Prototype-Guided Optimization of Quantum Data Embeddings with Linear Cost

Seokhoon Jeong; Daniel K. Park

quant-phOct 17, 2026

Prototype-Guided Optimization of Quantum Data Embeddings with Linear Cost

Authors: Seokhoon Jeong, Daniel K. Park

Organizations: Department of Statistics and Data Science Yonsei University Republic of Korea · Department of Statistics and Data Science Department of Applied Statistics Department of Quantum Information Yonsei University Republic of Korea

Abstract

In supervised quantum machine learning with classical data, classification performance critically depends on the geometry induced by quantum data embedding. Existing data-adaptive approaches rely on pairwise fidelity evaluations between samples, which scale as $O(B^2)$ for $B$ samples and create a computational bottleneck in iterative embedding optimization. To address this issue, we propose a prototype-guided framework for efficient data-adaptive quantum embedding. Using class-prototype separation and sample-to-prototype compactness, we derive a measurable lower certificate for class-ensemble distinguishability and construct a prototype-based objective requiring only $O(B)$ fidelity evaluations. We integrate this objective into an end-to-end pipeline that combines task-specific quantum circuit architecture optimization with continuous refinement of the data-embedding parameters. Across five benchmark datasets with up to 20 qubits, our method retains competitive quantum kernel support vector machine (SVM) performance while reducing fidelity-evaluation cost, with particularly strong gains over budget-matched pairwise optimization on T-MNIST. Geometric analyses examine how the learned embeddings relate to kernel alignment and classification margins, revealing dataset-dependent benefits and limitations. We further derive a prototype-based classifier in the reproducing kernel Hilbert space that provides an explicit witness for the SVM objective and a low-cost proxy for model selection, and extend the framework to multiple prototypes for multimodal class structures.

Figures & tables

Figure 1: Comparison of pairwise and single-prototype-guided quantum embedding optimization. (a) Pairwise optimization promotes high fidelity for same-class sample pairs and low fidelity for cross-class pairs, requiring

B(B-1)/2

evaluations. (b) The proposed method promotes high sample-to-class-prototype fidelity and low cross-class prototype fidelity, reducing the cost to

B+1

evaluations.

Stage	Method	IRIS	VERTEBRAL	T-CELL	GLASS	T-MNIST
Stage	Method	Test MCC / Test AUC
Fixed	ZZFeatureMap	.6470 / .8842	.1981 / .6749	.4542 / .7686	.3553 / .8025	.0703 / .5707
GA	Full pairs $[O(B^{2})]$	.9554 / .9994	.6152 / .9234	.5548 / .8034	.6057 / .8723	–
	Pairwise- $B$ $[O(B)]$	.9200 / .9966	.6403 / .9125	.5427 / .8141	.5907 / .8603	.6357 / .8883
	Prototype $[O(B)]$	.9387 / .9964	.5939 / .8851	.5502 / .8283	.5979 / .8837	.7606 / .9437
GA $\rightarrow$ NQE	Pairwise- $B$ $[O(B)]$	.9419 / .9976	.6471 / .9093	.5624 / .8007	.5826 / .8729	.6423 / .8962

Table 1: Held-out QKSVM performance (MCC/AUC, 10-seed means). The ZZfeaturemap applies

ZZ

rotations only between nearest-neighbor qubits on a rectangular lattice. Full pairs provides the

O(B^{2})

reference, while Pairwise-

B

is the budget-matched

O(B)

pairwise baseline. The GA

\rightarrow

NQE panel compares Pairwise-B and Prototype after NQE refinement of the GA-selected embeddings.

Figure 2: Prototype geometry and downstream behavior. VERT means VERTEBRAL. For each seed, ten unique GA circuits with the lowest and highest training prototype losses form the low- and high-loss groups, respectively. (a) Training

C_{\mathrm{prior}}

, where the dashed line marks the threshold for a non-vacuous positive certificate. (b) Test pairwise-macro CKTA. (c) Test QKSVM MCC. Markers and error bars denote the mean and 95% seed-bootstrap CI.

	Linear-cost selectors			Full-QKSVM reference
Dataset	P0 $(V)$	P1 $(CV)$	$\Delta$ MCC	P2 $(N^{2})$
IRIS	$.9387/.9964$	$.9387/.9967$	$+.0000$	$.9387/.9976$
VERTEBRAL	$.5939/.8851$	$.6189/.8920$	$\mathbf{+.0250}$	$.6201/.8946$
T-CELL	$.5502/.8283$	$.5502/.8283$	$+.0000$	$.5502/.8283$
GLASS	$.5979/.8837$	$.6073/.8800$	$\mathbf{+.0094}$	$.6001/.8865$
T-MNIST	$.7606/.9437$	$.7557/.9427$	$-0.0049$	$.7657/.9446$

Table 2: Held-out MCC/AUC after validation reranking of the same dataset-specific bank of

20

prototype-optimized circuits (10-seed means). Here,

N

and

V

denote the numbers of training and validation samples, respectively, and

C

denotes the number of classes, with one prototype per class. P0 selects using validation prototype loss, P1 using the exact prototype RKHS classifier in Proposition 3 , and P2 using a full validation QKSVM.

