponential growth of candidate structures limits exhaustive enumeration to moderate sizes; the implementation becomes memory-prohibitive beyond $b$2, and it is unclear whether the approach remains both sufficient and feasible for larger-distance codes such as the $b$3 BB code. The restriction to LETSs excludes ETSs with leaves and non-elementary TSs; the accuracy of the resulting lower bound for ImpulseBP and ELMS suggests—but does not prove—that unmodeled structures contribute meaningfully to their floors. The vast majority of enumerated LETSs are harmless, so membership in an $b$4 class alone is not predictive, and the roles of quantum degeneracy and nearby stabilizer structures remain unquantified. Finally, the applicability of the framework to AI-based decoders is left entirely open.

Conclusion

The paper establishes that exhaustive trapping-set analysis, originally developed for classical LDPC codes, transfers effectively to circuit-level detector error models of QLDPC codes. It delivers accurate error-floor prediction without Monte Carlo simulation at very low error rates, identifies a small shared core of highly vulnerable local graph structures across decoders with fundamentally different heuristics, and quantifies the shortfall of iterative decoding relative to code distance. The enumerated catalog of harmful LETSs provides a concrete target set for the design of trapping-set-aware decoders aimed at eliminating low-weight failures.