Can Entanglement-Induced Feature-Map Designs Improve Projected Quantum Kernels?

Ravi Kumar Jha; Hsuan-Yin Lin; Eirik Rosnes

quant-phOct 22, 2026

Can Entanglement-Induced Feature-Map Designs Improve Projected Quantum Kernels?

Authors: Ravi Kumar Jha, Hsuan-Yin Lin, Eirik Rosnes

Organizations: Simula UiB, N–5006 Bergen, Norway

Abstract

Projected quantum kernels (PQKs) have emerged as a notable approach to circumvent the hardness of machine learning tasks arising from the exponential dimensionality of the quantum Hilbert space. However, recent PQK studies often rely heavily on extensive classical hyperparameter tuning of the radial basis function (RBF) kernel parameter and the support vector machine regularization parameter. The extensive tuning can mask the true representational power of the underlying PQKs induced by quantum feature maps, which has not been explored in detail. Thus, this work aims to expose the structural capabilities of PQKs by isolating and evaluating their native inductive bias through comparisons between untuned and tuned PQKs against the RBF baseline. Furthermore, we propose novel quantum feature maps that leverage two-qubit entanglement to capture multivariate correlations in IoT data. The results show that extensive tuning can lead to competitive performance architectures, while our proposed entanglement topologies exhibit a significantly stronger native inductive bias. This reveals that superior quantum architectural design can yield structurally better PQKs with improvements arising from quantum-mechanical properties rather than from classical hyperparameter tuning.

Figures & tables

Panel of Figure 1 — Fig. 2: A schematic design of the PQK architecture, where the raw inputs are first mapped into quantum feature space using a quantum feature map $\phi$ , and finally projected back into a classical space through measurement.

Panel of Figure 2 — Fig. 3: Schematic diagram of the distinct feature maps used for PQKs: (a) $\textnormal{Rot}(\mathsf{X})$ , a single qubit $\mathsf{R}_{\textnormal{X}}$ rotation, inspired the designing of (e) $\textnormal{Rot}(\mathsf{X})^{\textnormal{en}}$ , $\textnormal{Rot}(\mathsf{X})$ with a specific pair of entanglement; (b) $\textnormal{Rot}(\mathsf{X},\mathsf{Y},\mathsf{Z})$ , $\mathsf{R}_{\textnormal{X}}$ , $\mathsf{R}_{\textnormal{Y}}$ , $\mathsf{R}_{\textnormal{Z}}$ rotations; (c) IQP , standard design, for simplicity, only an $n=3$ qubits circuit is shown; (d) $\textnormal{Rot}(\mathsf{X},\mathsf{Y},\mathsf{Z})\textnormal{+CNOT}$ , $\textnormal{Rot}(\mathsf{X},\mathsf{Y},\mathsf{Z})$ with circular entanglement, conceptually formed (f) $\textnormal{Rot}(\mathsf{X,Y,Z})^{\textnormal{en}}$ , with a specific pair of entanglement.

Models	Accuracy	F1-score	MCC
Lin	$0.792\pm 0.014$	$0.798\pm 0.015$	$0.626\pm 0.023$
RBF	$\color[rgb]{0,0,1}0.826\pm 0.017$	$\color[rgb]{0,0,1}0.820\pm 0.021$	$\color[rgb]{0,0,1}0.670\pm 0.033$
RBF	$0.893\pm 0.008$	$0.881\pm 0.009$	$0.785\pm 0.017$
LDA	$0.792\pm 0.012$	$0.798\pm 0.013$	$0.627\pm 0.017$
RF	$0.902\pm 0.006$	$0.891\pm 0.008$	$0.804\pm 0.013$
AB	$0.863\pm 0.010$	$0.839\pm 0.016$	$0.720\pm 0.023$

TABLE I: IoT data classification evaluation for classical models.