\Delta

MCC denotes P1–P0, and the leading fidelity cost of each selector is shown in the parentheses, the exact cost is in Appendix E .

Figure 3: Multi-prototype optimization on GLASS. (a) Vilolin plots of OOF minimum normalized one-vs-one margin distributions for the single-prototype (

K=1

) and multi-prototype (

K=(2,1,3)

) model, where the entries of

K

denote the class-specific numbers of prototypes. The white circle and the bar indicate the median and interquartile range (IQR) of each distribution, respectively. (b) OOF margin gain versus the reduction in sample-to-assigned-prototype trace-distance radius. Quantile-bin means

\pm

standard errors are shown.

\rho_{s}

means the pooled Spearman correlation.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Samples	#.Features	Qubits	Train / Val / Test
IRIS ( Fisher, 1936 )	150	4	4	84 / 21 / 45
VERTEBRAL ( Barreto & Neto, 2005 )	310	6	6	173 / 44 / 93
T-CELL ( Daniels et al., 2022 )	246	60 $\rightarrow$ 8	8	137 / 35 / 74
GLASS ( German, 1987 )	213	9	9	119 / 30 / 64
T-MNIST ( Lecun et al., 1998 )	550	784 $\rightarrow$ 20	20	150 / 150 / 250

Appendix

Table 3: Summary of the datasets used in the experiments. The # Features column reports the input dimensionality, and

\rightarrow

indicates dimensionality reduction prior to quantum embedding.

Dataset	$Q$	Lattice	$\|\mathcal{E}_{\mathrm{lat}}\|$	Population	Epochs/stage	$E_{\rm keep}$	Patience
IRIS	4	$2\times 2$	4	200	50	8	10
VERTEBRAL	6	$2\times 3$	7	300	100	5	15
T-CELL	8	$4\times 2$	10	300	100	5	15
GLASS	9	$3\times 3$	12	300	100	5	15
T-MNIST	20	$4\times 5$	31	300	100	5	15

Appendix

Table 4: Dataset-specific circuit layouts and GA hyperparameters.

Q

is the number of qubits, lattice specifies the rows and columns of the rectangular grid, and

|\mathcal{E}_{\mathrm{lat}}|

counts its nearest-neighbor CZ edges per layer. Population and Epochs/stage denote the population size and maximum number of generations at each of the five sequential stages.

E_{\mathrm{keep}}

is the number of tournament-selected parents protected from mutation. Patience is the number of consecutive generations without sufficient improvement in the stage’s monitored criterion before early stopping.

Dataset	Prototype loss	$\lambda_{\mathrm{P}}$	Prototype bound/LR	Pairwise-B bound/LR
IRIS	Mean	0.5	Unb./0.010	Unb./0.005
VERTEBRAL	Mean	0.7	Unb./0.010	Unb./0.010
T-CELL	Mean	0.3	Unb./0.003	Unb./0.010
GLASS	Dual Guard	0.5	$\pi/4$ /0.010	Unb./0.010
T-MNIST	Mean	0.5	$\pi/4$ /0.001	$\pi/4$ /0.001

Appendix

Table 5: Frozen NQE hyperparameters used for the main comparison.“Unb.” denotes an unbounded residual. Checkpoints are selected using each method’s validation objective.

Figure 4: Geometric analysis for the T-MNIST NQE endpoints in Table 1 . (a) Exact train class-pair ensemble distance

\Delta_{\Phi,ab}

. Each marker represents one of the ten class pairs and reports its average over ten seeds. (b) Mean Test margin gain,

\Delta\widetilde{m}=\widetilde{m}^{\mathrm{Prototype}}-\widetilde{m}^{\mathrm{Pairwise\text{-}B}}

, within deciles defined by the leave-one-seed-out Pairwise-B difficulty. Decile 1 contains the hardest 10% of Test samples. Error bars indicate 95% seed-bootstrap intervals.

Figure 5: T-MNIST lower-margin-tail diagnostic using the all-depth circuit bank from Figure 2 . (a) Mean paired change in RKHS-normalized geometric margin from High- to Low-loss circuits across Test-sample difficulty deciles. Decile 1 contains the hardest samples, as determined by the leave-one-seed-out High-loss reference. Error bars denote 95% seed-bootstrap intervals. (b) Fractions of paired seed–sample observations whose margin improves or worsens, evaluated over all Test samples and over the hardest decile. Circuit groups and QKSVM hyperparameters are fixed before Test evaluation.

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