Toward Joint Optimization of Circuit Depth and Training Data Size in Adaptively Grown Quantum Classifiers

The gist: No predictable relationship between training-set size and circuit size Q-FLAIR converges to, and circuit size and test accuracy both vary non monotonically with N, indicating that a straightforward joint depth/data scaling relationship does not emerge automatically from combining these two methods.

Background

Building a quantum model involves a tradeoff between how complex the circuit should be and how much training data it needs The first decision is how much training data to use, and the second is how complex the variational circuit should be Caro et al. show that models with fewer trainable gates need less training data to generalize well Q-FLAIR grows a quantum feature-map circuit gate-by-gate, stopping once further growth no longer improves the training loss

Methodology

The authors reimplement Q-FLAIR’s growth mechanism exactly, including its analytic reconstruction and exact stopping rule They run this on full-resolution (784-pixel) MNIST 3-vs-5 classification at five training set sizes from N = 2000 to 10000 They then fine-tune each resulting circuit to measure Caro et al.’s notion of active gates, K

Key Findings

The study found 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 However, the gap correlates only weakly with the bound’s value (r = 0.12)

Discussion and Conclusion

The absence of an N-dependent trend is attributed to the dimensionality of the candidate feature space, where a large pool of 784 raw-pixel candidates allows some gate to clear the stopping threshold at almost any N This suggests that combining an adaptive growth algorithm with a data-efficiency motivation does not, by itself, guarantee a joint scaling relationship The dimensionality of the candidate feature pool is a plausible contributing factor warranting further investigation Joint optimization of circuit depth and training data size cannot, therefore, be assumed to emerge automatically from an adaptive growth algorithm and a generalization bound alone

Limitations

The results come from a single dataset and task, limiting the generalizability of the observations The study also notes that different choices for the learning rate and active-gate threshold δ could change the measured value of K

References

[1] M Bilkis, M Cerezo, Guillaume Verdon, Patrick J Coles, and Lukasz Cincio. A semiagnostic ansatz with variable structure for quantum machine learning (2021). arXiv preprint ArXiv:2103.06712

[2] Matthias C Caro, Hsin-Yuan Huang, Marco Cerezo, Kunal Sharma, Andrew Sornborger, Lukasz Cincio, and Patrick J Coles. Generalization in quantum machine learning from few training data. Nature communications

[3] Harper R Grimsley, Sophia E Economou, Edwin Barnes, and Nicholas J Mayhall. An adaptive variational algorithm for exact molecular simulations on a quantum computer. Nature communications

[4] Vojtech Havlí ˇ cek, Antonio D Córcoles, Kristan Temme, Aram W Harrow, Abhinav Kandala, ˇ Jerry M Chow, and Jay M Gambetta. Supervised learning with quantum-enhanced feature spaces. Nature

[5] Jonas Jäger, Philipp Elsässer, and Elham Torabian. Quantum feature-map learning with reduced resource overhead. Physical Review Research

[6] Adrián Pérez-Salinas, Alba Cervera-Lierta, Elies Gil-Fuster, and José I Latorre. Data re-uploading for a universal quantum classifier. Quantum

[7] Maria Schuld and Nathan Killoran. Quantum machine learning in feature hilbert spaces. Physical review letters

--- Page 1 ---

Toward Joint Optimization of Circuit Depth and Training Data Size in Adaptively Grown Quantum Classifiers Saeefa Rubaiyet Nowmi Old Dominion University Norfolk, VA, USA snowm001@odu.edu Md Mahmuduzzaman Kamol Old Dominion University Norfolk, VA, USA mkamo001@odu.edu Mohammad Saidur Rahman University of Texas at El Paso El Paso, TX, USA msrahman3@utep.edu Abstract 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?

--- Page 2 ---

Adaptive circuit growth is not unique to Q-FLAIR ADAPT-VQE grows ansatze gate-by-gate for molecular simulation [3] and VAns grows and prunes circuit structure during training [1] Neither ADAPT-VQE nor VAns, however, is studied as a function of training-set size, and neither work asks the question their combination with a generalization bound raises That question is this: if Q-FLAIR’s growth algorithm is handed different amounts of training data, does the circuit size it converges to change in a way consistent with Caro et al.’s theory? A related question also follows: is the number of trainable gates required systematically correlated with the size of the training data? Answering these questions requires actually running Q-FLAIR’s mechanism, not a fixed circuit family, across a range of N This is what we do here, on full-resolution (unreduced, 784-pixel) MNIST 3-vs-5 classification This setting matches Q-FLAIR’s own “MNIST 28×28” benchmark dimensionality exactly [5]

--- Page 3 ---

Growth (Phase 1). We reimplement Q-FLAIR’s gate pool exactly (their Eq. 25): V = At each iteration, we consider every candidate gate For each one, we reconstruct its model output as a closed-form sinusoid of its rotation angle This takes only 2 additional quantum evaluations We then solve for the loss-minimizing weight classically At d = 784 candidate features, a brute-force grid search over feature index is too slow So we use a bounded scalar minimizer, with convergence tolerance 10−3 This cuts the classical cost by roughly an order of magnitude, solving the same one-dimensional bounded optimization problem that Q-FLAIR specifies Growth stops the first time the best candidate’s training-loss improvement falls below ∆L = 10−3 We place no additional cap on circuit size

--- Page 4 ---

Table 1: Per-N summary across 3 seeds, full-resolution (784-pixel) MNIST. N T∗ (mean ± SD) test acc. (mean ± SD) gen. gap (mean ± SD) 2000 22.67 ± 9.07 0.892 ± 0.028 0.0141 ± 15

4 N T∗ (cir cuit siz e) Circuit size N=5 T∗ (cir cuit siz e) N=15 T∗ (cir cuit siz e) N=20 T∗ (cir cuit siz e) N=25 T∗ (cir cuit siz e) N=30 2000 4000 6000 8.

1. A more robust stopping rule for Q-FLAIR growth could be implemented by scaling its threshold with both training data size (N) and candidate pool size to address the observed confounding factor. This correction may be necessary before a genuine joint depth-data scaling law can emerge on high-dimensional inputs.

2. The system can be optimized for joint circuit depth and training data size by explicitly incorporating the dimensionality of the feature space into the growth stopping criterion, moving beyond a fixed improvement threshold of ∆L = 10−3.

3. The generalization guarantee derived from Caro et al.'s bound can be made more predictive by developing a method to better correlate the empirical gap with its bound value, as the current correlation is only weakly correlated (r = 0.12).

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.