When Similarity Is Interaction-Driven: Quantum Kernels for Regime-Sensitive Learning

Hanqiu Peng; Jianlong Lu; Ying Chen

quant-phAug 25, 2026

When Similarity Is Interaction-Driven: Quantum Kernels for Regime-Sensitive Learning

Authors: Hanqiu Peng, Jianlong Lu, Ying Chen

Organizations: Centre for Quantitative Finance, Department of Mathematics Risk Management Institute, National University of Singapore, Singapore

Abstract

Similarity in many decision systems is governed not by distance alone but by interactions among variables. In fraud and anomaly detection, small local perturbations can cross interaction-sensitive decision boundaries while leaving ambient distance almost unchanged. Motivated by this setting, we introduce a thin-slab interaction model and an interaction-driven quantum kernel constructed from entangled Pauli-string feature maps. The feature map explicitly encodes sparse high-order block interactions. We show that the resulting fidelity kernel is positive semidefinite, admits an exact block-factorized formulation, and induces a geometry sensitive to changes in interaction regime. Across balanced and imbalanced synthetic experiments spanning third-, fourth-, sixth-, and eighth-order interactions, the proposed kernel consistently outperforms linear, radial basis function, Laplacian, and polynomial kernels, as well as an engineered-interaction linear baseline supplied with the planted block products. On real fraud-detection benchmarks, it achieves the highest mean accuracy and F1 on Credit Card Fraud Detection and ranks second on IEEE-CIS Fraud Detection. Executed on a 156-qubit IBM Quantum processor in a fourth-order setting, the hardware-estimated kernel matches the noise-free simulator within seed-to-seed variability and retains its advantage over the baselines. These findings show that quantum-kernel performance depends on alignment between feature-map geometry and the underlying predictive structure, rather than on Hilbert-space dimension alone. Because the prescribed block-factorized kernel can also be evaluated exactly on a classical computer, the results establish predictive and representational value rather than computational quantum speedup.

Figures & tables

Dataset/setting	$B$	Peak qubits	InterFea dim.	$H$	$R_{Z}$ : $\beta=0$ ( $\neq 0$ )	CNOTs	Block depth: $\beta=0$ ( $\neq 0$ )
Synthetic: $d=9,\ k=3$	3	3	12	9	3 (12)	12	6 (7)
Synthetic: $d=20,\ k=4$	5	4	25	20	5 (25)	30	8 (9)
Synthetic scaling: $d=60,\ k=6$	10	6	70	60	10 (70)	100	12 (13)
Synthetic scaling: $d=80,\ k=8$	10	8	90	80	10 (90)	140	16 (17)
Credit Card: $d=28,\ k=4$	7	4	35	28	7 (35)	42	8 (9)
IEEE-CIS: $d=60,\ k=6$	10	6	70	60	10 (70)	100	12 (13)

Table 1: Logical circuit resources per observation. Depth refers to one block circuit under a CNOT-ladder decomposition.

Model	$(0.05,0.00)$	$(0.05,0.10)$	$(0.15,0.00)$	$(0.15,0.10)$
Linear	$0.506\pm 0.025$	$0.494\pm 0.020$	$0.502\pm 0.023$	$0.504\pm 0.019$
RBF	$0.494\pm 0.027$	$0.497\pm 0.032$	$0.608\pm 0.020$	$0.593\pm 0.038$
Laplacian	$0.508\pm 0.031$	$0.510\pm 0.024$	$0.545\pm 0.035$	$0.538\pm 0.026$
Poly2	$0.490\pm 0.016$	$0.504\pm 0.019$	$0.490\pm 0.016$	$0.504\pm 0.011$
Poly4	$0.673\pm 0.021$	$0.641\pm 0.027$	$0.672\pm 0.018$	$0.644\pm 0.023$
Poly6	$0.593\pm 0.021$	$0.563\pm 0.027$	$0.639\pm 0.018$	$0.619\pm 0.029$

Table 2: Balanced synthetic results for

(d,k,B)=(9,3,3)

, reported as mean test accuracy

\pm

standard deviation over ten seeds. Columns correspond to

(\epsilon,p_{\mathrm{flip}})

