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Can the collapse of trajectory-noise training be prevented?

The study behind #347. #311 (trajectory-noise-study.md) found that training with noise_method="trajectories" at p = 0.05 after every gate left 3 of 20 runs below a test MCC of 0.5 within its training budget, while the exact "density" channel always trained. This tests four mitigations over twice the seeds, with the same budget.

Read every collapse count below as "had not trained within 30 epochs and a patience of 10". The study has no longer-budget control, so it cannot tell a run that starts late from one that never trains; see What a collapse means here. The control is #480.

Setup

  • Script: examples/study_trajectory_collapse.py, which reuses the data handling of examples/study_trajectory_noise.py. Raw results: docs/results/trajectory_collapse_study.jsonl (main grid, 380 runs) and docs/results/trajectory_collapse_settings.jsonl (the milder settings, 120 runs).
  • Same data, splits, model and training as #311: breast-cancer data, stratified 60/20/20 split per seed, HybridBinaryClassifier(n, n, 2) on default.qubit with backprop, focal loss, Adam, batch 32, up to 30 epochs with MCC early stopping (patience 10). The score is test MCC under the training noise, with the threshold re-tuned on the noisy validation rows.
  • Seeds 0 to 9, at 4 and 6 qubits. Seeds 0 to 4 are #311's, and the baseline runs reproduce #311 exactly, run for run (MCC equal to machine precision). That holds at the milder settings too, so pairing those with #311's density runs pairs the runs this script would have made.
  • Mitigations, each at k = 1, 4 and 8 draws per sample:
  • lr0.05: the baseline, as in #311;
  • lr0.02, lr0.01: a lower learning rate;
  • clip: gradient-norm clipping at 0.1 (lr 0.05). The median unclipped norm was 0.05 at k = 1 and 0.10–0.12 at k = 4 and 8. At k = 1 the bound acted on 6.5 % (4 qubits) and 8.8 % (6 qubits) of steps on average, between 0 and 20 % per run; at k = 4 on 43 % and 51 % (0.6–60 % per run) and at k = 8 on 52 % and 58 % (40–63 % per run);
  • warmup: noise ramped linearly from p/5 to p over the first 5 epochs (lr 0.05).
  • References: density at all three learning rates, and noiseless training, on the same seeds.
  • Collapse: test MCC under noise below 0.5. The collapses here scored 0.05–0.47, but the cut does not fall in a gap: 15 trajectory runs scored between 0.50 and 0.68, the lowest 0.504 (0.68 was the lowest successful score in #311). Sensitivity to the cut gives the counts at other cuts. It is a statement about the score at the end of this budget, not about whether the run could train.
  • Environment: PennyLane 0.45.1, torch 2.14, one 16-core CPU, 14 single-threaded worker processes.

Results

p = 0.05 after every gate

Collapsed runs of 10, and mean test MCC under noise:

qubits k lr 0.05 lr 0.02 lr 0.01 clip warm-up
4 1 1 · 0.809 0 · 0.853 0 · 0.690 1 · 0.769 0 · 0.862
4 4 1 · 0.800 0 · 0.879 0 · 0.867 1 · 0.780 0 · 0.900
4 8 0 · 0.904 0 · 0.898 0 · 0.892 0 · 0.890 0 · 0.882
6 1 2 · 0.774 1 · 0.834 2 · 0.748 2 · 0.748 1 · 0.773
6 4 0 · 0.911 2 · 0.790 2 · 0.734 0 · 0.894 0 · 0.897
6 8 0 · 0.888 0 · 0.926 0 · 0.881 0 · 0.914 0 · 0.921
qubits density lr 0.05 density lr 0.02 density lr 0.01 noiseless
4 0 · 0.891 0 · 0.867 0 · 0.871 0 · 0.850
6 0 · 0.853 0 · 0.896 1 · 0.844 0 · 0.872

In total, over the five variants and both sizes, k = 1 collapsed in 10 of 100 runs, k = 4 in 6 of 100 and k = 8 in 0 of 100. Density collapsed once in 60, at lr 0.01.

