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

Ravi Kumar Jha, Hsuan-Yin Lin, Eirik Rosnes

arXiv:2610.31260v1quant-phcs.AIcs.LG

TL;DR

PQK studies often rely on extensive tuning, leaving the native inductive bias of their feature maps insufficiently isolated. This work compares tuned and untuned kernels and proposes selected-entanglement maps; these outperform standard PQKs and are competitive with a classical baseline.

  • Problem

    PQK studies often tune the RBF parameter γ and SVM regularization parameter ξ extensively, while feature-map designs for real-world data lack structural explainability.

    Source: §I · p. 1, B. Quantum Feature Map Circuit · p. 3

  • Method

    The study compares tuned and untuned PQKs and RBF kernels, and evaluates feature maps with selected pairwise entanglement topologies.

    Source: C. Evaluation Criteria · p. 4, B. Quantum Feature Map Circuit · p. 4

  • Results

    Selected-entanglement maps outperform standard PQK designs and compete with RF; Rot(X)en reaches accuracy 0.903 ± 0.011 and MCC 0.806 ± 0.022 under m ∈ {1, 2}.

    Source: §IV · p. 6

  • Takeaways & Limitations

    The entanglement pattern matters: simply adding entanglement or using a dense entangling architecture does not automatically improve PQK performance.

    Source: §IV · p. 5

    The heuristic feature-map designs are evaluated on a single, relatively small dataset.

    Source: §V · p. 6

Abstract

from arXiv · show

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.

I. INTRODUCTION

Quantum kernels use quantum states to define similarities, while projected quantum kernels (PQKs) use local measurements to avoid relying on global state overlaps. This work isolates PQKs’ native inductive bias and proposes entanglement-based feature maps for IoT data.

I. INTRODUCTION

  • Quantum kernels compare inputs by mapping them to quantum states and evaluating state inner products, which serve as similarity measures for kernel methods.Source: §I · p. 1
  • Global quantum-state overlaps can concentrate toward zero as qubit counts grow, motivating PQKs that project states into a lower-dimensional classical feature space through local measurements.The authors state that concentration diminishes kernel-matrix variance and makes data points nearly indistinguishable.Source: §I · p. 1
  • The study compares tuned and untuned PQKs and RBF kernels to isolate how classical hyperparameter tuning influences their native inductive biases.Source: §I · p. 1
  • The authors propose two entanglement-induced feature maps intended to improve performance and reduce tuning sensitivity relative to untuned PQKs.Source: §I · p. 1

II. PRELIMINARIES

This section defines mathematical and quantum-operator notation used in the paper.

A. Notation

  • N and R denote the sets of positive integers and real numbers, respectively.Source: A. Notation · p. 2
  • Vectors are boldfaced, while matrices and sets use capital sans-serif and calligraphic uppercase letters, respectively.Source: A. Notation · p. 2
  • The notation RX, RY, and RZ denotes single-qubit rotations, H the Hadamard gate, and ρ a density operator.Source: A. Notation · p. 2
  • tr(X) denotes a trace and trA(X) a partial trace over subsystem A; XX abbreviates X ⊗ X.Source: A. Notation · p. 2

B. Internet of Things Data

The study evaluates occupancy classification using six sensor features from an open-access IoT dataset, whose classes remain partially overlapping in a PCA view.

B. Internet of Things Data

  • The open-access COGITO dataset contains 2,865 samples with six features for classifying room occupancy as occupied or nonoccupied.The features are illuminance, blinds, lamps, relative humidity, carbon dioxide, and temperature.Source: B. Internet of Things Data · p. 2
  • Preprocessing removes irrelevant variables and redundant records, and the binary occupancy data are normalized before classification.Source: B. Internet of Things Data · p. 2
  • The first two PCA components show classes that are partially separable but remain noticeably overlapping.Source: B. Internet of Things Data · p. 2

C. Data Encoding

Data encoding maps classical inputs to quantum states through a feature map, a central design component of quantum kernel models.

