Static Auxiliary-Graph Ensemble
Implementation & benchmark of arXiv:2607.12187 — Auxiliary Nodes for BP Decoding of Quantum LDPC Codes
Source paper: Auxiliary Nodes for BP Decoding of Quantum LDPC Codes arXiv:2607.12187 ⓘ · analyzed Aug 30, 2026
AI-generated research hypothesis, automatically tested. Not peer-reviewed.
Idea description
Run the same neural decoder over several algebraically equivalent augmented graphs and aggregate their variable-level predictions. Each graph exposes different cycle structure and message routes, providing structured architectural diversity rather than ordinary random-seed ensembling.
Formulas
Mathematical statement
For each admissible binary transformation A_q, the augmented pair satisfies H_q = A_q H and s_q = A_q s, where H is the original parity-check matrix, s is the syndrome, and q indexes graph variants. Thus H_q e = s_q whenever H e = s. Let z_qj be the decoder logit for variable j from graph q and let w_q be a nonnegative graph weight with sum_q w_q = 1. The ensemble logit is z_j = sum_q w_q z_qj and p_j = sigmoid(z_j). A residual-weighted variant uses r_q as the number or soft magnitude of violated augmented checks and sets w_q proportional to exp(-beta r_q), where beta is a tunable reliability coefficient. The paper's repeated-round evaluation uses p_L = 1 - (1 - LER)^(1/r), where LER is the logical error rate and r is the number of syndrome rounds.
Implementation notes
Integrate this as an inference-time ensemble wrapper around a neural BP or GNN decoder, using shared weights to limit parameter and memory overhead. Generate E sparse transformations A_q offline. Each should contain the identity rows plus independently selected low-weight combinations of existing checks, chosen to reduce length-4 cycles while keeping H_q sparse. Precompute H_q = (A_q H) mod 2 and compute s_q = (A_q s) mod 2 for every input syndrome. Pseudocode: for q in 1..E, run the shared decoder f_theta(channel, H_q, s_q) for T iterations and obtain variable logits z_q; set z = mean_q z_q and decode sign(z). Optionally compute a soft residual r_q from H_q and sigmoid(z_q), set w_q = softmax(-beta r_q), and use z = sum_q w_q z_q. Train with per-graph binary cross-entropy plus an ensemble loss and syndrome-consistency penalty. The paper-derived component is the algebraically equivalent graph family; logit aggregation and residual weighting are the neural adaptation. Begin with E=4 and E=8 on a small LDPC or quantum-DEM-inspired sparse binary benchmark. Compare against one decoder, a wider decoder with equal FLOPs, and ordinary seed ensembling. Measure block or logical error rate, calibration, error versus iteration count, and latency. Success is at least a 20% error reduction at fixed total message evaluations or equal error with fewer iterations per graph.
Verification
Stage 1 · Toy mechanism gate: Failed ✗
Stage 2 · Mechanism transferred to benchmark: Not tested
Stage 2 · Practical benchmark result: Not run
Methodology: Toy-system gate first; the benchmark stage runs only after a pass. How verification works
Stage 1 — Mechanism check agent confidence 9/10
Built a readable MVP for the Static Auxiliary-Graph Ensemble in experiment.py, including GF(2) row-transform generation, exact syndrome-equivalence checks, logit aggregation, and a sparse parity-decoding mini-experiment. The mechanism predictions were confirmed: transformed syndromes were exact, averaged independent-logit variance matched sigma^2/E within about 0.6%, and repeated-round LER inversion was numerically exact. However, the practical decoder gain was weak: BER was 0.06092 for one graph, 0.05779 for four auxiliary graphs, 0.05729 for eight, and 0.05823 for four ordinary seeds, so the stated 20% target and clear structured-diversity advantage were not observed.
- Agent confidence
- 9/10
- Baseline
- Single graph BER 0.0609187; ordinary 4-seed ensemble BER 0.0582262.
- Idea
- Static auxiliary graph ensemble: E=4 BER 0.0577937, a 5.1% relative reduction versus single; E=8 BER 0.0572857, a 6.0% relative reduction. Variance sweep observed/predicted ratios for E=1,2,4,8,16 were 1.006, 0.500, 0.250, 0.125, and 0.062.
Limitations: This is a synthetic, non-trained message-voting decoder rather than a trained BP/GNN decoder; no LDPC benchmark, calibration, latency/FLOP measurement, residual weighting, wider equal-FLOP model, or multi-round logical-error experiment was tested. The toy uses an equivalent transformed syndrome computed directly from the sampled error, and graph diversity was not optimized for cycle reduction.
How to run: python3 experiment.py
Verdict computed by deterministic test code from paired-seed statistics — not by the language model.
Artifacts
- experiment.py 3.7 KB View Raw
- report.md 1.7 KB View