Sensitivity to the cut

Runs below the cut, of 100 per k (five variants, both sizes) and of 60 for density:

cut k = 1 k = 4 k = 8 density
0.45 10 5 0 1
0.50 10 6 0 1
0.55 13 7 0 1
0.60 15 7 0 2
0.65 19 7 1 3
0.68 22 7 2 3

The order k = 1 > k = 4 > k = 8 holds at every cut, and k = 8 stays at zero up to 0.60; its two lowest runs scored 0.647 and 0.665. The comparisons between variants move more:

  • Of the 15 trajectory runs between 0.50 and 0.68, 12 are at k = 1 and 9 at lr 0.01, and 9 ran the full 30 epochs. The 4-qubit, k = 1, lr 0.01 cell (0 collapses, mean 0.690) holds six of them, at 0.50–0.63. Over all k, lr 0.01 counts 4 runs below 0.5 and 8 below 0.6.
  • The warm-up counts 1 run below 0.5, 2 below 0.55 and 3 below 0.6, against 4 for the baseline and for clipping at each of those cuts.

What a collapse means here

The 16 collapsed trajectory runs fall on 7 combinations of size, k and seed (k = 1 at 6 qubits on seed 0 accounts for 5 of them). Early stopping ended 13 of the 16 before epoch 30, between epochs 11 and 28; only 3 ran the full 30 epochs.

Three of the baseline collapses are #311's own runs (seeds 0 to 4 reproduce it exactly), and

311's review re-ran those three for 60 epochs without early stopping (not in the committed

data of either study):

run (lr 0.05) here, 30 epochs / patience 10 60 epochs, no early stopping
k = 4, 4 qubits, seed 1 stopped at epoch 15, test MCC 0.13 0.94
k = 1, 6 qubits, seed 4 stopped at epoch 28, test MCC 0.42 0.82
k = 1, 6 qubits, seed 0 30 epochs, test MCC 0.19 0.23, loss still on the initial plateau

So two of those three were late starts that the budget cut off, and one is a stall. The other 13 collapses have no such control. The counts are therefore an upper bound on stalls, and a mitigation that lengthens the initial plateau (a lower learning rate, a clipped step) is penalised by the budget whether or not it prevents stalls.

Cost

Median seconds per run at lr 0.05. They are comparable within this table only (14 runs shared the CPU):

qubits k = 1 k = 4 k = 8 density
4 27 30 30 87
6 75 87 102 1690

The milder settings

No run collapsed at p = 0.01 after every gate or at p = 0.05 before measurement, at any k. Mean test MCC under noise, against #311's density runs:

p position qubits density (5 seeds) k = 1 (10 seeds) k = 4 (10 seeds) k = 8 (10 seeds)
0.01 all 4 0.861 0.894 0.895 0.869
0.01 all 6 0.876 0.906 0.899 0.879
0.05 end 4 0.839 0.884 0.888 0.895
0.05 end 6 0.890 0.877 0.893 0.910

The density means cover #311's seeds 0 to 4, the trajectory means seeds 0 to 9, so the columns are not directly comparable. Paired by seed against density (seeds 0 to 4 only), k = 8's mean difference lies between −0.012 and +0.058.

What this shows

Like #311, everything below was measured on the breast-cancer proxy only: one small dataset, 4 and 6 qubits, 10 seeds. It has not been checked on the credit-card or UCI benchmark data; that re-run is #414.