Model	$(0.05,0.00)$	$(0.05,0.10)$	$(0.15,0.00)$	$(0.15,0.10)$
Linear	$0.520\pm 0.019$	$0.509\pm 0.018$	$0.515\pm 0.015$	$0.513\pm 0.018$
RBF	$0.504\pm 0.023$	$0.504\pm 0.007$	$0.512\pm 0.020$	$0.502\pm 0.022$
Laplacian	$0.502\pm 0.011$	$0.510\pm 0.019$	$0.508\pm 0.012$	$0.505\pm 0.015$
Poly2	$0.512\pm 0.024$	$0.514\pm 0.019$	$0.507\pm 0.018$	$0.506\pm 0.022$
Poly4	$0.516\pm 0.020$	$0.496\pm 0.005$	$0.502\pm 0.014$	$0.487\pm 0.025$
Poly6	$0.492\pm 0.029$	$0.501\pm 0.017$	$0.499\pm 0.020$	$0.488\pm 0.026$

Table 3: Balanced synthetic results for

(d,k,B)=(20,4,5)

, reported as mean test accuracy

\pm

standard deviation over ten seeds. Columns correspond to

(\epsilon,p_{\mathrm{flip}})

Figure 1: Learning curves for the balanced $(d,k)=(20,4)$ setting. Lines and bands show mean test accuracy and $\pm 1$ standard deviation.

Model	$(0.05,0.00)$		$(0.05,0.10)$		$(0.15,0.00)$		$(0.15,0.10)$
Model	Acc.	F1	Acc.	F1	Acc.	F1	Acc.	F1
Linear	$0.506\pm 0.012$	$0.200\pm 0.027$	$0.497\pm 0.025$	$0.194\pm 0.036$	$0.499\pm 0.014$	$0.197\pm 0.033$	$0.493\pm 0.021$	$0.191\pm 0.040$
RBF	$0.588\pm 0.032$	$0.179\pm 0.041$	$0.568\pm 0.019$	$0.191\pm 0.032$	$0.748\pm 0.050$	$0.268\pm 0.065$	$0.608\pm 0.079$	$0.215\pm 0.039$
Laplacian	$\mathbf{0.834\pm 0.024}$	$0.101\pm 0.054$	$\mathbf{0.773\pm 0.018}$	$0.161\pm 0.029$	$\mathbf{0.848\pm 0.021}$	$0.106\pm 0.040$	$\mathbf{0.791\pm 0.016}$	$0.192\pm 0.031$
Poly2	$0.539\pm 0.028$	$0.180\pm 0.037$	$0.522\pm 0.015$	$0.180\pm 0.026$	$0.533\pm 0.024$	$0.181\pm 0.039$	$0.522\pm 0.021$	$0.183\pm 0.031$
Poly4	$0.739\pm 0.030$	$0.320\pm 0.082$	$0.660\pm 0.021$	$0.290\pm 0.039$	$0.752\pm 0.026$	$0.351\pm 0.066$	$0.654\pm 0.041$	$0.277\pm 0.051$

Table 4: Imbalanced synthetic results for

(d,k,B)=(9,3,3)

: mean test accuracy and F1

\pm

standard deviation over ten seeds. Columns correspond to

(\epsilon,p_{\mathrm{flip}})

. Bold and underlined values denote the best and second-best results.

Model	$(0.05,0.00)$		$(0.05,0.10)$		$(0.15,0.00)$		$(0.15,0.10)$
Model	Acc.	F1	Acc.	F1	Acc.	F1	Acc.	F1
Linear	$0.502\pm 0.038$	$0.281\pm 0.044$	$0.514\pm 0.026$	$0.296\pm 0.032$	$0.506\pm 0.036$	$0.284\pm 0.050$	$0.511\pm 0.028$	$0.289\pm 0.041$
RBF	$0.646\pm 0.023$	$0.207\pm 0.033$	$0.619\pm 0.012$	$0.242\pm 0.023$	$0.678\pm 0.030$	$0.193\pm 0.040$	$0.672\pm 0.030$	$0.221\pm 0.047$
Laplacian	$0.728\pm 0.023$	$0.188\pm 0.024$	$0.691\pm 0.019$	$0.211\pm 0.059$	$0.735\pm 0.018$	$0.183\pm 0.058$	$0.700\pm 0.016$	$0.205\pm 0.048$
Poly2	$0.573\pm 0.019$	$0.273\pm 0.034$	$0.542\pm 0.019$	$0.267\pm 0.019$	$0.568\pm 0.026$	$0.258\pm 0.031$	$0.547\pm 0.006$	$0.271\pm 0.019$
Poly4	$0.592\pm 0.026$	$0.237\pm 0.039$	$0.581\pm 0.034$	$0.233\pm 0.029$	$0.618\pm 0.021$	$0.221\pm 0.041$	$0.586\pm 0.028$	$0.229\pm 0.035$

Table 5: Imbalanced synthetic results for

(d,k,B)=(20,4,5)

: mean test accuracy and F1

\pm

standard deviation over ten seeds. Columns correspond to

(\epsilon,p_{\mathrm{flip}})

. Bold and underlined values denote the best and second-best results.

