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Toward Joint Optimization of Circuit Depth and Training Data Size in Adaptively Grown Quantum Classifiers
Saeefa Rubaiyet Nowmi, Md Mahmuduzzaman Kamol, Mohammad Saidur Rahman
TL;DR
Caro et al.’s gate-efficiency theory and Q-FLAIR’s adaptive circuit growth leave open whether training-set size predicts the circuit selected by growth. This study tests that question on full-resolution MNIST 3-vs-5 and finds no predictable scaling relationship; the bound is valid in most runs but weakly tracks the observed gap.
Problem
It is unknown whether Q-FLAIR’s converged circuit size changes with training-set size in a way consistent with Caro et al.’s theory, limiting evidence about joint depth-data scaling.
Source: §1 · pp. 1–2Method
Results
Takeaways & Limitations
A joint depth-data scaling relationship cannot be assumed to emerge automatically from adaptive growth and a generalization bound alone.
Source: §1 · p. 2, §8 · p. 4The results cover one dataset and task with three seeds per N, and the study does not test alternative fine-tuning learning rates or active-gate thresholds.
Source: §7 · p. 4
Abstract
from arXiv · showhide
Building a quantum model involves a tradeoff: how complex the circuit should be, and how much training data it needs. Caro et al. show that models with fewer trainable gates need less training data to generalize well. Q-FLAIR shows that a quantum feature-map circuit can be grown gate-by-gate, stopping once further growth stops improving the training loss. We ask whether these two results combine into a predictable scaling law. Does Q-FLAIR's own stopping rule pick larger or smaller circuits as training data grows? Does the resulting generalization behavior track Caro et al.'s bound? We reimplement Q-FLAIR's growth mechanism faithfully, including its analytic reconstruction and exact stopping rule. We run it on full-resolution (784-pixel) MNIST 3-vs-5 classification, at five training-set sizes from N = 2000 to 10000. We then fine-tune each resulting circuit, so we can measure Caro et al.'s notion of active gates, K. We find no predictable relationship between training-set size and the circuit size Q-FLAIR converges to. Circuit size and test accuracy both vary non-monotonically with N, and seed-to-seed variance is nearly as large as any trend across N. The empirical generalization gap never exceeds Caro et al.'s bound in 14 of 15 runs, so the bound holds as a valid guarantee in those runs. But the gap correlates only weakly with the bound's value (r = 0.12). This shows that K does not explain most of the variation we observe. Why a valid guarantee can coexist with such weak predictive power remains an open question, and answering it may be necessary before circuit depth and training data size can be jointly optimized in practice.
1 Introduction
This work asks whether Q-FLAIR’s adaptive growth and Caro et al.’s generalization theory combine into a predictable relationship between training-set size and circuit complexity. It studies that question by running Q-FLAIR across training-set sizes on full-resolution MNIST 3-vs-5 classification.
1 Introduction
- Quantum feature maps embed classical data into quantum states, and model design involves choosing both training-set size and variational-circuit complexity.Re-encoding data across multiple layers is known as data re-uploading [6].Source: §1 · p. 1
- Caro et al. [2] relate generalization to the number of gates that change substantially during training, while Q-FLAIR [5] grows a feature-map circuit gate-by-gate until training loss stops improving.Q-FLAIR uses analytic reconstruction to choose each gate’s weight and data feature classically.Source: §1 · p. 1
- Other adaptive methods include ADAPT-VQE [3] and VAns [1], but neither is studied as a function of training-set size in the cited work.Caro et al.’s experiments also do not ask how adaptive growth responds to different amounts of training data.Source: §1 · p. 2
- The study tests whether Q-FLAIR’s converged circuit size changes with training-set size in a way consistent with Caro et al.’s theory.It uses full-resolution, unreduced 784-pixel MNIST 3-vs-5 classification, matching Q-FLAIR’s stated MNIST 28×28 benchmark dimensionality [5].Source: §1 · p. 2
- Circuit size and accuracy vary non-monotonically and noisily with training-set size, with no discernible scaling trend.The study asks whether combining adaptive growth with a generalization bound yields a joint depth-data scaling relationship.Source: §1 · p. 2
2 Related Work
Prior studies leave open how an adaptive circuit-growth algorithm responds to different training-set sizes. This work runs Q-FLAIR across values of N and adds fine-tuning to measure active gates in its resulting circuits.
2 Related Work
- Caro et al.’s numerical studies use fixed architectures or circuits grown once and then held constant, without varying N for a given architecture.Those studies therefore do not test how adaptive growth output responds to available training data.Source: §2 · p. 2
- Q-FLAIR [5] grows a feature-map circuit gate-by-gate using analytic reconstruction and stops when further growth ceases to improve training loss.Its reported experiments hold N fixed for each task and do not address how the converged output depends on N.Source: §2 · p. 2
- This work runs Q-FLAIR’s growth mechanism across values of N and adds post-hoc fine-tuning to measure Caro et al.’s active-gate quantity K.Neither source method alone provides this measurement for Q-FLAIR-grown circuits.Source: §2 · p. 2
3 Method
The method reproduces Q-FLAIR’s gate-by-gate growth, then fine-tunes each finished circuit so active gates can be measured. It retains the better validation-loss result from the original and fine-tuned weights.