Models	$m=1$			$m=2$			$m\in\{1,2\}$
Models	Accuracy	F1-score	MCC	Accuracy	F1-score	MCC	Accuracy	F1-score	MCC
$\footnotesize\textnormal{Rot}(\mathsf{X})$	$\color[rgb]{0,0,1}0.853\pm 0.015$	$\color[rgb]{0,0,1}0.842\pm 0.017$	$\color[rgb]{0,0,1}0.711\pm 0.030$	$\color[rgb]{0,0,1}0.855\pm 0.013$	$\color[rgb]{0,0,1}0.847\pm 0.012$	$\color[rgb]{0,0,1}0.720\pm 0.026$	$\color[rgb]{0,0,1}0.859\pm 0.015$	$\color[rgb]{0,0,1}0.850\pm 0.016$	$\color[rgb]{0,0,1}0.726\pm 0.030$
$\footnotesize\textnormal{Rot}(\mathsf{X})$	$0.893\pm 0.009$	$0.881\pm 0.009$	$0.786\pm 0.018$	$0.895\pm 0.009$	$0.883\pm 0.010$	$0.789\pm 0.018$	$0.896\pm 0.009$	$0.884\pm 0.010$	$0.792\pm 0.019$
$\footnotesize\textnormal{Rot}(\mathsf{X,Y,Z})$	$\color[rgb]{0,0,1}0.864\pm 0.012$	$\color[rgb]{0,0,1}0.858\pm 0.015$	$\color[rgb]{0,0,1}0.742\pm 0.023$	$\color[rgb]{0,0,1}0.852\pm 0.013$	$\color[rgb]{0,0,1}0.843\pm 0.015$	$\color[rgb]{0,0,1}0.714\pm 0.026$	$\color[rgb]{0,0,1}0.864\pm 0.013$	$\color[rgb]{0,0,1}0.857\pm 0.015$	$\color[rgb]{0,0,1}0.740\pm 0.026$
$\footnotesize\textnormal{Rot}(\mathsf{X,Y,Z})$	$0.895\pm 0.011$	$0.883\pm 0.012$	$0.789\pm 0.022$	$0.892\pm 0.010$	$0.880\pm 0.012$	$0.784\pm 0.021$	$0.897\pm 0.010$	$0.885\pm 0.013$	$0.793\pm 0.021$
$\footnotesize\textnormal{Rot}(\mathsf{X,Y,Z})$ +CNOT	$\color[rgb]{0,0,1}0.844\pm 0.013$	$\color[rgb]{0,0,1}0.832\pm 0.015$	$\color[rgb]{0,0,1}0.693\pm 0.027$	$\color[rgb]{0,0,1}0.839\pm 0.013$	$\color[rgb]{0,0,1}0.829\pm 0.016$	$\color[rgb]{0,0,1}0.688\pm 0.022$	$\color[rgb]{0,0,1}0.857\pm 0.014$	$\color[rgb]{0,0,1}0.849\pm 0.017$	$\color[rgb]{0,0,1}0.724\pm 0.027$

TABLE II: IoT data classification accuracy, F1-score, and MCC value for PQK models before and after hyperparameter tuning.

Panel of Figure 5 — Fig. 4: Sensitivity analysis of PQK and RBF hyperparameter tuning.

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Expressivity of IQP Feature-Map Quantum Kernels via Reproducing Kernel Hilbert Spaces

Valeria Repetto, Stefano Melchionna, Romina D'Aurizio

National Research Council of Italy (CNR), Institute of Informatics and Telematics (IIT), via Giuseppe Moruzzi 1, Pisa, Italy · Computational Biomedicine Unit, Department of Medical Sciences, University of Turin, via Santena 19, Torino, Italy · Kipu Quantum, Greifswalderstrasse 212, 10405 Berlin, Germany

Quantum Kernel Methods Quantum Feature Maps Reproducing Kernel Hilbert Spaces Frequency-Domain Feature Learning

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Quantum kernels are among the most promising methods in quantum machine learning, providing a natural way to embed data into high-dimensional Hilbert spaces via quantum circuits and capturing non-linear relationships in data. Their learning behaviour is governed by the function class they represent, which is formally characterized by the structure of the associated Reproducing Kernel Hilbert Space (RKHS). However, a theoretical characterization of this function class, and of how it depends on the choice of feature map and its hyperparameters, is still lacking. In this work, we focus on the Instantaneous Quantum Polynomial (IQP) feature map and derive a Fourier representation of the associated RKHS generators as finite sums of trigonometric functions evaluated on polynomially transformed input variables, with a finite set of accessible frequencies determined by the circuit construction and scaled by the bandwidth hyperparameter. This representation identifies three bandwidth regimes with qualitatively distinct expressive properties from localised, through multimodal, to periodic. Numerical experiments on synthetic datasets with controlled periodic and multimodal structure, across data dimensions and feature-map configurations, reproduce the three regimes in both supervised classification and unsupervised anomaly detection, benchmarked against a classical Gaussian kernel. These results make the relationship between expressivity and data structure explicit for IQP kernels: expressive power translates into a practical benefit only when the accessible frequencies match with the structure of the data, which provides a criterion for selecting the bandwidth when the periodic or multimodal structure of data is known.

Abstract

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