  1. At k = 8 every run trained within the budget; no other mitigation achieved that. k = 8 never collapsed, under any variant, at either size, at any cut up to 0.60. Against k = 1 that is 0 of 100 against 10 of 100 (Fisher's exact test p = 0.002), and against k = 4, 0 against 6 (p = 0.03). Both p-values pool the five variants, which share seeds and are therefore not independent (k = 1 at 6 qubits collapsed on seed 0 under every variant), so they overstate the evidence. Read them as indicative. The pattern is consistent, though: at neither size do collapses rise with k, and both reach zero at k = 8 (2, 2 and 0 of 50 at k = 1, 4 and 8 on 4 qubits; 8, 4 and 0 of 50 on 6 qubits).
  2. A lower learning rate did not help within this budget, and at k = 4 it did worse. At 6 qubits, k = 4 went from 0 collapses at lr 0.05 to 2 at lr 0.02 and 2 at lr 0.01. At k = 1 no learning rate brought the collapses to zero: 3, 1 and 2 of 20 at lr 0.05, 0.02 and 0.01. This does not show that a smaller step fails to prevent stalls. A smaller step also takes longer to leave the plateau, and 6 of the 7 collapses at the lower rates were ended by early stopping between epochs 14 and 25. Whether they would have trained is #480.
  3. Clipping did not help within this budget, and the warm-up helped only partly. Clipping at 0.1 collapsed on exactly the four combinations of size, k and seed on which the baseline did (4 of 60 each). Two are the baseline's runs unchanged, score for score (k = 1, 4 qubits, seed 6, where no step was clipped, and k = 4, 4 qubits, seed 1, where 0.6 % were), and two scored lower (k = 1 at 6 qubits: 0.10 against 0.19 on seed 0, 0.23 against 0.42 on seed 4). In those four runs the bound acted on at most 1.1 % of steps, so the clipping was all but absent where it was needed. The warm-up removed three of the four, including k = 4's (1 of 60 left, k = 1 at 6 qubits on seed 0), but that margin depends on the cut: below 0.6 it counts 3 runs against the baseline's 4.
  4. k = 8 matches density, and costs a fraction of it. At lr 0.05, paired over 20 runs (10 seeds × 2 sizes, so not 20 independent seeds), k = 8 scored +0.024 MCC above density on average (Wilcoxon p = 0.23, no significant difference). It took 1.1× (4 qubits) and 1.4× (6 qubits) the time of k = 1, not 8×, because on default.qubit with backprop the k draws run as one vectorised batch. At 6 qubits it was about 17× faster than density.
  5. Density is not immune either. It collapsed once, at lr 0.01 on seed 0 at 6 qubits (stopped at epoch 27), the seed where k = 1 collapsed under every variant. Some splits are slow or hard to train at this noise level, and with fewer draws more runs had not trained by the end of the budget.

Recommendation

On the evidence above (breast-cancer proxy, 4 and 6 qubits, backprop; the benchmark re-run is

414):

  • Train with noise_trajectories ≥ 8 when using noise_method="trajectories" at noise of a few percent per gate. It was the one setting at which every run trained within the same budget as density, and on backprop devices it cost little more than k = 1 here.
  • With fewer draws, give training a longer budget (more epochs, and a patience well above 10) and check the runs. Two of the three runs re-run for 60 epochs trained; whether a longer budget is enough in general is #480.
  • Keep lr 0.05 unless the budget is lengthened too: within 30 epochs a lower rate did not help.
  • Warm-up and clipping aren't worth adding to the library on this evidence: k = 8 does better alone, and adds no option.
  • The default stays noise_trajectories=1. On the adjoint path, which device_name="auto" picks above 12 qubits (#349), each sample runs separately, so k = 8 is expected to cost about 8× there (not measured). This study measured only 4 and 6 qubits on backprop, so it does not justify that cost as a default. The recommendation is in the hqnn_forge.noise docstring.
  • density remains the default noise method where it fits (up to about 6 qubits).

Limits

  • One small proxy dataset (breast cancer; the benchmark datasets are #414), 4 and 6 qubits, one noise model (depolarizing). The ">12 qubit" regime, where trajectories matter most, is extrapolated, not measured.
  • Training budget. At most 30 epochs, with early stopping (patience 10) and checkpoint selection on the clean validation rows, and no longer-budget control. A collapse is a run that had not trained by then; the study cannot separate late starts from stalls, and that weakens the learning-rate and clipping findings most (#480).
  • The collapse cut. 0.5 does not separate two groups of runs: 15 trajectory runs scored 0.50–0.68. k = 8's zero holds up to a cut of 0.60, but the counts for lr 0.01 and for the warm-up rise with the cut (Sensitivity to the cut).
  • 10 seeds per cell. A collapse rate of a few percent at k = 8 cannot be ruled out: 0 of 100 bounds it below about 3 % (95 %, one-sided), assuming independent runs, and they are not fully independent.
  • Timings come from a shared CPU and are only comparable within this study.