3 Method
- The growth phase uses Q-FLAIR’s gate pool on a five-qubit ring, matching its stated QNN qubit ceiling.The pool includes single-qubit rotations, data-encoding rotations, two-qubit rotations, and H.Source: §3 · p. 2
- For each candidate gate, analytic reconstruction expresses model output as a sinusoid of its rotation angle; two additional quantum evaluations support classical loss minimization.This selects the candidate gate and its loss-minimizing weight at each iteration.Source: §3 · p. 2
- At 784 candidate features, a bounded scalar minimizer replaces the too-slow feature-index grid search, and growth stops when the best candidate improves training loss by less than ∆L = 10−3.The minimizer uses tolerance 10−3; no additional circuit-size cap is imposed.Source: §3 · p. 2
- The method compares training-set size N with circuit size, test accuracy, generalization gap, and the full Caro et al. bound; Table 1 summarizes per-N results across three seeds.The figure’s shaded region indicates where the bound holds.Source: Figure 1 · p. 3, Table 1 · p. 3
- Joint Adam fine-tuning starts from the finished circuit and defines K as the count of gates whose weights move by more than δ = 0.05.Q-FLAIR freezes weights once found, so active gates cannot be measured from Phase 1 alone.Source: §3 · p. 3
- The reported model keeps whichever original or fine-tuned weights produce lower validation loss, so Phase 2 cannot report a worse model than Phase 1.The original Phase-1 validation loss is recorded as the baseline.Source: §3 · p. 3
K log(MT)/N + P
The section reports the full Caro et al. Theorem 3 bound and specifies when a zero-valued result is not treated as an applicable guarantee.
K log(MT)/N + P
- The method reports the full Caro et al. Theorem 3 bound, expressed through terms M∆k for k > K.The bound is evaluated with a minimum over K.Source: K log(MT)/N + P · p. 3
- When fine-tuning makes no improvement, every ∆k is exactly zero, making the computed bound trivially zero.This zero does not reflect a real generalization guarantee.Source: K log(MT)/N + P · p. 3
- Cases with no fine-tuning improvement are flagged as not applicable for the generalization bound.This avoids interpreting the trivial zero as a real guarantee.Source: K log(MT)/N + P · p. 3
4 Experimental Setup
The experiments use raw, full-resolution MNIST 3-vs-5 data across five training-set sizes and evaluate every run on the same held-out test set.
4 Experimental Setup
- The sweep uses N ∈ {2000, 4000, 6000, 8000, 10000}, three seeds per size, and a shared 1000-example test set.Inputs are unreduced 784-pixel intensities rescaled to [−π, π] using training-set statistics.Source: §4 · p. 3
- The loss is negative log-likelihood, and a classical ROC-style threshold search maximizes balanced accuracy on the training set.Circuits are simulated exactly; runs take minutes to tens of minutes depending on N and gate count.Source: §4 · p. 3
- Circuit construction, statevector simulation, and Phase-2 fine-tuning use PennyLane, while Phase 1 feature-weight search uses SciPy’s bounded scalar minimizer.Source: §4 · p. 3
5 Results
Across the tested training sizes, circuit size and accuracy show no monotonic trend, while the empirical gap remains below the bound in applicable runs even though their correlation is weak.
5.1 No discernible scaling trend
- Neither T* nor test accuracy varies monotonically with N: circuits span 16–23 gates, accuracy 84%–89%, and seed standard deviation reaches ±9 gates.The seed-to-seed variation is comparable to the total spread across all five training-set sizes.Source: §5.1 · pp. 3–4
5.2 Consistency with Caro et al.’s bound
- In 14 applicable runs, the empirical gap never exceeded the full bound, but bound–gap correlation was r = 0.12 versus r = 0.05 for raw T alone.One N=10000 run had no fine-tuning improvement and was excluded because its K = 0 made the bound trivially zero.Source: §5.2 · p. 4
6 Discussion
The authors attribute the absent N-dependent trend to the size of the candidate feature space and suggest accounting for N and pool size in the stopping threshold.
6 Discussion
- The authors attribute the missing N trend to 784 raw-pixel candidates: some gate clears the fixed threshold at almost any N, decoupling stopping from data availability.Source: §6 · p. 4
- A more principled stopping rule might scale its threshold with both N and candidate-pool size, such as by correcting for per-iteration comparisons.The authors suggest this may be necessary before a genuine joint depth–data scaling law can emerge on high-dimensional inputs.Source: §6 · p. 4
7 Limitations
The findings are limited by a single task, only three seeds per training size, and untested fine-tuning choices.
7 Limitations
- The study covers one dataset and task, so generalization to other high-dimensional inputs or Q-FLAIR’s QSVM variant is unknown.Source: §7 · p. 4
- Three seeds per N limit statistical power, leaving trends smaller than the observed seed variance potentially undetected.Source: §7 · p. 4
- The Phase-2 learning rate and active-gate threshold δ were each chosen once; other choices could change K and its reported correlation with the bound.Q-FLAIR itself does not include a fine-tuning stage.Source: §7 · p. 4
8 Conclusion
On full-resolution MNIST 3-vs-5 classification, no predictable joint depth-data relationship emerged: circuit size and accuracy varied non-monotonically with training-set size, and the bound tracked the observed generalization gap only weakly.
8 Conclusion
- Circuit size and accuracy varied non-monotonically with training-set size, so no predictable joint depth-data scaling relationship was observed.The study used Q-FLAIR’s gate-by-gate growth mechanism at five training-set sizes.Source: §8 · p. 4
- In 14 of 15 runs, the empirical generalization gap did not exceed Caro et al.’s bound, but the gap-bound correlation was only r = 0.12.The weak correlation leaves open why a valid guarantee can have weak predictive power.Source: §8 · p. 4
- Jointly optimizing circuit depth and training-data size cannot be assumed to follow automatically from adaptive growth and a generalization bound alone.The dimensionality of the candidate feature pool is a plausible contributing factor that warrants further investigation.Source: §8 · p. 4