C. Data Encoding

  • A quantum feature map encodes a d-dimensional input x into an n-qubit state |ϕ(x)⟩ using a data-dependent unitary U(x).The state is written as |ϕ(x)⟩ ≜ U(x)|0⟩⊗n, with corresponding density matrix ρ(x).Source: C. Data Encoding · p. 2
  • Feature maps transform input data into a high-dimensional Hilbert space, and selecting a suitable map is central to quantum kernel models.Source: C. Data Encoding · p. 2
  • For this IoT setting, the number of qubits is set to six, matching the number of input features, although these dimensions need not generally coincide.Source: C. Data Encoding · p. 2

D. Projected Quantum Kernel

Projected quantum kernels (PQKs) project quantum states into an approximate classical space, addressing challenges from the exponential dimensionality of standard quantum kernels. Their similarity measure uses selected observables of reduced states, and the resulting projected features can be used with an RBF kernel.

D. Projected Quantum Kernel

  • PQKs measure selected observables on reduced quantum states and project them into an approximate classical space, rather than computing state fidelity.This formulation was intended to address the exponential-dimensionality challenges of standard quantum kernels.Source: D. Projected Quantum Kernel · p. 2
  • The kernel uses γ, a set of subsystems S, and nontrivial Pauli observables P whose locality matches those subsystems.For subsystem M, ρM(x) is the reduced density operator obtained by tracing out its complement.Source: D. Projected Quantum Kernel · p. 2
  • For six qubits, the measurement settings use 18 single-qubit observables, 18 adjacent-pair observables, or 36 concatenated observables.The example computes a two-sample PQK of approximately 0.6376 from two local Z-expectation values per sample with γ = 1.Source: D. Projected Quantum Kernel · p. 3

A. Projected Quantum Kernel Classifier

The PQK classifier computes kernel values from quantum-projected features and then uses a classical soft-margin SVM. The classical RBF version retains the same optimization formulation with a different kernel.

A. Projected Quantum Kernel Classifier

  • Six-qubit feature-map outputs are measured with local Pauli observables, concatenated into classical feature vectors, and used to compute pairwise PQK values before classical classification.The remaining classification stage follows the standard SVM-kernel pipeline, with the classical feature map replaced by the PQK pipeline.Source: A. Projected Quantum Kernel Classifier · p. 3
  • A classical soft-margin SVM receives the pairwise PQK values and learns its dual coefficients by solving an optimization problem.The training data are binary-labeled with y_i ∈ {−1, +1}.Source: A. Projected Quantum Kernel Classifier · p. 3
  • For the classical RBF baseline, the SVM optimization is unchanged, but κPQK is replaced by κRBF before hyperparameter tuning.The RBF kernel is exp(−γ∥x_i − x_j∥²).Source: A. Projected Quantum Kernel Classifier · p. 3

B. Quantum Feature Map Circuit

The paper proposes two IoT-focused feature maps with selected pairwise entanglement, against a background of existing rotation and IQP designs. Their design tests whether chosen feature-pair interactions improve PQKs on real-world IoT data.

B. Quantum Feature Map Circuit

  • Earlier designs include Rot(X), IQP, Rot(X, Y, Z), and Rot(X, Y, Z)+CNOT, which use rotations and different entanglement patterns.The common design approach broadens entanglement patterns or adds rotation-gate complexity.Source: B. Quantum Feature Map Circuit · p. 3
  • The paper identifies extensive tuning of γ and ξ and limited structural explainability for real-world feature-map designs as shortcomings in prior PQK studies.Its stated aim is to assess feature maps without tuning these hyperparameters to isolate their representational capabilities.Source: B. Quantum Feature Map Circuit · pp. 3–4
  • Rot(X)en and Rot(X, Y, Z)en add selected pairwise entanglement to existing rotation-based designs for IoT data classification.The figure contrasts these proposals with Rot(X), Rot(X, Y, Z), IQP, and Rot(X, Y, Z)+CNOT designs.Source: B. Quantum Feature Map Circuit · p. 4, Figure 3 · p. 4
  • The proposed maps couple selected qubit pairs through feature-dependent two-qubit interactions to encode multivariate input-feature correlations.Candidate topologies are constructed from possible two-qubit feature-interaction pairs, then the topology with the best empirical performance is selected.Source: B. Quantum Feature Map Circuit · p. 4

C. Evaluation Criteria

The evaluation compares tuned and untuned PQKs and a classical RBF baseline, reporting accuracy, F1-score, and MCC. The supplied passages specify the baseline heuristics and evaluation context but do not provide table values.

