INTW2026, Kemer, Antalya, TÜRKİYE

Comparison of Classical and Quantum Circuit-Based Machine Learning Approaches Supported by Feature Engineering in Nuclear Radius Prediction

A hybrid quantum feature extraction approach

Serkan Akkoyun

Department of Physics · Sivas Cumhuriyet University

WHY THIS PROBLEM MATTERS

Charge radii reveal rich nuclear structure

Serkan Akkoyun · Hybrid QFE for Nuclear Charge Radii

02

Rch = √⟨r²⟩p

The global A¹ᐟ³ trend is only the beginning. Local departures reveal the physics that models must learn.

Shell closures

kinks and changes of slope

Pairing

odd–even staggering

Deformation

collective and shape effects

Classical + Quantum ML GUI Platform

22/26

1

Data Config

Load AME2020 Excel or generate synthetic data. Select input features, target variable (BE/A). Live data preview. Missing value detection.

2

Feature Eng.

Physics-aware templates: Basic Ratios, Asymmetry, Surface/Volume, Magic Numbers. Add custom algebraic expressions via formula editor.

3

Model Config

Select Classical / Quantum-Inspired / Quantum models. Adjust hyperparameters (n_trees, depth, VQC layers) via sliders. PennyLane toggled here.

4

Training

Choose mass region (All/Light/Medium/Heavy). Set train-test split. Launch training. Real-time log shows R², RMSE, training time per model.

5

Results & Viz

Summary table (R², R²-gap, RMSE, time). Prediction vs Actual scatter plots. Overfitting & performance bar charts. Export results & figures.

https://physiqdynamic.com/pd-aitool/

https://physiqdynamic.com/pd-aitool/

https://physiqdynamic.com/pd-aitool/

https://physiqdynamic.com/pd-aitool/

https://physiqdynamic.com/pd-aitool/

https://physiqdynamic.com/pd-aitool/

https://physiqdynamic.com/pd-aitool/

Thank You

Prof. Dr. Serkan Akkoyun

University of Surrey

Sivas Cumhuriyet University

sakkoyun@cumhuriyet.edu.tr

Quantum-Inspired Kernel Regression (QI-Kernel)

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1. Encoding block — Each classical feature xᵢ is embedded in the quantum state with an angle encoding gate Ry(xᵢ). This corresponds to the angle encoding strategy; In amplitude encoding, post-Hadamard Ry angles are proportional to normalized vector components.

2. Feature map U(x) block — It consists of three layers:

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Rz(xᵢ²) gates: quadratic (nonlinear) contribution corresponds to the ‖x‖² term in the classical RBF kernel

CNOT gates: create entanglement between qubits, enabling cross-correlations

Rzz(xᵢ·xⱼ) gates: multiplicative terms combining two qubits; encodes the cross term xᵢxⱼ, which is the kernel component based on the tensor product

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3. SWAP test block — Calculates the inner product between the prepared state |ψ(x)⟩ and |ψ(x′)⟩ for two different inputs x and x′. Controlled post-SWAP measurement on the auxiliary (ancilla) qubit gives K(x,x′) = |⟨ψ(x)|ψ(x′)⟩|²; This value is directly an element of the kernel matrix.

The classical simulation approach you refer to in your article is based on the following limits: It can be shown that in the limit γ → 0 the trigonometric kernel converges to cos(x·x′) and for large γ it converges to the exponential RBF kernel—without the need for quantum hardware.

Quantum-Inspired Hybrid Model (QI-Hybrid)

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The key architectural distinction from QI-Kernel is worth noting for your paper: while QI-Kernel computes pairwise inner products K(x,x′) between data points (requiring the full training set at inference time), QI-Hybrid transforms each point independently into φ(x) and then trains a standard regressor — making it more scalable to large nuclear databases like AME2020 and more amenable to the optimization routines of RF, GBM, and MLP backends.

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The three channels map naturally onto nuclear physics structure: Channel A captures periodic shell-closure behaviour through its trigonometric basis; Channel B encodes N–Z correlations and pairing interactions via pairwise products; Channel C's Rzz-inspired exponential terms model longer-range deformation–shell coupling effects that standard polynomial features would miss.

