Does trajectory noise train as well as the exact channel?

The study behind #311. noise_method="trajectories" (#229) samples depolarizing noise as Pauli trajectories at pure-state cost; its gradient is unbiased but noisier than the exact "density" one. This compares the two, and noiseless training, on held-out data.

Setup

• Script: examples/study_trajectory_noise.py (since the run, only the class balance in its docstring was corrected). Raw results, one JSON line per run: docs/results/trajectory_noise_study.jsonl.
• Data: scikit-learn's breast-cancer set (569 samples, 63 % positive: benign is class 1). It is bundled with scikit-learn, so the study runs offline; the credit-card and UCI benchmarks named in the issue were not used (see Limits).
• Splits: for each of 5 seeds, a stratified 60/20/20 train/validation/test split, with standardisation and PCA fitted on the training rows. The seed also fixes the initial weights, so every method starts from the same point.
• Model: HybridBinaryClassifier(n_qubits, n_qubits, 2) with 4 and 6 qubits, on default.qubit with backprop, trained with focal loss, Adam (lr 0.05), batch 32, up to 30 epochs, and MCC early stopping (patience 10).
• Methods: noiseless, density, trajectories with k = 1 and k = 4 draws per sample, at p = 0.01 and 0.05 after every gate ("all"), and p = 0.05 before measurement ("end").
• Scores: test MCC clean (validation threshold), and test MCC under the training noise (threshold re-tuned on the validation rows under that noise, as a model deployed on a noisy device would be).
• Environment: PennyLane 0.45.1, torch 2.14, one laptop CPU; 120 runs.

Results

Test MCC under the training noise, mean ± sd over 5 seeds, and seconds per run. In the end rows this score equals the clean one by construction (see Limits), so those rows compare the training methods only:

qubits	p	position	noiseless	density	trajectories k=1	trajectories k=4
4	0.01	all	0.850 ± 0.071	0.861 ± 0.094 (17 s)	0.902 ± 0.024 (8 s)	0.893 ± 0.064 (12 s)
4	0.05	all	0.848 ± 0.079	0.872 ± 0.050 (22 s)	0.825 ± 0.085 (10 s)	0.713 ± 0.326 (10 s)
4	0.05	end	0.846 ± 0.072	0.839 ± 0.039 (13 s)	0.856 ± 0.059 (6 s)	0.882 ± 0.037 (6 s)
6	0.01	all	0.875 ± 0.068	0.876 ± 0.087 (137 s)	0.927 ± 0.039 (13 s)	0.905 ± 0.067 (16 s)
6	0.05	all	0.855 ± 0.065	0.883 ± 0.053 (120 s)	0.673 ± 0.351 (16 s)	0.922 ± 0.040 (17 s)
6	0.05	end	0.847 ± 0.072	0.890 ± 0.035 (60 s)	0.884 ± 0.058 (6 s)	0.879 ± 0.101 (7 s)

The noiseless model is trained once per seed, taking about 5 s. Total training time: density 1843 s, trajectories 298 s (k=1) and 340 s (k=4).

What this shows

Everything below was measured on the breast-cancer proxy only: one small dataset, 4 and 6 qubits, 5 seeds. It has not been checked on the credit-card or UCI data that #311 names; that is #414.

1. At p = 0.01, and with noise only before measurement, trajectories train as well as the exact channel. Paired by seed, the mean difference from density is between −0.01 and +0.05, which is within the seed-to-seed spread.
2. At p = 0.05 after every gate, trajectory training was slow to start within this budget, and at least once did not start. Calling a run collapsed when its test MCC under the noise is below 0.5 (the cut-off of the follow-up study, #347), 3 of 20 trajectory runs collapsed, against 0 of 10 density runs:
3. k=1 at 6 qubits, seeds 0 and 4: test MCC 0.19 and 0.42;
4. k=4 at 4 qubits, seed 1: test MCC 0.13.

These are not three failures to train. All three were still on the initial loss plateau (training loss about 0.07) when the 30-epoch, patience-10 budget ended them; the k=4 run was stopped at epoch 15 and restored to its epoch-5 weights. Re-run during review with no early stopping and 60 epochs (not in the committed data): - k=4, 4 qubits, seed 1 left the plateau near epoch 25 and reached test MCC 0.94; - k=1, 6 qubits, seed 4 reached test MCC 0.82; - k=1, 6 qubits, seed 0 stayed on the plateau (loss 0.071, test MCC 0.23).

So one of the three is a stall and two are slow starts that the budget cut off. Every density run trained within the same budget. In the other runs trajectories matched density, and k=4 at 6 qubits beat it on 4 of 5 seeds. The extra gradient variance is the likely cause of the longer plateau; whether k=4 prevents stalls cannot be told from one stall. Separating the two properly is #480. 3. Cost. Per epoch, trajectories trained about 2× faster at 4 qubits (1.8–2.5× across the settings) and 7–11× faster at 6. Density becomes impractical past about 6 qubits (#229), where trajectories keep pure-state memory. 4. Noise-aware training did not beat noiseless training here, even under noise: the noiseless model scores within 0.05 of density in every setting. For the two end settings that is a statement about clean accuracy only. At these noise levels on this data the noise is too weak to hurt a noiselessly trained model much, so this study says nothing either way about noise-aware training's value at stronger noise.

Recommendation

• Keep noise_method="density" as the default wherever it fits in memory (up to about 6 qubits). It never failed here.
• Beyond that, use "trajectories" with noise_trajectories ≥ 4, give it a longer training budget than density needs (more epochs, and a patience well above 10), and check the runs, especially at noise strengths of a few percent per gate. With 5 seeds, a run that has not left the plateau is visible as an outlier in the seed spread. The follow-up in trajectory-collapse-study.md (#347) raises this to ≥ 8: at k = 8 every run trained within the same budget as density.
• Do not switch the default automatically by qubit count on this evidence. A slower start and an occasional stall are behaviour changes that a default should not introduce silently.

Limits

• One small, easy dataset (noiseless MCC ≈ 0.85), 4 and 6 qubits, 5 seeds. With 5 paired seeds, a Wilcoxon signed-rank test cannot go below p = 0.0625, so none of the differences above is statistically significant. The slower start of trajectory training at p = 0.05 after every gate is the robust observation.
• Training budget. Every run had at most 30 epochs, with early stopping (patience 10) and checkpoint selection on the clean validation rows: train_model validates in eval mode, which is the noiseless circuit. There is no longer-budget control in the committed data, so it cannot separate "starts later" from "does not train"; the three re-runs under finding 2 are the only evidence, and #480 is the full control.
• The end rows do not test evaluation under noise. Depolarizing before measurement scales every ⟨Z⟩ by the same factor 1 − 4p/3, and the head is one linear layer, so the samples keep their order and the re-tuned threshold returns the clean labels: the noisy and the clean score agree in all 40 end rows.
• The issue asked for the credit-card and UCI benchmarks through run_benchmark. When the study ran, the loaders (#266) and the benchmark runner (#269–#297) were on other open stacks. They have since been merged into this branch's base, but the study has not been re-run on them, for two reasons. First, cost: a 6-qubit density step at batch 256 took 9.1 s, and a credit-card training fold is about 364k rows after SMOTE balancing (about 1,420 steps), so one epoch of one fold would take about 3.6 h. Second, run_benchmark does not return its trained models, so noise_sweep cannot score them. The re-run is #414.
• The learning rate and schedule were not tuned per method. #347 tried lower learning rates within the same budget (trajectory-collapse-study.md): they did not help there, which a longer plateau would also explain. Whether they shorten or lengthen it is #480.