Model	$(d,k,B)=(60,6,10)$		$(d,k,B)=(80,8,10)$
Model	Acc.	F1	Acc.	F1
Linear	$0.494\pm 0.024$	$0.247\pm 0.017$	$0.508\pm 0.020$	$0.252\pm 0.043$
RBF	$0.597\pm 0.046$	$0.225\pm 0.034$	$0.580\pm 0.058$	$0.237\pm 0.047$
Laplacian	$0.651\pm 0.054$	$0.200\pm 0.043$	$\underline{0.661\pm 0.058}$	$0.224\pm 0.054$
Poly2	$0.574\pm 0.036$	$0.241\pm 0.037$	$0.551\pm 0.067$	$0.256\pm 0.046$
Poly4	$0.603\pm 0.079$	$0.221\pm 0.048$	$0.558\pm 0.054$	$0.252\pm 0.046$

Table 6: Higher-order imbalanced scaling results at

\epsilon=0.15

and

p_{\mathrm{flip}}=0

: mean test accuracy and F1

\pm

standard deviation over ten seeds. Bold and underlined values denote the best and second-best results within each column.

	Balanced: Acc.		Imbalanced, $p_{\mathrm{flip}}=0.00$		Imbalanced, $p_{\mathrm{flip}}=0.10$
Model	$p_{\mathrm{flip}}=0.00$	$p_{\mathrm{flip}}=0.10$	Acc.	F1	Acc.	F1
InterFea	$0.734\pm 0.027$	$0.721\pm 0.025$	$0.753\pm 0.017$	$0.505\pm 0.036$	$0.710\pm 0.022$	$0.491\pm 0.026$
Quantum, simulator	$0.753\pm 0.016$	$0.752\pm 0.017$	$0.770\pm 0.016$	$0.534\pm 0.029$	$0.732\pm 0.017$	$0.516\pm 0.025$
Quantum, IBM hardware	$0.749\pm 0.013$	$0.745\pm 0.015$	$0.760\pm 0.017$	$0.531\pm 0.030$	$0.728\pm 0.021$	$0.510\pm 0.034$

Table 7: IBM Quantum hardware results for

(d,k,B)=(20,4,5)

\epsilon=0.15

: mean

\pm

standard deviation over ten seeds. Balanced tasks report test accuracy; imbalanced tasks report test accuracy and F1. The simulator row is a noise-free simulation on the same seeds.

Model	Acc.	F1
Linear	$0.983\pm 0.007$	$0.124\pm 0.050$
RBF	$0.995\pm 0.001$	$0.335\pm 0.085$
Laplacian	$0.994\pm 0.000$	$0.275\pm 0.013$
Poly2	$\underline{0.996\pm 0.000}$	$\underline{0.348\pm 0.021}$
Poly4	$0.983\pm 0.002$	$0.119\pm 0.015$
Poly6	$0.977\pm 0.011$	$0.105\pm 0.035$

Table 8: Credit Card Fraud results on the complete chronological FDB test partition: mean test accuracy and F1

\pm

standard deviation over ten training-resampling seeds. Bold and underlined values denote the best and second-best results.

Model	Acc.	F1
Linear	$0.804\pm 0.005$	$0.216\pm 0.004$
RBF	$0.812\pm 0.007$	$0.231\pm 0.008$
Laplacian	$\mathbf{0.831\pm 0.008}$	$\mathbf{0.267\pm 0.007}$
Poly2	$0.806\pm 0.008$	$0.224\pm 0.008$
Poly4	$0.809\pm 0.007$	$0.227\pm 0.004$
Poly6	$0.815\pm 0.011$	$0.229\pm 0.009$

Table 9: IEEE-CIS results on the complete FDB test partition: mean test accuracy and F1

\pm

standard deviation over ten seeds. Bold and underlined values denote the best and second-best results.

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