Hybrid Quantum Machine Learning Model (Q-Hybrid)

The dashed purple arrow on the right is the defining feature of Q-Hybrid: a closed feedback loop where the classical optimizer updates θ after each forward pass through the circuit. Now the variational quantum circuit itself in detail — showing exactly what one layer looks like on n qubits:

The circuit detail reveals what makes Q-Hybrid structurally distinct from the previous two models. Each qubit gets three trainable rotation gates (Rx, Ry, Rz) forming a full SU(2) rotation — so with n qubits and L layers the parameter count is only P = 3nL, which stays tractable on NISQ hardware. The CNOT ladder creates nearest-neighbour entanglement without long-range gates, keeping circuit depth minimal and decoherence manageable.

The double classical lines from the measurement boxes represent the quantum-to-classical interface: Pauli-Z expectation values ⟨Zᵢ⟩ ∈ [−1, +1] are real numbers that feed directly into the MSE loss. The gradient can be computed analytically via the parameter-shift rule (∂L/∂θ = [L(θ+π/2) − L(θ−π/2)]/2), or approximated gradient-free with SPSA or COBYLA — the latter being preferable on real hardware where shot noise makes finite-difference gradients unreliable.

For your paper, the key contrast with QI-Hybrid is worth making explicit: QI-Hybrid uses a fixed classical simulation of quantum feature maps (no hardware, no optimization of circuit parameters), whereas Q-Hybrid trains the circuit parameters θ end-to-end through the quantum-classical loop — making it a true variational quantum algorithm and the only one of the three models that can, in principle, run on NISQ hardware.

Physical Interpretation of Results

Akkoyun & Yeşilkanat | Computer Physics Communications | 2025

23/26

Q Why does Linear Regression dominate the Medium Zone?

In the 40 ≤ A ≤ 120 region, BE/A is nearly a linear combination of our engineered features (A^(2/3), Z²/A, (N−Z)²/A, etc.) — which explicitly encode Bethe-Weizsäcker structure. A linear model therefore naturally captures most variance. Complex nonlinear models add parameters without adding useful signal in this regime.

Q Why does MLP outperform quantum models in the Light Zone?

Light nuclei (A<40) show stochastic shell effects — pairing, clustering, halo phenomena — highly specific to individual nuclei. With only 292 data points, quantum models cannot tune their many parameters effectively. MLP's depth and regularization provide better noise tolerance, while VQCs face the barren plateau problem with sparse gradients.

Q What makes the Heavy Zone ideal for Quantum ML?

With 2102 heavy nuclei, the dataset is rich enough that quantum-inspired feature geometry can be exploited. Heavy nuclei exhibit smooth collective phenomena (deformation, rotational bands) where nonlinear correlations across many features are systematic rather than random — exactly the pattern Hilbert space feature maps capture naturally.

Q What does QI-Hybrid's 30× RMSE improvement tell us?

QI-Hybrid's RMSE drops from 256.9 keV (Light) to 8.75 keV (Heavy) — a 30-fold improvement. This is not hardware noise but data noise: tensor product expansions amplify measurement variance in sparse regions. In the Heavy Zone, the same expansions capture genuine physical feature interactions. Data quality matters more than circuit architecture for QML.

Future Directions & Open Challenges

Akkoyun & Yeşilkanat | Computer Physics Communications | 2025

24/26

⚛ Real Quantum Hardware

Execute Q-Hybrid circuits on IBM Quantum, IonQ, or Rigetti devices. Quantify decoherence and shot noise effects on nuclear regression accuracy. Benchmark simulator vs hardware results.

🔬 Extended Observables

Apply the GUI framework to nuclear charge radii, β-decay half-lives, quadrupole deformation, neutron-capture cross-sections. Platform is observable-agnostic by design.

🌌 Astrophysical R-Process

Predict masses of exotic neutron-rich nuclei beyond AME2020 for r-process nucleosynthesis. QML's Hilbert space extrapolation may reduce uncertainty in unexplored nuclear chart regions.

📈 Uncertainty Quantification

Integrate Bayesian neural networks and Gaussian processes alongside QML models. Provide prediction uncertainty bands — essential for astrophysical and reactor applications.

🔗 Quantum Kernel Hardware

Implement quantum kernel estimation on real hardware. Bypass classical kernel approximation. Potential quantum advantage when n_features approaches hardware qubit limits.

⚡ Fault-Tolerant QML

Transition from NISQ to fault-tolerant quantum computing will transform QML capabilities. Models developed here are designed to scale naturally onto future quantum processors.

The transition from NISQ to fault-tolerant QC will be transformative. The hybrid framework developed here is designed to scale naturally onto future quantum processors.