C. Evaluation Criteria

  • Untuned PQKs and the classical RBF baseline use ξ = 1 and data-dependent γ heuristics based on projected-feature or input-feature variance, respectively.For the combined PQK setting, the heuristic uses the 36-dimensional concatenated training representation.Source: C. Evaluation Criteria · p. 4
  • The models are evaluated with 10-fold cross-validation and 95% confidence intervals, with test accuracy reported.The tuning procedure uses grid searches over specified sets of ξ and γ values.Source: C. Evaluation Criteria · p. 4
  • PQK evaluation reports accuracy, F1-score, and MCC, and includes linear and RBF kernels plus tuned LDA, random forest, AdaBoost, and MLP comparisons.Experiments use a noiseless quantum PennyLane simulator; MCC is included given slight IoT data imbalance.Source: C. Evaluation Criteria · p. 4
  • Table II is scoped to IoT classification accuracy, F1-score, and MCC for PQK models before and after hyperparameter tuning.The supplied table passage gives the comparison scope but no values.Source: Table II · p. 5

IV. NUMERICAL RESULTS AND DISCUSSION

The proposed selected-entanglement feature maps produced the strongest PQK results and were competitive with the strongest classical baseline. The analysis also compares classical and PQK performance across untuned and tuned settings to assess tuning sensitivity and native inductive bias.

Evaluation structure

  • The evaluation separates classical-model baselines, emphasizing the RBF, from PQK feature-map comparisons with and without hyperparameter tuning.Untuned results are highlighted in blue in both reported tables.Source: §IV · p. 5

Classical baselines

  • The Random Forest was the strongest classical model, achieving 0.902 ± 0.006 accuracy, 0.891±0.008 F1-score, and 0.804±0.013 MCC.The comparison frames PQK competitiveness against a classical ensemble that can capture nonlinear and higher-order feature interactions.Source: §IV · p. 5

Tuning sensitivity and native inductive bias

  • All untuned PQKs induced useful geometric separation relative to the untuned RBF, while the proposed maps required less tuning-induced correction than the RBF.Fig. 4 analyzes tuning sensitivity through performance differences between tuned and untuned settings.Source: Figure 4 · p. 6, §IV · p. 6

PQK and classical model results

  • Rot(X)en reached 0.902 ± 0.009 accuracy and 0.803 ± 0.019 MCC under m = 1, while Rot(X, Y, Z)en reached 0.903 ± 0.008 accuracy and 0.806 ± 0.016 MCC under m ∈{1, 2}; both were competitive with the strongest classical baseline.The proposed maps yielded the strongest PQK results; Rot(X, Y, Z)en also achieved 0.892±0.010 F1-score under m ∈{1, 2}.Source: §IV · pp. 5–6

V. CONCLUSION AND FUTURE WORK

The study compares six quantum feature maps for IoT classification before and after classical hyperparameter tuning. The proposed circuits achieved the strongest PQK performance, while the study’s conclusions are bounded by heuristic designs evaluated on one relatively small dataset.

Conclusion

  • The study compared six quantum feature maps for IoT classification before and after classical hyperparameter tuning.Source: §V · p. 6
  • The proposed Rot(X)en and Rot(X, Y, Z)en circuits achieved the strongest PQK performance and were competitive with the strongest classical baseline.Under prescribed untuned settings, PQKs outperformed the untuned RBF and generally showed smaller tuning-related gains.Source: §V · p. 6

Future work

  • The study is limited to heuristic feature-map designs evaluated on a single, relatively small dataset.The authors call for assessment across datasets and baseline hyperparameter choices.Source: §V · p. 6
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